Proof. Resolve the rational maps and use smooth projective models. Compose \(V\dashrightarrow W\) with the maximal rationally connected quotient of \(W\). A general fibre of the composite fibres over a rationally connected general fibre of \(W\dashrightarrow R(W)\), with rationally connected general fibre. It is rationally connected by the fibration theorem of Graber–Harris–Starr (Graber et al. 2003, Corollary 1.3). Since \(R(W)\) is not uniruled, this composite is a maximal rationally connected quotient of \(V\). Indeed, a family of rational curves through general points of \(V\) that is not vertical would induce a covering family of rational curves on \(R(W)\). Uniqueness of the quotient proves the first assertion. A projective bundle has a projective-space generic fibre. The remaining statements follow from the dimension of the quotient, and from the fact that a non-uniruled surface is its own quotient. ◻

Proof. The field \(k\) is rational. If \(u(\ell)=1\), the surface field \(\ell\) is the function field of a member of \(\mathcal S\), by Lemma 6. A nonidentity field automorphism would give a nonidentity birational, hence regular, automorphism of that K3 surface. This contradicts the choice of \(\mathcal S\). A quadratic extension in characteristic zero has its nonidentity involution. ◻

Proof. Choose dense open subsets on which \(f\) is an isomorphism. Every divisor whose generic point belongs to these opens has a corresponding divisor on the other model with exactly the same valuation. Any valuation in the symmetric difference must therefore be represented by a divisorial component of one of the two closed complements. There are finitely many such components. This also proves that an isomorphism in codimension one has value zero.

For the composition use the common field \(F=\mathbb C(X)\), transporting \(\mathbb C(Y)\) by \(f^*\) and \(\mathbb C(Z)\) by \(f^*g^*\). Put \(D_X=\mathcal D_F(X)\), \(D_Y=\mathcal D_F(Y)\) and \(D_Z=\mathcal D_F(Z)\). Their pairwise symmetric differences are finite. On the set of all prime divisorial valuations of \(F\), \[\mathbf 1_{D_Z}-\mathbf 1_{D_X} = (\mathbf 1_{D_Y}-\mathbf 1_{D_X}) +(\mathbf 1_{D_Z}-\mathbf 1_{D_Y}).\] All three differences have finite support. Multiplying by \(u(\kappa(\nu))\) and summing proves additivity. Transport by a \(\mathbb C\)-field isomorphism identifies residue fields, so the last term is precisely \(c_u(g)\). Reversing the two valuation sets gives the inverse formula.

For the blow-up, the target of the extraction \(\pi^{-1}:X\dashrightarrow\widetilde X\) retains every original divisorial valuation and adds exactly one exceptional divisor \(E_i=\mathbb P_{Z_i}(N_{Z_i/X})\) for each \(Z_i\). Its residue field is purely transcendental over \(\mathbb C(Z_i)\), of transcendence degree \(\operatorname{codim}_X(Z_i)-1\). Lemma 6 gives \(u(E_i)=u(Z_i)\). The stated signs now follow from the definition and the inverse formula. ◻

Proof. Rational connectedness gives \(H^{p,0}(M)=0\) for \(p>0\) (Araujo and Kollár 2002, Theorem 30). We use the Hodge decomposition, functoriality and Lefschetz polarizations in (Voisin 2016, secs. 2.1–2.2); the Hodge-complement statement is Lemma 1 there. Thus \(H^2(M,\mathbb Q)\) is of type \((1,1)\); hard Lefschetz gives the analogous statement in degree six. The remaining assertion about types follows from Hodge symmetry. In degree four all nonprimitive Lefschetz summands are rational Hodge classes. Hodge–Riemann therefore makes their orthogonal primitive transcendental part nondegenerate. The same argument in degree two applies to a surface, using an ample class to split off its nonprimitive part. Alternatively, the nondegeneracy on the Hodge-class space follows on each primitive Lefschetz summand from definiteness.

By semisimplicity of polarizable rational Hodge structures, the substructure generated by the stated off-diagonal types has a Hodge complement in the transcendental structure. That complement would have only diagonal type, and hence would consist of rational Hodge classes. It is zero by the definition and nondegeneracy of the transcendental part. Finally, \(i^*\) takes the off-diagonal part of \(H^4(M)\) into \(H^4(Z)\), which has only type \((2,2)\); and \(i_*\) takes the off-diagonal part of \(H^2(Z)\) into \(H^6(M)\), which has only type \((3,3)\). Both maps vanish on the generating types and therefore on the whole corresponding rational Hodge structure. ◻

Proof. There is nothing to prove if \(W=0\). Otherwise \(p_g(Z)>0\), so \(Z\) is not uniruled. A fixed-degree connected genus-zero stable map with positive projected degree has projected image a connected union of rational curves. For each irreducible component of its finite-type moduli space the evaluation image in \(Z\) has dimension at most one: a dominating evaluation image would give a covering family of rational curves on \(Z\). There are finitely many such components for the fixed discrete data. A holomorphic two-form pulls back to zero through each of their evaluation maps, since their images lie in curves or points. The bilinear tensor, with all other data fixed, is rational and respects Hodge structures. Its associated Hodge endomorphism therefore annihilates the \((2,0)\) generating subspace, and its conjugate, and hence all of \(W\) by Lemma 10. This proves vanishing as a tensor, independently of the values of the two variables. The argument also applies after projecting an expanded target to \(Z\).

If the projected degree is zero, all evaluations in \(Z\) agree. Factoring out base classes by the projection formula, the two variable classes occur as \(u\smile v\). They already have top degree on the surface. Every additional positive-degree base factor kills their product; the remaining fibre integral is a rational number multiplying \(\int_Zuv\). This reasoning is independent of \(u,v\) and allows the specified descendants. ◻

Proof. Cut an edge outside the path. The detached connected subtree has only rational diagonal input classes. Its single output is rational and of Hodge difference zero, because its virtual class and all structure maps respect Hodge difference. At the retained vertex the attachment remains its existing relative root, with the same contact and the resulting diagonal evaluation weight; it is not an additional ordinary marking. Repeating this operation removes all side branches from the tensor contraction. Conservation of Hodge difference at the remaining vertices forces difference two along the path. Lemma 10 and the projective bundle formula identify the indicated variable spaces. These path classes and the diagonal branch outputs have even cohomological degree, so their permutations introduce no odd-degree signs. For the first row, contraction of consecutive tensors written in path order as \(q_M(Au,x)\) and \(q_M(By,v)\) pairs \(x\) and \(y\) with the inverse cup form and gives \(q_M(BAu,v)\). Thus contraction composes operators without taking adjoints. In the second row it composes scalar maps between copies of \(W\). A path changing between the two sorts has a mixed vertex, whose tensor vanishes by the stated vertex hypothesis. The diagonal of a projective bundle gives the same identities between its copies of \(W\), with the relevant nonzero pairing constants. The conclusion follows from closure under composition and rational linear combination. If the two variables meet on one local piece, Lemma 13 supplies the second row directly. ◻

Proof. Replace a relative weight \((k,\xi^bp^*\delta)\) by the ordinary insertion \[ \tau_{r(k-1)+b}(i_*\delta) \tag{9}\] to form the absolute tensor \(B_{\mathrm{abs}}(u,v)\) on \(M\) in numerical class \(\pi_*\beta\), retaining all original ordinary insertions and their descendant powers. We apply degeneration to the normal cone using the following specific lifts. In \[\mathcal W=\operatorname{Bl}_{Z\times\{0\}} (M\times\mathbb A^1)\] the strict transform \(\mathcal Z\) of \(Z\times\mathbb A^1\) is still \(Z\times\mathbb A^1\), because the blowup ideal restricts to the Cartier ideal \((t)\). Lift \(i_*\delta\) by \((i_{\mathcal Z})_*(\delta\times1)\). On the special fibre its restriction to \(Y\) is zero and its restriction to \(P=\mathbb P_Z(N_{Z/M}\oplus\mathcal O_Z)\) is the zero-section Gysin class. These are transverse fibre restrictions. Original ordinary classes use pullbacks from \(M\).

This lift is valid even when \(i_*\delta=0\). A class on \(\mathcal W\) may restrict to zero on the smooth fibre and nontrivially to its special components. For a relative variable in \(W\), Lemma 10 says that the associated absolute insertion is zero, so the absolute bilinear tensor is zero. For two ordinary \(T_M\) variables the associated absolute tensor is a generator allowed in the first row. Thus the absolute starting tensor has the required restriction in all three cases.

We prove the relative restriction by simultaneous strong induction, keeping the two variables throughout. Fix an integral ample divisor \(H\) on \(M\). The outer order is first \(H\cdot\pi_*\beta\), then the number of original ordinary markings. Within fixed outer data, use, in succession, the length of the relative list, the sum of its contacts, and the sum of its hyperplane exponents, with smaller values first. At every smaller outer index the assertion is assumed simultaneously for all numerical curve classes at that index, relative lists, evaluation labels, placements of the two formal variable slots, and ordinary descendant exponents. At equal outer indices the inner induction is likewise simultaneous in the curve classes, labels, variable placements, and descendant exponents. Cutting a side branch only replaces the weight at an existing relative root, so it does not increase the number of original ordinary markings.

A connected relative vertex on \(Y\) of zero projected degree cannot have joining markings. Indeed every nonzero effective curve contracted by \(\pi\) has class a positive multiple of an exceptional line, and has negative \(D\)-degree; relative admissibility would require its \(D\)-degree to equal a sum of positive contacts. A zero-class vertex likewise has no contacts. Thus zero projected degree contributes only a constant invariant with no joining marks. Its two middle-degree ordinary variable classes already have total top degree; the invariant is a scalar cup pairing, or zero. This supplies the initial case.

For a noninitial term of the degeneration formula, cut away the branches with no variables, as in Lemma 14. Every \(Y\)-vertex in a nontrivial connected tree has a joining edge, so the zero-degree argument above makes its projected ample degree positive. The projected classes of all components are effective and add to the original class. Hence, if there is more than one \(Y\)-vertex, every one has smaller projected degree and its path tensor is controlled by induction. A unique \(Y\)-vertex of smaller projected degree is controlled in the same way. If a \(Y\)-vertex has the full projected degree, it is the unique \(Y\)-vertex; its tensor is still an earlier case if it retains fewer original ordinary markings. Local path vertices are controlled by Lemma 13. In these cases every path vertex therefore satisfies its row, so the conditional contraction rule of Lemma 14 controls the term. A path containing no \(Y\)-vertex is controlled entirely by the local lemma.

It remains to consider a unique \(Y\)-vertex of the full projected class carrying all original ordinary markings. Every \(P\)-vertex is then a fibre tail with one joining marking, since the graph is a tree. Let such a tail have joining contact \(k'\) and receive the translated markings indexed by a set \(J\). All its evaluations in \(Z\) agree. Its pushforward to the joining divisor therefore has the form \[\left(\prod_{j\in J}p^*\delta_j\right)\eta,\] where \(\eta\) is a rational diagonal class independent of the base labels. The projection formula gives this identity also when a label is one of the two variables. The complex codimension of \(\eta\) is \[ c=r\left(\sum_{j\in J}k_j-k'\right) +\sum_{j\in J}b_j-|J|+1. \tag{10}\] Indeed the relative fibre virtual dimension is \(r-2+rk'+|J|\); each translated zero-section insertion has fibre codimension \(r+r(k_j-1)+b_j=rk_j+b_j\), and pushing to the relative fibre divisor of dimension \(r-1\) gives the displayed difference.

If one tail contains both variable markings, both are relative \(W\) variables, since all original ordinary markings remain on the main vertex. Their product is \(q_Z(u,v)\) times the point class on \(Z\), so the tail output is \(q_Z(u,v)\) times a fixed class. The remaining vertices then have only fixed insertions, so the graph is a rational multiple of \(q_Z\). It already lies in the allowed \(W,W\) row, without using induction on its main tensor. For every other nonzero term, expand the tail outputs in the projective bundle basis. A tail with one \(W\) variable multiplies it only by fixed diagonal classes, which act by scalars between the allowed copies or annihilate it. Thus each remaining summand retains two separate variable slots in the prescribed spaces, and the following induction applies to it.

A negative \(c\) gives zero. An empty tail has \(c=1-rk'<0\), so every tail receives a translated marking. The new relative list therefore has no more entries than the old one. A strict decrease is an earlier induction case. If the lengths agree, each tail has exactly one translated marking. Then \(c\geq0\) and \(b_j<r\) imply \(k'\leq k_j\). A strict decrease of the total contact is again earlier. At equal contacts, expand \(\eta\) in the projective bundle basis. Its fibre exponent is at most \(b_j\), so either the total hyperplane exponent decreases or all the original indices agree. A decrease of a fibre exponent only multiplies its base label by a rational diagonal class, and preserves the stated variable spaces. Thus all smaller terms with separate variable slots retain the required tensor restriction.

For one translated mark and equal indices, the coefficient of the highest fibre power is obtained by restricting to a fibre. The relative invariant computed in (Hu et al. 2008, Theorem 7.1) is \[ \left\langle\tau_{rk-1-j}(\mathrm{pt})\mid H^j\right\rangle^% { (\mathbb P^r,\mathbb P^{r-1})}_{0,k[\mathrm{line}]} =\frac{1}{k^{r-j}((k-1)!)^r}>0, \qquad 0\leq j\leq r-1. \tag{11}\] The proof of that theorem identifies the descendant line with the cotangent line of the source at the ordinary marked point, so its convention is the one fixed above. Taking \(j=r-1-b\) gives exactly the descendant in (9). Gluing multiplies these numbers by positive contact and labeling factors. Consequently the remaining term is a nonzero rational scalar times the original relative tensor. At fixed projection, the total contact fixes the numerical class on the blowup, since the kernel on curve classes is generated by the exceptional line.

We have an identity \(B_{\mathrm{abs}}=C B_{\mathrm{rel}}+R\), with \(C\in\mathbb Q^*\) independent of the two variable vectors. The starting tensor and the remainder belong to the relevant row of Lemma 14; hence so does \(B_{\mathrm{rel}}\). The induction is simultaneous in the three rows. It is a tensor identity, not a separate choice of a scalar for each pair of vectors. Testing on opposite off-diagonal types and using Lemma 10 extends it to the full rational transcendental structures. This completes the induction. ◻

Proof. Fix a relative tensor \(B_{\mathrm{rel}}(u,v)\) in numerical curve class \(\beta\) on \(Y\), with relative list \(\mu=((k_j,\delta_j))_{j=1}^{L}\). Thus \(D\cdot\beta=\sum_{j=1}^{L}k_j\). Use degeneration of \(Y\) along \(D\), with the same source cotangent convention, and translate a relative weight \((k,\delta)\) into \(\tau_{k-1}(j_*\delta)\) to form an absolute tensor \(B_{\mathrm{abs}}(u,v)\) on \(Y\) in the same numerical class \(\beta\), retaining the original ordinary insertions and descendants. Use the strict divisor-family lift for translated classes and pullbacks for original ordinary classes. This is the classical divisor correspondence of (Maulik and Pandharipande 2006, sec. 2.4, Lemma 4); we give the ordering needed for the restricted tensors.

The ordinary \(T_M\) part restricts to zero on \(D\). The rational Hodge structures generated by boundary classes of Hodge difference \(\pm2\) are copies of \(W\): for a surface center they are \(p^*W\subset H^2(D)\) and \(\xi p^*W\subset H^4(D)\). Gysin carries the first into the exceptional \(W\) summand of \(H^4(Y)\) and annihilates the second, since \(H^6(Y)\) is diagonal. Restriction of the ordinary exceptional class is \(j^*j_*p^*u=-\xi p^*u\). It follows from (12) that every associated absolute tensor has the claimed restriction. Local tensors have it by Lemma 13. This also treats a translated variable with zero Gysin class; the absolute tensor is then zero.

Fix an integral ample divisor on \(Y\) and perform outer strong induction on its curve degree, then on the number of original ordinary markings. As before, these are the ordinary markings before translation, and the induction is simultaneous for all numerical classes, relative data, labels, variable placements, and descendant exponents at the given index. Pruning replaces existing relative weights as in Lemma 14.

Every main \(Y\)-vertex in a nontrivial tree has a joining root. Its curve class cannot be zero, since a zero class has total contact zero; its ample degree is therefore positive. Under collapse of the degeneration to \(Y\), the projected component classes are effective and add to the original class \(\beta\). If there are several main vertices, all have smaller ample degree. A unique main vertex of smaller degree is also an earlier case. A main vertex with full degree is unique, and is an earlier case unless it retains all original ordinary markings. Thus in these cases induction controls every main path vertex and Lemma 13 controls every local one, as required by the conditional path rule. A path with no main vertex is controlled by the local lemma after pruning.

At equal outer data the unique main vertex has curve class \(\beta\): all other components project to points of \(Y\). They are fibre tails of \(P_D\to D\), since the graph is a tree. This argument uses the ample degree on \(Y\), without requiring \(D\) to be nef. Hence \(K=D\cdot\beta\), the total contact, is fixed. If \(K<0\) there is no admissible relative datum. A zero curve class has \(K=0\) and no relative marks; its constant tensors are scalar cup pairings, zero on mixed summands. More generally, for \(K=0\) the following inner induction has just the empty-list case.

Let the new main relative list have length \(t\), with tail contacts \(k'_a\). If \(J_a\) is the set of translated markings on tail \(a\), the tail calculation (10) now applies with \(r=1\): the center of this normal-cone degeneration is the divisor \(D\subset Y\). Thus \[ c_a=\sum_{j\in J_a}(k_j-1)-k'_a+1, \qquad \sum_a c_a=t-L. \tag{14}\] Before applying the list order, use the tail-output factorization from the first reconstruction. A tail containing both variable slots gives \(q_Z(u,v)\) times fixed classes, so its whole graph contribution is already an allowed scalar pairing. In every other nonzero summand the variable slots remain separate: fixed diagonal classes act by scalars between the copies of \(W\). The following inner induction applies to these summands.

Each nonzero tail has \(c_a\geq0\). Therefore \(t\geq L\). Use decreasing length as the next order; it terminates because \(t\leq K\). This is an inner induction for each fixed \(\beta\), so no unbounded reversal of an order is being used.

At equal lengths every \(c_a\) is zero. Each tail has \[k'_a-1=\sum_{j\in J_a}(k_j-1),\qquad \delta'_a=\text{a scalar times }\prod_{j\in J_a}\delta_j.\] Take all fixed labels homogeneous. Give \((k_j,\delta_j)\) weight \(k_j-1+\operatorname{codim}\delta_j\). These weights are nonnegative. Each formal variable slot has a fixed cohomological degree, unchanged when its vector is specialized to zero or to the other vector. A boundary \(W\) slot has positive codimension, so it always has positive weight. Combining two positive-weight labels on a tail decreases their number, so use that number, with smaller values first, as the final order. A strict decrease of this count is an earlier case with two separately labelled variable slots. If the number of positive-weight labels is unchanged, each tail has at most one positive-weight label. All other labels are \((1,1)\), so every nontrivial contact and label is unchanged. Empty tails have degree one and identity output; redistribution of the trivial labels, at the fixed total length, consequently gives the original weighted partition, with each formal variable slot still distinguished.

These unchanged-data terms cannot cancel. A tail carrying its one positive-weight label has the leading fibre invariant in (11), now for \(r=1\), with its positive gluing contact factor. An additional \((1,1)\) label translates to an ordinary point-divisor insertion on \(\mathbb P^1\). By the divisor equation it multiplies the leading number by \(k'_a\). The descendant correction contains the square of the fibre point class and vanishes. For the source cotangent convention this is the ordinary relative divisor equation with the divisor supported in the interior, disjoint from the relative boundary. A degree-one tail without translated markings contributes one. Thus every unchanged-data coefficient is a product of the same positive leading numbers and positive contact, divisor, and labeling factors. All path labels are even, so no permutation signs occur. The graph with one leading fibre tail for each relative entry is among these terms. Their sum is therefore a nonzero rational coefficient \(C\), depending only on the fixed discrete data and the formal labels, not on the two variable vectors. For the empty list the coefficient is one. This is the nonvanishing part of the triangular coefficient calculation in (Maulik and Pandharipande 2006, sec. 2.4, Lemma 4); its exact normalization is not needed here.

As before, the formula is an identity of bilinear tensors. All earlier terms are allowed by the simultaneous induction and the path lemma; division by the nonzero, vector-independent coefficient proves the three stated restrictions. Ordinary descendants at retained markings have not changed, and those at markings moved to a fibre occur only in the smaller ordinary-mark cases. Hence no bound or extra hypothesis on original descendants or ordinary diagonal insertions has been used. ◻

Proof. At \(\mathbb P^4\) the transcendental lattice is zero. A blowup with a nonzero surface contribution adds exactly one summand by Theorem 12; a blowup with no transcendental contribution does nothing. A blowdown with zero transcendental contribution also leaves the lattice and algebra unchanged by the same theorem. Suppose instead that the next step is a blowdown with nonzero surface contribution. The theorem, read in reverse, exhibits the removed center as a factor \(\mathbb Q\) of the operator algebra of the current variety. Its identity is a primitive central idempotent. The primitive central idempotents of the already identified algebra \(\mathbb Q^m\) are precisely the projectors onto its existing whole summands. Thus the removed rational subspace is one of those summands, even if some summands are mutually Hodge-isomorphic or a surface transcendental structure is reducible.

The lattice of each summand is its intersection with the ambient integral lattice: the historical decomposition and the geometric blowup decomposition are integral direct sums. Equality of their rational subspaces therefore identifies their saturated integral lattices and cup forms. The residual summands are unchanged. This proves the induction. ◻

Proof. Geometric genus and the transcendental lattice are unchanged by point blowups. Pass to a minimal model using (Beauville 1978, Proposition II.16). In the following numerical calculation, \(K\), \(b_2\), and \(q=h^{1,0}\) refer to that minimal surface. Since \(p_g=1\), it is not ruled and has nonnegative Kodaira dimension (Beauville 1978, Propositions III.20–III.21 and Example VII.3). Its canonical class is nef (Beauville 1978, Corollary VI.18), so \(K^2\geq0\). Noether’s formula (Beauville 1978, sec. I.14 and III.19) gives \(c_2=12(1-q+p_g)-K^2=24-12q-K^2\). On the other hand its topological Euler characteristic is \(c_2=2-4q+b_2\). Hence \[b_2=22-8q-K^2.\] Projectivity gives Picard rank at least one, while the rank assumption gives \(b_2\geq22\). Thus \(q=0\), \(K^2=0\), and the Picard rank is one. Kodaira dimension two is excluded by \(K^2=0\) (Beauville 1978, Proposition X.1). In Kodaira dimension one the elliptic fibration (Beauville 1978, Proposition IX.2) supplies a nonzero isotropic divisor class, independent of an ample class, contradicting Picard rank one. The classification in Kodaira dimension zero leaves a K3 surface, since \(p_g=1\) and \(q=0\) (Beauville 1978, Theorem VIII.2).

The K3 lattice is unimodular, and its Néron–Severi lattice is primitive by the integral Lefschetz \((1,1)\) theorem (Voisin 2016, Theorem 2). For a primitive sublattice of a unimodular lattice, its discriminant group and that of its orthogonal have equal orders. The Néron–Severi lattice here is generated by a primitive ample class \(L\), so its discriminant is \(L^2\). This proves the degree assertion.

Any automorphism fixes \(L\) and hence acts trivially on this rank-one Néron–Severi lattice. Its action on \(T_{Z,\mathbb Q}\) is a rational scalar by the endomorphism hypothesis, and preservation of the nondegenerate form makes that scalar \(+1\) or \(-1\). The unimodular gluing identifies the cyclic discriminant group of \(T_Z\) with that of \(\mathbb ZL\), of order \(d\); see (Nikulin 1980, Proposition 1.6.1) for the compatible gluing of primitive orthogonal sublattices of an even unimodular lattice. The action \(-1\) on the first and \(+1\) on the second is compatible only if \(2\) annihilates this cyclic group, contrary to \(d>2\). Thus the action is \(+1\) on both lattices, and hence on the whole integral second cohomology. The faithful cohomological action for K3 automorphisms, a consequence of global Torelli (Huybrechts 2016, Proposition 15.2.1), makes the automorphism the identity. ◻

Proof. Under the rationality assumption all varieties in the smooth factorization of Theorem 9 are rationally connected. Lemma 17 tracks integral whole blocks through the factorization. A nonzero transcendental rational Hodge structure of this kind has a nonzero off-diagonal part by Lemma 10. Since the final \(h^{3,1}\) is one, its transcendental structure is simple: a nontrivial semisimple decomposition would require at least two nonzero \((3,1)\) parts. Thus the final lattice is one whole block. The number of additions minus removals of blocks in its integral polarized Hodge-isometry class is one.

Every center in this class has geometric genus one, transcendental rank \(21\), determinant of absolute value \(d\), and rational Hodge endomorphism ring \(\mathbb Q\). Lemma 18 makes it birational to a member of \(\mathcal S\), and proves the degree and automorphism assertions. Conversely, a smooth surface center counted by \(u\) is birational to a member of \(\mathcal S\): in dimension two an MRC base which is a K3 has the same function field as the surface. Its entire transcendental lattice therefore has exactly the required class. Point and curve centers are not counted.

The exceptional divisor of a smooth blowup is a projective bundle over its center and has the same MRC base, by Lemma 6. The sign convention and additivity in Lemma 8 therefore identify \(c_u(f)\) with the above number of block additions minus removals. This number is one. ◻

Proof. Resolve the map by point blow-ups \(\pi:Y\to S\). Since \(h^1(Y,\mathcal O_Y)=0\), pullback of differentials shows that the base of its Stein factorization has genus zero. Thus, after Stein factorization, we have a morphism \(f:Y\to\mathbb P^1\) with smooth connected general fibre \(F\). The images on \(S\) of these fibres form a pencil without fixed components, of class \(nL\) for some integer \(n\geq1\). In the total-transform basis of the successive exceptional curves write \[F=\pi^*(nL)-\sum_i m_iE_i^*,\qquad K_Y=\sum_iE_i^*,\qquad m_i\geq0.\] The exceptional basis is orthogonal, with \((E_i^*)^2=-1\). Consequently adjunction and \(F^2=0\) give \[\sum_i m_i^2=n^2d,\qquad 2g-2=K_Y\cdot F=\sum_i m_i.\] It follows that \(2g-2\geq(\sum_i m_i^2)^{1/2}=n\sqrt d\). Stein factorization accounts for the geometric components in the last assertion. ◻

Proof. First consider a bounded smooth projective surface \(Y\) birational to \(S\). Its minimal-model morphism \(\mu:Y\to S\) is a succession of point blow-downs. Let \(A\) be a very ample divisor from the bounded family. In the total-transform exceptional basis, \[A=\mu^*B-\sum_i a_iE_i^*,\qquad K_Y=\sum_iE_i^*,\qquad a_i=A\cdot E_i^*>0.\] The pushforward \(B\) is ample: at each blow-down its square increases and its intersection with every curve is positive, so the surface Nakai–Moishezon criterion applies (Cartier 1966, Theorem 1). It is an integral multiple of \(L\). Hence \[ d\leq B^2=A^2+\sum_i a_i^2 \leq A^2+(A\cdot K_Y)^2. \tag{15}\] The right-hand side takes only finitely many values in a bounded smooth projective family after a finite stratification. This proves the assertion in dimension two; in smaller dimensions a K3 MRC base is impossible.

It remains to bound the birational types of the K3 bases of bounded smooth projective threefolds. Put these threefolds into finitely many smooth projective families \(\mathcal X\to T\), with relatively very ample divisor \(H\). We use the fixed-polynomial Hilbert space and its open graph locus (The Stacks Project Authors 2026c, Tags 0DPH, 0D1A and 0D1B), together with semicontinuity and base change (The Stacks Project Authors 2026c, Tags 0BDN and 0B91). For \(e\geq1\), the free locus in the relative scheme of degree-\(e\) maps \(\mathbb P^1\to\mathcal X/T\) is open. Its morphism to \(T\) is smooth: for a free map \(a\) the obstruction space \(H^1(\mathbb P^1,a^*T_{\mathcal X_t})\) vanishes. Its image is therefore open. If \(U_D\) is the union of these images for \(1\leq e\leq D\), then the \(U_D\) are ascending open subsets of the noetherian space \(T\). Their union is precisely the locus of uniruled geometric fibres, since in characteristic zero a smooth projective variety is uniruled if and only if it has a nonconstant free rational curve. The union is quasi-compact, so \(U_D\) equals that union for some \(D\). These deformation statements follow, for example, from (Araujo and Kollár 2002, Remark 9, Proposition 10 and Theorem 15); this argument proves the uniformity rather than assuming that the uniruled fibres themselves form a separately specified family.

Suppose now that the MRC quotient of \(X=\mathcal X_t\) is a surface birational to \(S\). Use its almost-holomorphic presentation \(r:U\to S^\circ\) with proper smooth rationally connected fibres; see (Kollár 1996, IV, Section 5). Its fibres are smooth copies of \(\mathbb P^1\). A degree-at-most-\(D\) free map on \(X\) deforms in a covering family. A very general member meets a very general fibre of \(r\) and is contained in it, by the MRC property. Since that fibre is a complete integral curve, the image of the map is the whole fibre. The \(H\)-degree \(e\) of the fibre is thus at most \(D\).

For such a fibre \(F\), smoothness of \(r\) gives \(N_{F/X}\simeq\mathcal O_{\mathbb P^1}^{\oplus2}\). Its Hilbert scheme is therefore smooth of dimension two at \([F]\). The family of fibres defines an injective map from \(S^\circ\) to this Hilbert scheme, with isomorphic tangent spaces; its closure is an irreducible component of \(\operatorname{Hilb}^{en+1}(X)\) birational to \(S\). The components so obtained are components of fibres of the finite-type projective scheme \[\coprod_{e=1}^{D} \operatorname{Hilb}^{en+1}(\mathcal X/T).\] They have bounded smooth projective birational models by the preceding bounded-family reduction. Apply (15) to these surfaces. This proves a uniform bound independent of the threefold and of its MRC quotient map. ◻

Proof. Part (i) is Lemma 21. For (ii), pass to bounded smooth projective models of the pairs, including the exceptional locus in the boundary. The normalized base-changed covers are still étale on the complementary open. For a smooth curve \(A\) of genus \(g_A\) with at most \(r\) marked points, and a connected degree-\(e\) cover \(A'\to A\) unramified elsewhere, Riemann–Hurwitz gives \[ 2g(A')-2\leq e(2g_A-2)+(e-1)r. \tag{16}\] Indeed, over any one branch point the ramification contribution is \(e\) minus the number of points above it, and is at most \(e-1\). This proves the curve assertion uniformly for \(e\leq N\).

For a bounded smooth surface pair choose a pencil of hyperplane sections for a relatively very ample divisor \(H\). A general section \(A\) is smooth, has uniformly bounded genus by adjunction, and meets the divisorial part of \(D\) in uniformly boundedly many points. Choose it to avoid the zero-dimensional part of \(D\). For any one cover, general members can also avoid the images of the singular points of its normalization; normal surfaces have only finitely many singular points. Each normalized component over \(A\) has degree at most \(N\) and is ramified only over \(A\cap D\), so (16) gives a uniform genus bound \(G\). Resolve the pulled-back pencil on a smooth model of the covering surface and take its Stein factorization. Its smooth connected general fibres are precisely these normalized components. If the surface field were that of \(S\), Lemma 20 would give \(\sqrt d\leq2G-2\). This proves the surface assertion. The pencil can depend on the individual cover; only its genus and boundary-intersection bounds need to be uniform.

The same curve estimate over \(\overline{\mathbb C(C)}\) bounds the genera of the geometric generic components in (iii). If \(T\) were birational to \(S\), the given map to \(C\), after resolution and Stein factorization, would give a pencil on \(S\). Lemma 20 again bounds \(d\). No genus bound for the parameter curve \(C\) is needed.

For (iv), suppose the MRC base of \(T\) is birational to \(S\), and write \(r:T\dashrightarrow S\) and \(f:T\dashrightarrow C\). Put \(F=\mathbb C(C)\) and choose an algebraic closure \(\overline F\). Resolve the maps when necessary. If \(f\) were nonconstant on a general fibre of \(r\), the map \((r,f):T\dashrightarrow S\times C\) would be dominant and generically finite. A component of a geometric generic surface fibre of \(f\) would then dominate \(S_{\overline F}\) generically finitely. Such a component is uniruled, whereas a generically finite dominant image of an uniruled surface is uniruled. This contradicts the fact that a K3 surface is not uniruled in characteristic zero.

Consequently \(f=h\circ r\) for a dominant rational map \(h:S\dashrightarrow C\). This is descent along the geometrically integral general fibres of the MRC quotient, or equivalently the relative algebraic closedness of \(\mathbb C(S)\) in \(\mathbb C(T)\). Resolve \(h\) and take its Stein factorization. Over \(\overline F\), let \(\Gamma\) be a smooth geometric generic fibre of the resulting connected-fibre pencil, of genus \(g\). A geometric generic component \(P\) of \(f\) dominates the corresponding \(\Gamma\). For a smooth projective model \(\widetilde P\) over \(\overline F\) this implies \[g\leq h^0(\widetilde P,\Omega^1_{\widetilde P}) =h^1(\widetilde P,\mathcal O_{\widetilde P}).\] To see the inequality, resolve the rational map to \(\Gamma\) and pull back regular one-forms; pullback is injective for a dominant map in characteristic zero, and irregularity is a birational invariant of smooth projective surfaces. The irregularities of \(\widetilde P\) are uniformly bounded by bounded smooth projective birational models and upper semicontinuity. Lemma 20 therefore bounds \(d\).

All genus, intersection, and coherent-cohomology bounds used here are preserved by extension of algebraically closed fields. The same proofs therefore apply to the geometric generic data specified in (iii) and (iv). Taking the maximum of the finitely many bounds proves all the assertions. ◻

Proof. Take \(N=N_1\) in the displayed wall diagram, let \(A=\Theta-\Phi\) be the ample difference on \(W\), and write \(A_N,\Theta_N,\Phi_N\) for transforms. We first transfer the discrepancy bound from \(W\) to \(N\). We then make the original log-anticanonical class nef and pass to its ample model. That last map must be small: a divisorial contraction would lose one of the prime valuations we need to retain. On a graph resolution with projections \(p\) to \(W\) and \(q\) to \(N\), \[\begin{align*} p^*(K_W+\Theta)-q^*(K_N+\Theta_N)&=E_\Theta\geq0, \tag{18}\\ p^*A-q^*A_N&\leq0. \tag{19}\end{align*}\] For this comparison at the wall, approach \(\Theta\) by interior parameters \(\Theta_i\) in the chamber defining \(N\). By (Hacon and McKernan 2013, Theorem 3.3(3)), \(W\dashrightarrow N\) is a log terminal model for \(K_W+\Theta_i\), so the exceptional difference on the same graph resolution is effective. Its finitely many coefficients depend linearly on the boundary. Taking their limits gives \(E_\Theta\geq0\); (Hacon and McKernan 2013, Theorem 3.3(2)) supplies the contraction \(N\to Q\). Thus the first difference is exceptional and nonnegative at the wall. The second is also \(q\)-exceptional, since the map does not extract. On a \(q\)-contracted curve its degree equals that of \(p^*A\), so it is \(q\)-nef; the negativity lemma proves (19). Subtracting gives \[ p^*(K_W+\Phi)-q^*(K_N+\Phi_N)\geq0. \tag{20}\] It follows valuation by valuation that the original \(\epsilon\)-lc bound is retained on \(N\).

There is a useful extension of this argument. The divisor \(K_N+\Theta_N\) is pulled back from \(Q\). Any small \(\mathbb Q\)-factorial modification \(N'\to Q\) has the same codimension-one divisor and the same pullback relation. Thus \(N\dashrightarrow N'\) is crepant for \(K_N+\Theta_N\). The comparison (18) remains nonnegative for \(W\dashrightarrow N'\), and the ample subtraction proves (20) for \(N'\) as well. In particular, the bound holds on any required relative small chamber; there is no need to locate all those chambers in one chosen two-dimensional slice.

The pair \((N,\Theta_N)\) is klt, its boundary is big over \(Q\), and its log-canonical divisor is trivial over \(Q\). Writing that boundary, up to relative rational linear equivalence, as an ample divisor plus an effective divisor, a sufficiently small perturbation makes a klt log-Fano boundary. Therefore \(N/Q\) is of Fano type. Furthermore, \[-(K_N+\Phi_N)\equiv_Q A_N.\] The transform of an ample class by a birational contraction is in the interior of the movable cone: choose divisor classes on \(W\) mapping onto a basis on \(N\), perturb \(A\) slightly in both directions in this basis, and push their ample systems forward. No prime divisor is fixed, because none is extracted. The same argument works after passing to the relative numerical space over \(Q\).

Finite generation now makes this movable class nef on a small model \(N^{\mathrm c}/Q\). Only small steps are needed: a negative divisorial contraction would force its exceptional divisor to be fixed in the corresponding system. The class is big, and its semiample contraction cannot be divisorial because it lies in the interior of the movable cone. Indeed a nef class vanishing on curves covering a contracted divisor lies on the boundary of the movable cone, as follows by intersecting those curves with effective divisors avoiding that divisor. Its ample-model map \(N^{\mathrm c}\to Z^{\mathrm c}\) is therefore small. The log-canonical divisor is crepant for this last contraction, so the discrepancy bound passes to \(Z^{\mathrm c}\). Its small \(\mathbb Q\)-factorializations have precisely the common prime valuations of the link. In particular, \(N^{\mathrm c}\to Z^{\mathrm c}\) is one such small \(\mathbb Q\)-factorialization, and \(N^{\mathrm c}\dashrightarrow N_i\) is small for \(i=1,2\). Composing with the contraction towers \(N_i\to V_i\to B_i\to Q\) in the wall diagram gives exactly the rational contractions asserted in the statement.

When \(\Phi=0\), exceptional valuations above \(W\) already have log discrepancy greater than one. A divisor of \(W\) that is contracted has a strictly negative coefficient in (19): intersect with a general curve in its positive-dimensional contracted fibre, on which \(A\) has positive degree. Its coefficient in (20) is consequently strictly positive. This gives terminality also for these remaining exceptional valuations. The same reasoning applies to each small model as above. ◻

Proof. Write \((W',\Phi')\) for the higher smooth start with its new marking. On the original smooth SNC pair, every exceptional valuation has log discrepancy greater than one: (21) gives the bound \(2(1-c)>1\) for rational \(c\). For irrational \(c\), apply it to rational coefficients decreasing to \(c\) and use the affine dependence of log discrepancy on the boundary coefficients. The non-exceptional divisors have discrepancy at least \(1-c\). These facts also hold on the new smooth SNC start directly, without comparing the number of its exceptional divisors.

Fix one endpoint \(V\) and write \(f:W'\to V\) for the resolved map. For a divisor of \(W\) contracted by its negative program, discrepancy strictly improves. If its retained marking is \(b\in\{0,c\}\), then on the higher start its coefficient in the difference from the pullback of \(K_V+\Phi_V\) is \[a(F,V,\Phi_V)-1+b>0.\] For a divisor exceptional over \(W\), monotonicity gives \(a(F,V,\Phi_V)\geq a(F,W,\Phi)>1\), and adding a new mark \(b\geq0\) again makes this coefficient positive. Thus the entire difference over \(V\) is effective with strictly positive coefficients on every endpoint-exceptional divisor.

Run the relative program over \(V\). Existence follows from relative bigness for a birational morphism (Birkar et al. 2010, Theorem 1.2). At a relative minimal model the exceptional difference is nef over \(V\) and hence nonpositive by negativity. Since all the above positive exceptional coefficients must have been contracted, the resulting map to \(V\) is small. A projective small birational morphism to a \(\mathbb Q\)-factorial variety is an isomorphism. This realizes the same endpoint, after which its Mori contraction can be used. The argument works for both endpoints.

We spell out the endpoint perturbations from (Hacon and McKernan 2013, Lemma 4.1). Write \(f_i:W'\to V_i\) for the two resolved maps and \(\pi_i:V_i\to B_i\) for their Mori contractions. Choose small general effective ample rational divisors \(A,H_1,\ldots,H_s\), with the classes of the \(H_j\) spanning \(N^1(W')\), and put \(H=A+\sum_j H_j\). If \(c\) is irrational, also choose a rational boundary \(\Phi_0\leq\Phi'\) sufficiently close to \(\Phi'\); for rational \(c\) take \(\Phi_0=\Phi'\). Make these choices so that \(A+\Phi_0-\Phi'\) is ample, both \(f_i\) are \((K_{W'}+\Phi_0+H)\)-negative, and \(-(K_{V_i}+(f_i)_*\Phi_0+(f_i)_*H)\) is \(\pi_i\)-ample. The exceptional differences have finitely many strictly positive coefficients, which stay positive under these small changes; the ampleness conditions are open as well. For a sufficiently positive ample rational divisor \(C_i\) on \(B_i\), the divisor \[L_i=-(K_{V_i}+(f_i)_*\Phi_0+(f_i)_*H)+\pi_i^*C_i\] is ample. Choose a general effective rational representative \(F_i^{V_i}\sim_{\mathbb Q}L_i\) and put \(F_i=f_i^*F_i^{V_i}\). Then \[\Theta_i=\Phi_0+H+F_i,\qquad K_{V_i}+(f_i)_*\Theta_i\sim_{\mathbb Q}\pi_i^*C_i.\] Here \(\Theta_i-\Phi'\) is ample: it is the sum of the ample divisor \(A+\Phi_0-\Phi'\) and the nef divisors \(H_j,F_i\). Simultaneous general representatives of sufficiently divisible multiples make the dominating pair \((W',\Phi'+H+F_1+F_2)\) klt. The negativity of \(f_i\) therefore makes \(B_i\) the required endpoint ample model. The spanning classes \(H_j\) and general two-dimensional slice in (Hacon and McKernan 2013, Lemma 4.1) complete these two perturbations to one geography, retaining \(\Phi'\) as the fixed original boundary. Lemma 23 now uses the unchanged lower bound \(1-c\). With zero boundary the same proof starts from the strictly positive discrepancies of terminal endpoints. ◻

Proof. We give the generic argument, and then apply noetherian induction. Take the closure of the parameter points under consideration and an irreducible component of this closure. After shrinking and a generically finite base change, we have flat normal geometric fibres \(\mathcal Y_t\), a family \(\pi:\mathcal R\to\mathcal Y\) of resolutions, and smooth projective fibres \(\mathcal R_t\) on which the marked strict transforms and exceptional divisors form a relative SNC divisor. The individual exceptional components can also be separated by this base change. These are the usual spreading operations on a resolution of the geometric generic fibre.

For the members in question, rational singularities and vanishing give \[H^i(\mathcal Y_t,\mathcal O_{\mathcal Y_t}) =H^i(\mathcal R_t,\mathcal O_{\mathcal R_t})=0\quad(i=1,2).\] For log-Fano pairs, the downstairs vanishing follows from Kawamata–Viehweg vanishing and the equality follows from rational singularities; see (Fujino 2014, Theorem 3.13.1 and Corollary 5.7.7). The parameter points of these members are Zariski dense in the closure chosen above. Thus they satisfy the dense-subset hypothesis of the relative Picard-space theorem (Choi et al. 2026, Theorem 1.3): rational singularities and \(H^1(\mathcal O)=H^2(\mathcal O)=0\) are required on that dense subset. Upper semicontinuity also gives the displayed vanishings on an open set. The theorem permits a further generically finite base change and shrinking such that the rational numerical divisor spaces of \(\mathcal Y_t\) and \(\mathcal R_t\), their pullback map, and the exceptional divisor classes are identified with fixed rational linear data. In particular, choose relative Cartier divisors whose classes span the downstairs space on every fibre.

Here is the required \(\mathbb Q\)-Cartier criterion. For a rational Weil divisor \(D\) on a fibre, let \(\widetilde D\) be its strict transform. Then \[ D\text{ is }\mathbb Q\text{-Cartier} \quad\Longleftrightarrow\quad [\widetilde D]\in \pi^*N^1(\mathcal Y_t)_{\mathbb Q} +\sum_E\mathbb Q[E]\ \subset N^1(\mathcal R_t)_{\mathbb Q}. \tag{22}\] The forward implication follows by pullback. For the converse, numerical equivalence of rational divisors on \(\mathcal R_t\) is rational linear equivalence: its \(\operatorname{Pic}^0\) vanishes, and the numerically trivial part of the Néron–Severi group is torsion. Thus the asserted relation, after clearing denominators, is a linear-equivalence relation with a Cartier divisor pulled back from \(\mathcal Y_t\) and exceptional divisors. Pushing down proves that a multiple of \(D\) is linearly equivalent to a Cartier divisor. This is precisely the converse in (22).

Apply the criterion to the fixed rational classes on the resolution. For the log-canonical class one may use \(K_{\mathcal R_t}+\widetilde B_t\) modulo exceptional classes; this does not presuppose that \(K_{\mathcal Y_t}+B_t\) is \(\mathbb Q\)-Cartier. The membership in (22) holds on the dense set of members under consideration, and the linear data are fixed. It therefore holds at the geometric generic point, with fixed rational coefficients in a basis of relative Cartier classes. Rational linear equivalence upstairs, followed by pushdown, proves that the generic log-canonical divisor is \(\mathbb Q\)-Cartier. Clear denominators and spread this linear equivalence. Its possible vertical errors disappear after shrinking, giving a relative Cartier multiple. This argument applies simultaneously to each additional specified class.

Exceptional coefficients in a pullback are unique. Indeed, an exceptional rational divisor numerically trivial over its target is zero, by applying the negativity lemma to it and its negative. They are thus the solutions of fixed rational linear equations on the resolution, and are constant after shrinking. The klt condition is now the finite set of strict inequalities saying that the crepant SNC coefficients are less than one. Relative log-anticanonical ampleness is open. Thus an open set consists entirely of the required log-Fano pairs, with all the asserted properties.

Finally, all Weil divisors downstairs are \(\mathbb Q\)-Cartier exactly when the right side of (22) exhausts \(N^1(\mathcal R_t)_{\mathbb Q}\): every divisor upstairs pushes to a Weil divisor, and the same argument applies. This is again a condition on the fixed linear data. Applying the construction to the remaining closed parameter sets proves all the assertions by noetherian induction. ◻

Proof. Lemma 25 supplies families of Fano-type fibres. Take smooth parameter strata and remove the images of any vertical exceptional components of the simultaneous resolution. The relative \(\mathbb Q\)-Cartier equality and the crepant coefficients less than one then make the total-space pair klt as well. This verifies the total-space hypothesis of (Choi et al. 2026, Theorem 1.4). After a generically finite base change and shrinking, (Choi et al. 2026, Theorems 1.4–1.6) gives fibrewise small \(\mathbb Q\)-factorializations and compatible numerical spaces, nef cones, Mori chambers, and MMP steps. Its hypotheses hold: the fibres have rational singularities and the stated vanishings, and every fibre is of Fano type. Thus the very-general-fibre alternative in (Choi et al. 2026, Theorem 1.2) applies.

For clarity, the finiteness-to-boundedness step is as follows. On the geometric generic fibre of one stratum choose a small \(\mathbb Q\)-factorialization. Finite generation supplies finitely many small models, and finitely many faces and semiample contractions on each of them. All this finite data spreads after a finite extension and shrinking. The two inclusions needed for constancy of a nef cone can be seen directly: spread its finitely many semiample generators, which remain semiample on an open set, and spread effective curves generating the dual polyhedral cone. The divisor generators give one inclusion; the curve generators give the reverse. The fibrewise \(\mathbb Q\)-factorializations and compatible numerical spaces ensure that these comparisons take place in the same space. The relative chamber/MMP compatibility cited above then rules out further chambers on a fibre. Here the source has first been made fibrewise \(\mathbb Q\)-factorial, so the partial-MMP chamber description agrees with that from small modifications; the converse direction in (Choi et al. 2026, Theorem 1.6(3)) applies. The generic finite list of graphs consequently supplies the complete list on an open set. Apply noetherian induction to the remaining strata. This proves boundedness of the diagrams, not just of their vertices.

We also check the last assertion. If \(f:Y\dashrightarrow Z\) is a non-extracting birational map, take a sufficiently ample Cartier divisor \(H\) on \(Z\) and a general member avoiding the generic points of the finitely many centers of divisors contracted by \(f\). Write \(D\) for its strict transform on \(Y\). The corresponding linear system has no fixed prime component. Moreover, \[R(Y,D)=R(Z,H).\] At a prime divisor retained by the map this is the same valuation test on both sides. At a prime divisor contracted on \(Y\), a regular section of \(\mathcal O_Z(mH)\) pulls back regularly, and the chosen \(H\) has order zero at that valuation. Normality then gives equality of the section spaces. Hence \(Z\) is the ample model of a movable system on \(Y\). A subsequent contraction is handled by taking the pullback of an ample divisor from its base. The Mori dream space chamber description therefore includes these maps too. If an intermediate target is not \(\mathbb Q\)-factorial, apply this description to the composite from the initial \(\mathbb Q\)-factorial source; the diagram also records the intermediate maps.

Finally take closures of the marked transforms on the finitely many graph families, and resolve their union with the exceptional loci. Proper images, inverse images, and irreducible components of these finite-type families remain bounded after stratification. This gives the stated marked and exceptional data. ◻

Proof. Apply the geography above to a common smooth resolution of the specified map in (i), and to the given common-start maps in (ii), retaining their function-field identifications. Lemmas 23 and 24 give a central ample model \(Z^{\mathrm c}/Q\), reached by a small map from \(N^{\mathrm c}/Q\), with \(\epsilon=1\) in the first case and \(\epsilon=1-c\) in the second. Put \(Z=Z^{\mathrm c}_{\overline F}\). Its dimension is at most four, the negative log-canonical divisor is ample, and the same discrepancy bound holds. Birkar’s boundedness theorem (Birkar 2021, Theorem 1.1) applies: projective varieties admitting an \(\epsilon\)-lc boundary with nef and big negative log-canonical divisor form a bounded family. In particular it bounds the underlying varieties \(Z\) just constructed.

The marked boundary is bounded as well. In case (ii), write its restriction to \(Z\) as \(cD_Z\), and let \(H\) be one of the bounded very ample polarizations supplied by boundedness. In fibre dimension \(r\), \[cD_Z\cdot H^{r-1}\leq -K_Z\cdot H^{r-1}.\] The right side is bounded by the coefficient of the Hilbert polynomial of the bounded polarized varieties. Thus \(D_Z\) has bounded degree. Reduced cycles of bounded degree in a fixed projective space form a bounded family, so the marked central pairs are bounded. Marks vertical over \(Q\) have empty restriction to the generic fibre. Horizontal components may split over \(\overline F\), but their nonzero coefficients remain \(c\) and their total degree is still bounded.

All the geometric generic link vertices and bases are rational contractions of small \(\mathbb Q\)-factorializations of these central pairs. Lemma 26 supplies bounded diagrams, and Remark 27 supplies the closed bad loci. Small loci do not acquire divisorial components under the algebraic extension to the geometric generic field. The same constructions apply over algebraically closed fields of characteristic zero. For the family results stated over \(\mathbb C\), we use their finite-data spreading arguments and algebraically closed base change to obtain the corresponding geometric generic families here. Every bound was taken from one of these fixed bounded families, not by accumulating constants along a path.

For the assertion about base contractions, the central rank-two model is of Fano type over \(Q\) by Lemma 23. This property survives small modifications and descends under contractions, including fibre-type contractions (Prokhorov and Shokurov 2009, Lemma 2.8, with the relative convention after Lemma–Definition 2.6). It therefore holds for every end and its base \(B\to Q\). These are contractions of normal varieties, so their generic fibres are geometrically integral in characteristic zero. The rational connectedness theorem for klt log-Fano contractions (Hacon and McKernan 2007), applied also to a resolution, gives the asserted rationally connected smooth proper models. ◻

Proof. At the generic point of the center of \(F\), of codimension \(r\), use the regular parameters and boundary coefficients in (21). If the center is not a boundary stratum, at least one \(b_i\) is zero; since an exceptional center has \(r\geq2\), the right side is strictly greater than one. Thus a valuation in question centers at a boundary stratum. Write its orders there as \(w_i\). They satisfy \[\sum_i(1-b_i)w_i\leq1.\] There are finitely many possibilities because every \(1-b_i\) is positive.

Blow up the stratum. In a chart containing the center of the valuation, choose an index with minimal order; the new orders are that minimum and the differences from it. Zero differences correspond to units unless the center lies in a further proper subvariety of the exceptional stratum. The latter would be a non-stratum center and is excluded by (21) applied to the crepant SNC transform. This transform is still klt, so its coefficients remain strictly less than one and the same non-stratum exclusion applies at every step. Thus, until the valuation itself becomes a divisor, its center continues to be a stratum and the sum of its positive orders strictly decreases. This is the usual monomial subtraction algorithm, and proves both that the valuation is a stratum valuation and that finitely many blow-up sequences suffice. In a fixed family there are finitely many strata and bounded orders; separate their geometric components and perform the same finite sequences relatively. ◻

Proof. Use Lemma 25 to obtain simultaneous crepant log resolutions. Their coefficients are in a fixed finite set strictly below one on each stratum. The exceptional divisors already on the resolution are a finite list; all additional valuations of log discrepancy at most one are in the uniformly finite list of Lemma 29. Extract all of them on a common resolution.

The extraction theorem (Birkar et al. 2010, Corollary 1.4.3) produces a \(\mathbb Q\)-factorial model \(\tau:T^{\mathrm t}\to T\) extracting exactly this set. The extra restriction concerning non-klt centers in that theorem is empty here. Its crepant boundary \(\Lambda^{\mathrm t}\) has coefficient \(1-a(F,T,\Lambda)\) on an extracted divisor, and hence is effective. By construction the pair is terminal. Also \[-(K_{T^{\mathrm t}}+\Lambda^{\mathrm t}) =\tau^*\bigl(-(K_T+\Lambda)\bigr)\] is nef and big, so \(T^{\mathrm t}\) is of Fano type. Apply this construction on a geometric generic member and spread it; the fixed extraction list, crepant equality, and the discrepancy inequalities persist after shrinking. Thus the terminalizations and all their marks form bounded families. This construction does not resolve an unbounded set of arbitrarily chosen higher models.

For a model \(T'\) extracting a subset, every prime divisor on \(T'\) is already a valuation appearing on \(T^{\mathrm t}\). Consequently \(T^{\mathrm t}\dashrightarrow T'\) is a birational contraction. Lemma 26, applied to the bounded Fano-type terminalization families, supplies its diagram and every indicated subsequent contraction. In applying that lemma one can either use the displayed weak log-Fano pair and a small ample perturbation of its boundary on each stratum, or spread a Fano-type boundary chosen on its generic member. Both give families of klt log-Fano pairs on an open set. Include all the extracted divisors as marks, even those with crepant coefficient zero. The marked support on every output is then among the bounded transforms already considered. ◻

Proof. Zero-dimensional members have \(D=0\) and impose no restriction on \(c\); we treat the positive-dimensional members below. Apply Lemma 25 to the log-Fano pairs \((T,0)\) with additional marked divisor \(D\). On finitely many simultaneous log resolutions the crepant coefficients for \((T,0)\) and the coefficients of \(\pi^*D\) are fixed. Write \(a_E=a(E,T,0)>0\) and \(m_E\) for the coefficient of \(\pi^*D\) on an exceptional component. Choosing \(c<1/2\) and \(cm_E<a_E/2\) for every \(m_E>0\) makes every coefficient of the crepant pullback of \(cD\) strictly less than one. These finitely many inequalities give a uniform positive klt bound for \(c\).

For ampleness use a bounded very ample divisor \(H\) and let \(r=\dim T\). A common multiple \(mK_T\) is Cartier on each of the finitely many strata. Thus \[k=-K_T\cdot H^{r-1}>0,\qquad mk\in\mathbb Z_{>0}.\] The degree \(e=D\cdot H^{r-1}\) is a bounded nonnegative integer. Numerical proportionality says \(D\equiv(e/k)(-K_T)\). Since \(e/k\leq me\), choosing \(cme<1/2\) for all these data gives \[-(K_T+cD)\equiv (1-ce/k)(-K_T),\] which is ample. Taking the minimum of the finitely many rational choices gives \(c_*\).

Finally numerical equivalence between divisors defined over the original field persists geometrically. Any geometric curve is defined over a finite extension; all its conjugates have the same intersection with such a divisor. Its orbit descends to a curve cycle over the original field, so a zero intersection downstairs forces zero on every conjugate. The preceding argument therefore applies after the geometric base change. ◻

Proof. First use only the ordinary part of Theorem 28. At the indicated end the tower is \(N_i\xrightarrow{e_i}V\to T\to Q\). Since \(\dim T=3\) and \(\dim Q\leq1\), the last contraction has positive relative dimension and \(\rho(T/Q)\geq1\). Equation (17) gives \[2=\rho(N_i/V)+1+\rho(T/Q).\] Consequently \(\rho(N_i/V)=0\) and \(\rho(T/Q)=1\): the arrow \(e_i\) is the identity and \(T\to Q\) is an elementary contraction of fibre type. This calculation takes place over \(Q\) before geometric generic base change. The geometric generic model \(T_{\overline F}\) and its conic projection occur in the bounded diagram. The restriction of the original visible ramification is contained in the bounded bad locus of this projection, and its divisorial support is a union of the finitely many divisorial components there. These marked supports are therefore bounded.

Apply (Druel 2016, Lemma 4.6) to the elementary conic Mori contraction \(V\to T\). Its source is terminal and \(\mathbb Q\)-factorial, so its base \(T\) is \(\mathbb Q\)-factorial. Divisors on \(T_F\) extend by closure to Weil divisors on \(T\), which are \(\mathbb Q\)-Cartier. Restriction therefore surjects onto the generic numerical divisor space. The relative rank-one condition and the nonzero restriction of an ample class give \(\rho(T_F/F)=1\). Fano type descends under contractions (Prokhorov and Shokurov 2009, Lemma 2.8); hence \(T_F\) and \(T_{\overline F}\) are of Fano type. Over the original generic field \(F\), rank one implies that \(-K_{T_F}\) is ample: it is the sum of an ample log-anticanonical class and an effective class. It also implies numerical proportionality of \((D_T)_F\) and \(-K_{T_F}\). These two divisors are \(\mathbb Q\)-Cartier before, and hence after, base change. Their numerical proportionality persists over \(\overline F\) by Lemma 31, even if the geometric Picard rank increases. That lemma provides a uniform \(c_*\) from these ordinary diagrams.

Choose a rational \(c<\min\{1/4,c_*\}\) once. With this fixed value the marked log links have the uniform bound of Theorem 28, and the bounded \((T_{\overline F},c(D_T)_{\overline F})\) data have the extraction bound of Theorem 30. Only finitely many bounded families, dimensions, and bounded-degree cover constructions are used subsequently: the exceptional and bad-locus components in Section 5, and the ramification and del Pezzo class covers in Section 6. Their branch and degree bounds are checked there. Take the maximum of their thresholds in Theorem 22. Every bound depends only on those families and this chosen \(c\), proving the asserted independence. ◻

Proof. Choose a rational boundary \(\Theta\) making \((Y,\Theta)\) klt and \(-(K_Y+\Theta)\) relatively ample. Then \(-K_Y=-(K_Y+\Theta)+\Theta\) is relatively big, and a sufficiently divisible multiple of \(-(K_Y+\Theta)\) is generated over \(Z\). Theorem 1.2 and Corollary 1.4 of (Hacon and McKernan 2007) give rational chain connectedness of every fibre; the non-klt locus is empty. The statement after passage to a characteristic-zero geometric generic field follows by descent of the finite data and base extension.

Put \(T=f(E)\) and choose a general point \(e\) of \(E\) outside the other components of \(f^{-1}(T)\). The fibre of \(E\to T\) through \(e\) has positive dimension. A chain of rational curves in the fibre of \(f\) joining \(e\) to another point starts with a nonconstant rational curve through \(e\). That curve is contained in \(E\): its image is in \(f^{-1}(T)\), and an irreducible curve through a point lying on only the component \(E\) must lie in that component. Thus rational curves cover \(E\), proving uniruledness. Equivalently, one can select a dominating component of the parameter space of these rational curves. The argument can be made after an uncountable algebraically closed extension and descended, so it does not require an uncountability assumption on the original field.

For (i), \(f(E)\) is proper and \(f^{-1}(f(E))\) is a proper closed subset of the integral variety \(Y\). Its component containing \(E\) must equal \(E\), and \(\dim E-\dim f(E)\ge\dim Y-\dim Z>0\). The same component argument works in (ii), where exceptionalness gives \(\dim E-\dim f(E)\ge1\). Finally uniruledness is birationally invariant. ◻

Proof. The assertion is immediate for an isomorphism. For a birational contraction, nonexceptional prime divisors cancel with their birational transforms on \(Q\). The remaining divisors are uniruled by Lemma 33. Since \(\dim B\le3\), these divisors have dimension at most two, and their MRC quotients have dimension at most one; they have test value zero.

If the contraction has positive relative dimension, then \(\dim Q\le2\). Every prime divisor of \(Q\) has test value zero for dimension reasons. Every divisor of \(B\) vertical over \(Q\) is uniruled by Lemma 33, and again has dimension at most two, so its test value is zero. This includes \(Q\) a point, when there are no vertical divisors and no base divisors to sum. ◻

Proof. In the difference \(v(V'/Q)-v(V/Q)\) the base-divisor sums cancel. Common vertical prime valuations cancel as well, leaving precisely the signed exceptional divisors vertical over \(Q\). This proves (26). Substitute (25) and use Lemma 34 on both ends to obtain (27). ◻

Proof. Suppose first that \(\dim Q=3\). Each Mori base has dimension at most three and dominates \(Q\), so its morphism to \(Q\) is birational. No divisor of either Mori base, and no divisor of \(V\) vertical over that base, dominates \(Q\). Therefore both \(h_Q\) terms vanish. Normal proper generic curves over \(\mathbb C(Q)\) are regular and hence smooth in characteristic zero. A birational map between them is an isomorphism. Their closed points are exactly the prime divisors of the fourfolds dominating \(Q\), so there are no horizontal divisorial contributions either.

For \(\dim Q=0\), the entire marked contraction diagram is bounded by Theorem 28. The exceptional prime divisors contributing to \(c_u(f)\) therefore have bounded birational models of dimension three. Choose a marked good open for each contraction \(V\to B\). After cancellation, the divisors contributing to \(h_Q(V/B)=v(V/B)\) are components of the marked bad locus on \(B\) or its preimage on \(V\). These components have bounded birational models by the marked-locus assertion of Theorem 28. The bounded-variety exclusion in Theorem 22(i) gives a single threshold beyond which every term has test value zero.

Now suppose \(\dim Q=1\), and put \(F=\mathbb C(Q)\). We use the bounded diagrams after extension to \(\overline F\). If \(E\) is a horizontal exceptional prime divisor of the link, the geometric generic fibre components of \(E\to Q\) are surfaces. On the side where the divisor is extracted, each is exceptional for a divisorial contraction of the generic threefold. They are uniruled by Lemma 33; any intervening small modifications preserve their birational types. Their smooth birational models are bounded by Theorem 28. The threefold-over-a-curve exclusion in Theorem 22(iv) therefore makes \(u(E)=0\) uniformly for large \(d\). Here \(u\) is evaluated on the original complex threefold field \(\mathbb C(E)\); only its geometric generic surface components, rather than \(E\) itself, are being placed in bounded families.

It remains to treat \(h_Q(V/B)\), the other end being identical. Work with a good open for the generic contraction \(V_F\to B_F\) and spread it over an open of \(Q\). Shrinking \(Q\) changes no horizontal prime field. The cancellations defining \(h_Q\) remove all divisors meeting that good open. Consequently the geometric generic components of the remaining prime divisors lie in the bounded marked bad locus or its bounded preimage.

A remaining positive term is a divisor \(P\subset V\) vertical over \(B\) and dominant over \(Q\). Its geometric generic fibre components are vertical divisors of \(V_{\overline F}\to B_{\overline F}\). They are uniruled surfaces by Lemma 33, with bounded birational models. Theorem 22(iv) therefore gives \(u(P)=0\) for large \(d\). A remaining negative term is a divisor \(D\subset B\) dominant over \(Q\). If \(\dim D\le1\), its test value is zero automatically. If \(\dim D=2\), its geometric generic fibre components over \(Q\) are bounded curves. Were \(u(D)=1\), the surface \(D\) would be birational to a member of \(\mathcal S\); the induced pencil, after Stein factorization, would have general fibre genus at least \(1+\sqrt d/2\). This contradicts the uniform genus bound for those components when \(d\) is large.

Only finitely many bounded families and types of marked components have been used. Take the maximum of their exclusion thresholds. All exclusions concern the full collection of degree-\(d\) Picard-number-one K3 surfaces and therefore hold for every subcollection \(\mathcal S\). This proves the stated uniformity. ◻

Proof. A smooth projective model of \(Q\) is a dominant rational image of the rationally connected fourfold, hence is rationally connected. A smooth projective rationally connected complex surface is rational: it is uniruled, its minimal model is ruled over a curve, and rational connectedness forces that curve to have genus zero; see (Araujo and Kollár 2002, Theorem 30) and (Beauville 1978, Theorem V.1). Consequently \(k\simeq\mathbb C(x,y)\).

A terminal fourfold is nonsingular in codimension two (Kollár 1992, Exercise 4.9.1.1), so its singular locus has dimension at most one and cannot dominate \(Q\). The generic surface is regular, as a localization of its regular total space near the generic fibre, and is smooth because \(k\) is perfect. The contractions have connected fibres and rationally connected geometric generic fibres as above, giving geometric integrality. Projectivity is inherited by base change. A small birational map between smooth proper surfaces is an isomorphism: the exceptional locus of a non-isomorphic birational morphism of smooth surfaces contains a curve, and resolving a birational map gives the same conclusion if it is non-isomorphic in either direction. Such a curve would spread to a divisor exceptional for the original small map.

If \(\dim B=2\), the contraction \(B\to Q\) is birational. The generic end \(V_k\) has ample anticanonical divisor and Picard number one over \(k\), hence is a del Pezzo surface. If \(\dim B=3\), the generic base \(B_k\) is a normal proper curve with geometrically integral rationally connected generic model, so it is a smooth genus-zero curve. The generic morphism \(V_k\to B_k\) has relatively ample anticanonical divisor and relative Picard number one, hence is a smooth-surface Mori conic bundle.

Here Picard numbers mean dimensions of numerical divisor spaces over \(k\), not over its algebraic closure. To justify their restriction from the fourfold, a divisor on \(V_k\) closes up to a Weil divisor on \(V\), which is \(\mathbb Q\)-Cartier. This gives surjectivity onto the generic numerical divisor space. A class numerically zero over the indicated original base remains numerically zero on the generic fibre: a generic curve and its intersection degrees spread over an open of \(Q\). Thus the relative rank is at most the original rank one. The anticanonical class is nonzero, so it is exactly one. The same argument bounds the numerical rank of the generic middle surface by two. No passage to an algebraic closure is made in this rank calculation. In the conic case the base curve has numerical Picard number one, its pullback to the surface is nonzero, and the relative numerical divisor space has dimension one. Thus the total numerical divisor space has dimension two. ◻

Proof. By Lemma 37, \(\rho(W/k)\leq2\). If a divisorial contraction \(W\to S\) survives on the generic fibre, then \(\rho(W/k)=\rho(S/k)+1\). The end \(S\) therefore has rank one and is a del Pezzo surface. Extraction from a conic end, which has rank two, would force \(\rho(W/k)\geq3\). Thus any surviving extraction forces \(\rho(W/k)=2\). If the opposite end has no surviving extraction, its surface is \(W\) itself and cannot be a rank-one del Pezzo. It must instead carry its conic contraction. If both ends have surviving extractions, both are divisorial contractions from \(W\) to rank-one del Pezzo surfaces.

If neither side has a surviving extraction, both generic end surfaces are \(W\). Rank one then gives two structures over a point, and the restricted map is a structured isomorphism. Rank two gives two conic contractions. These may agree or may be distinct; the latter case is retained as a two-conic move.

In all the rank-two cases, compare the two end contractions on \(W\). If they contract the same extremal ray, uniqueness of the contraction identifies their normal targets. Indeed, each is constant on the connected fibres of the other, hence factors through it, and the two factorizations are inverse. For conic contractions this identifies their pullback curve fields inside \(k(W)\), not merely the abstract isomorphism classes of the curves. For identical birational contractions the common exceptional orbit cancels from the horizontal cocycle. Thus the restricted birational map is a structured isomorphism in either case, with equal vertical divisor data and \(H_Q=0\).

Otherwise the two contractions give distinct \(K_W\)-negative extremal rays. These are the two boundary rays of the two-dimensional cone of curves of the projective surface \(W\). The anticanonical divisor is positive on both and is therefore ample. This proves (ii). The argument uses numerical classes and contractions over \(k\) and does not require the generic surfaces or conic bases to split. ◻

Proof. The morphism from the smooth surface to the smooth curve is flat, so its fibres have the arithmetic genus of the generic conic. Over \(\overline k\), every fibre is connected, has arithmetic genus zero, and has anticanonical degree two. The anticanonical degree of each irreducible component is a positive integer. An irreducible reduced fibre therefore has arithmetic genus zero and is a smooth rational curve. A double fibre \(2R\) would have \(R^2=0\) and \(K_S\cdot R=-1\), contradicting adjunction parity. The only remaining possibility is a reduced pair \(R_1+R_2\) with \(-K_S\cdot R_i=1\). Put \(m=R_1\cdot R_2\). Since \((R_1+R_2)\cdot R_i=0\) and the fibre is connected, adjunction gives \[2p_a(R_i)-2=-m-1,\qquad m\geq1.\] Thus \(m=1\), both curves are smooth rational curves of self-intersection \(-1\), and they meet transversely.

If the component cover over an orbit were split, the sum of one chosen component over each point of that orbit would descend to a divisor on \(S\). It has zero intersection with a general fibre and negative intersection with every component in that chosen orbit. This is impossible when the relative numerical divisor space is one-dimensional: the general fibre is a nonzero relative curve class and spans its dual. Consequently every component orbit has field \(L\) quadratic over the corresponding discriminant field \(K\).

The relative anticanonical system embeds the fibres as plane conics: on a smooth fibre it has degree two, and on a reducible fibre it has degree one on each component, with vanishing first cohomology. Cohomology and base change give the local rank-three system. At a singular fibre the total surface is regular, so after diagonalizing the quadratic form over the local discrete valuation ring, its nonsingular binary part has unit determinant and the remaining coefficient has valuation one. More explicitly, with uniformizer \(t\) the equation is \[ax^2+by^2+tc z^2=0,\qquad a,b,c\text{ units}.\] An exponent of \(t\) greater than one would make the total space singular at the geometric node. We use the Kummer identification and quaternion residue formula of (Auel et al. 2020, sec. 2, equations (1)–(4)). The associated quaternion can be written \((-ab,-act)\); its tame residue is the square class of \(-ab\) in the residue field. This is exactly the extension separating the two lines \(ax^2+by^2=0\). At a smooth fibre all three diagonal coefficients are units, so the residue vanishes.

The corresponding vertical divisor has function field \(L(t)\), since a line is split over its component field. The involution of \(L/K\) excludes \(u(L)=1\) by the automorphism clause of Lemma 7; the degree of \(L\) over \(k\) need not be two. All smooth fibres cancel against their base points by the MRC rule. This proves (30). On a del Pezzo side the generic base is a point, so there are no vertical primes dominating \(Q\). ◻

Proof. Use a plane blowup basis \(L,E_1,\ldots,E_n\) with \(n=9-K_S^2\). A class \(xL-\sum y_iE_i\) belongs to \(A(S)\) exactly when \[\sum y_i=3x-2,\qquad \sum y_i^2=x^2.\] Cauchy’s inequality gives \((3x-2)^2\leq nx^2\). For \(n=4\) this restricts the integer \(x\) to \(1,2\). At \(x=1\) there is one coefficient \(y_i=1\) and all others vanish; at \(x=2\) all four coefficients are one. These are the four line pencils and the conic pencil. For \(n=3\) the inequality forces \(x=1\), giving the three line pencils. The same calculation for \(A_2(S)\) gives \(x=1,2\) and the two classes \[L,\qquad 2L-E_1-E_2-E_3.\] The three pencil classes \(L-E_i\) and these two plane classes span the degree-six Picard space. Their only linear relation is the equality of the two class sums, each equal to \(-K_S\). The relation is invariant under \(G_k\); semisimplicity yields (31). Taking invariants gives \[2=\#(A(S)/G_k)+\#(A_2(S)/G_k).\] The Brauer obstruction to descending an invariant line bundle is torsion (Auel and Bernardara 2018, Remark 2.7). Thus invariant rational Picard classes descend, and the invariant Picard space has dimension one. Both finite sets are nonempty, proving transitivity. ◻

Proof. By Lemma 38, the common smooth surface \(W\) is del Pezzo with a two-dimensional invariant numerical plane and the two end contractions generate its two rays. Each is either divisorial to a smooth surface or a conic contraction. For a divisorial contraction, pass first to \(\overline k\). Its exceptional intersection matrix is negative definite. An exceptional component \(R\) has \(R^2<0\) and \(K_W\cdot R<0\); adjunction \(2p_a(R)-2=R^2+K_W\cdot R\) forces both intersections to equal \(-1\) and \(R\) to be a smooth rational curve. Two such curves must be disjoint, since otherwise their two-by-two intersection matrix would not be negative definite. Thus the contraction is the blowdown of disjoint \((-1)\)-curves to distinct smooth points. Each Galois orbit of these curves contributes an independent invariant relative divisor class. Relative Picard number one therefore gives a single orbit. The contraction is consequently the blowup of one closed point over \(k\), and the equivariant correspondence between the geometric points and their exceptional curves identifies their orbit fields. Put \(D=K_W^2>0\).

Every fibre class used below is the integral class of a geometric fibre in \(\operatorname{Pic}(W_{\overline k})\). It is \(G_k\)-invariant, has square zero and anticanonical degree two, and lies in the rational numerical plane over \(k\) by the torsion descent argument used in Lemma 41. A closed fibre over a point of degree \(r\) on the original genus-zero curve has class \(r\) times this geometric class. Thus none of the following calculations requires a \(k\)-point on the conic base.

For a divisorial end let \(E\) denote its orbit of exceptional curves, write \(e=|E|\), and put \(E_\Sigma=\sum_{R\in E}R\) on \(W_{\overline k}\). Thus \(u(E)=u(k(E))\) counts one orbit field, whereas \(E_\Sigma^2=-e\) counts geometric curves in an intersection calculation. When both ends are divisorial, use \(F\), \(f=|F|\), and \(F_\Sigma\) for the corresponding data at the primed end. In the diagram \[\begin{tikzcd}[column sep=large] &W\arrow[dl,"\pi"']\arrow[dr,"\pi'"]&\\ S\arrow[rr,dashed,"\varphi_k"']&&S' \end{tikzcd}\] \(E\) is the exceptional set of \(\pi\) and \(F\) that of \(\pi'\). The factorization \(\varphi_k=\pi'\circ\pi^{-1}\) therefore contributes \(u(E)-u(F)\) by Lemma 8: the extraction has positive sign and the contraction negative sign. Neither term is multiplied by its orbit size.

Proof. Resolve the diagram and marked support in bounded families, marking all resolution exceptional divisors as well. Write \(D\) for the resulting SNC support on the smooth total space. It is enough that \(D\) contain the ramification; its unramified components do no harm. Every prime on the resolution is either the strict transform of a prime on the old model or an exceptional prime. Marking all exceptional primes therefore includes any newly visible ramification, even over the singular locus of the old model.

We form residues and their quadratic covers before extending constants: the ground field \(F\) is \(\mathbb C\), or \(\mathbb C(Q)\) on the ordinary generic fibre over \(Q\). After finding their unramified open, we will base change these same covers to \(\overline{\mathbb C(Q)}\) to use the geometric bounds. The Kummer isomorphism identifies the Brauer \(2\)-torsion of the total function field with its \(H^2\) with coefficients in \(\mu_2\) (Auel et al. 2020, sec. 2, equations (1)–(4)). On the smooth total model, the first two residue maps form a complex \[H^2(F(Y),\mu_2) \xrightarrow{\partial} \bigoplus_{E\in Y^{(1)}}H^1(F(E),\mathbb Z/2) \xrightarrow{\partial} \bigoplus_{z\in Y^{(2)}}H^0(F(z),\mu_2^{-1}), \qquad \partial^2=0.\] Here \(Y^{(j)}\) denotes the codimension-\(j\) points. These are the coniveau differentials of (Bloch and Ogus 1974, Example 2.1 and Proposition 3.9); the same complex applies over the characteristic-zero function field of \(Q\). Consider a ramification component \(D_i\) and a codimension-one point \(z\) of \(D_i\) outside its intersections with the other components of \(D\). Every term except that from \(D_i\) is zero in the displayed secondary residue at \(z\). Since \(D_i\) is smooth, that term is its ordinary DVR residue, so \(\partial_z(\partial_{D_i}\alpha)=0\). If the quadratic residue is represented by \(a\in F(D_i)^*/F(D_i)^{*2}\), this says that \(\operatorname{ord}_z(a)\) is even. After multiplication by a square, \(a\) is a unit at the DVR, and adjoining its square root is unramified there. Normalize the indicated open of \(D_i\) in the quadratic algebra, componentwise in the split case. This normalization is finite, is unramified at every codimension-one point, and has regular target. Purity of the branch locus makes it étale throughout that open (The Stacks Project Authors 2026a, Lemma 58.21.4, Tag 0BMB).

The horizontal ramification components are generically finite over \(S\). Delete their nonfinite and branch loci on \(S\), the images of their pairwise intersections, the images of their intersections with the remaining marked support, and the singular and resolution bad loci of the diagram. These images are proper closed subsets: over the generic point of \(S\) the distinct horizontal components are distinct points of the smooth proper curve. The deleted sets form a bounded embedded boundary after stratifying the family. Over the resulting smooth open \(U\subset S\), the horizontal union is finite étale, its components are smooth and disjoint from the other marked divisors, and its residue double cover is finite étale. Their degrees over \(U\) are \(r_k\) and \(2r_k\). The scheme of sections defining \(P_k\) is then a finite étale scheme of degree eight over \(U\). Thus all the fields in the statement have the asserted degree and branch bounds.

These degree and branch bounds exclude the large-degree K3 fields. First consider a bounded diagram over \(\mathbb C\). Fix an orbit field \(E/k\) in the statement, and let \(S_E\to S\) be the normalization in \(E\). Take a general pencil in a uniformly bounded very ample system on the resolved base surface \(S\). Let \(h\) bound the genus of its normalized general member and let \(b\) bound its intersections with the deleted boundary. Each normalized component of the pullback of that member to \(S_E\) has degree \(n\leq14\) and, by Hurwitz (The Stacks Project Authors 2026b, Lemmas 53.12.2 and 53.12.4), \[ 2g-2\leq n(2h-2)+b(n-1). \tag{39}\] If \(u(E)=1\), a smooth projective model of \(S_E\) is birational to a test K3 surface. Resolve the pulled-back pencil on this model and take its Stein factorization. Its connected general fibre is one of these normalized components. This contradicts the bound \(2g-2\geq\sqrt d\) in Lemma 20 for large \(d\).

When the family is given only on the geometric generic fibre of \(Q\), the same construction is performed over \(\overline{\mathbb C(Q)}\) on the base curve there. Finite étale covers remain finite étale after this extension, although they may split. The same degree and Hurwitz bounds therefore hold on every geometric component. We retain the base changes of the original ramification support and residue covers; we do not recompute the support from the extended Brauer class, whose residues may have become zero. For any orbit field \(E/k\), the inclusions \(\mathbb C(Q)\subset k\subset E\) give a rational map from its surface model to \(Q\). If \(u(E)=1\), resolve that map on the corresponding K3 surface and take its Stein factorization. Then the geometric general fibres of that map are precisely the components whose genera were bounded. Lemma 20 again gives the contradiction. No assertion about the complexity of an unmarked Brauer class is used. ◻

Proof. Let \(k'\subset L\) be any other eligible field; its candidate need not be nonzero. Resolve the two presentations on a common start (41). By (42), its log MMP has an output with base field \(k\) and another with base field \(k'\). Connect these outputs by the log Sarkisov links of Theorem 28, and write their successive nodes as \(Z_i/S_i\) for \(0\leq i\leq N\). The birational maps from the common start identify every \(\mathbb C(Z_i)\) with \(L\); set \(K_i=\mathbb C(S_i)\subset L\). Thus \(K_0=k\) and \(K_N=k'\). The class \(\alpha\in\operatorname{Br}(L)[2]\) is fixed throughout this path, and the support of each boundary is its full visible ramification on the corresponding model.

Every Mori base in this three-dimensional programme has dimension at most two. If a common base of a link has dimension two, the endpoint bases also have dimension two and their contractions to the common base are birational: they are connected-fibre contractions between normal varieties of the same dimension. The induced identifications are the ones coming from \(L\). Such a link therefore preserves the embedded surface field.

If \(k'\ne k\), let \(m\) be the first index for which \(S_{m+1}\) is not a surface with \(K_{m+1}=k\). Its common base \(Q\) has dimension at most one, by the preceding paragraph. At each preceding node the data \((k\subset L,\alpha)\) are unchanged. Hence its proper generic curve, residue covers, and candidate are the original \(C_k\), the original covers, and \(\sigma_k(L,\alpha)\). The relevant part of the path is \[\begin{tikzcd}[column sep=large,row sep=large] Z_m \arrow[rr,dashed,"\text{first departure}"] \arrow[d] && Z_{m+1}\arrow[d] \\ S_m\arrow[dr] && S_{m+1}\arrow[dl] \\ & Q \end{tikzcd} \qquad \begin{gathered} \mathbb C(S_0)=\cdots=\mathbb C(S_m)=k\subset L,\\ \dim Q\leq1. \end{gathered}\] By Theorem 28, on the geometric generic fibre of \(Q\) its whole marked diagram is uniformly bounded, including the projection to the endpoint surface base. All ramification points on the proper generic curve are visible as divisors on this total model: their closures dominate the surface base and have codimension one. They are among the marked transforms by the full visible convention. If \(Q\) is a curve, its function field is contained in \(k\) because the endpoint base contracts to \(Q\). Lemma 44 applies and gives \(\sigma_k(L,\alpha)=0\), a contradiction. Thus \(k'=k\). ◻

Proof. The relative generic diagram over \(Q\) is bounded by Theorem 28. On the smooth del Pezzo locus, let \(a\in\{5,6\}\) be the fibre degree. The vanishings \(H^1(\mathcal O)=H^2(\mathcal O)=0\) make the relative Picard scheme étale (Kleiman 2005, Theorem 4.8, Corollary 5.13 and Proposition 5.19). We identify its finite part that parametrizes conic-pencil classes. For a line-bundle class \(F\) on a geometric fibre \(S\), Riemann–Roch gives \[\chi\bigl(\mathcal O_S(F-mK_S)\bigr) =1+\tfrac12\bigl(F^2-(2m+1)F\cdot K_S+a m(m+1)\bigr).\] Consequently the equations \(F^2=0\) and \(-K_S\cdot F=2\) are equivalent to having the fixed anticanonical Hilbert polynomial \(\tfrac a2m(m+1)+2m+2\). This fixed-polynomial locus in the relative Picard scheme is open and closed and of finite type; in the smooth projective family it is also projective (Kleiman 2005, Theorem 6.20, Exercise 5.7 and its answer). It is therefore proper and quasi-finite, hence finite étale. Lemma 41 identifies its geometric fibres with \(A(S)\), of cardinality five or three according as \(a=5\) or \(6\). These are statements about the Picard scheme and require no universal line bundle on the original family. The complement of this smooth family is bounded as an embedded locus of the diagram. Each field factor of the finite scheme is therefore a cover of bounded degree unramified outside this bounded locus. Apply the bounded-cover exclusion of Theorem 22, or the Hurwitz calculation (39). When \(Q\) is a curve its field is contained in each tested surface field. Thus no factor is counted by \(u\) and \(w=0\). ◻

Proof. Put \(L=\mathbb C(T)\) and suppose an eligible \(k\subset L\) has nonzero candidate. The bounded diagram on \(T\) does not yet include the presentation over \(k\); in particular, \(k\) need not contain \(\mathbb C(Q)\). We will construct an output with bounded marked geometric generic diagram over \(Q\) and compare it with the output over \(k\). Choose a common smooth SNC start \(Y\) mapping to \(T\) and to a projective surface model of \(k\), with the full visible boundary (41). One log MMP output has base field \(k\) by (42).

Proof. If there is a nonzero candidate, Proposition 45 says that its eligible field is the only eligible field. If there is no nonzero candidate, all eligible candidates are zero and agree with the definition. In particular a zero candidate cannot be ignored in favour of a nonzero candidate from another field. The last assertion follows from Proposition 47, Lemma 46, and the definition when \(\dim B\leq1\). ◻

Proof. Choose an ordinary Sarkisov factorization supplied by Theorem 28, and orient each link from \(V/B\) to \(V'/B'\). Write \(\Delta\) for final minus initial value of a function on its nodes. Proposition 35 gives \[ c_u(\text{link})-\Delta v =H_Q=c_{u,\mathrm{hor}/Q}-\Delta h_Q. \tag{46}\] We check \(H_Q=\Delta s\) in every dimension of the common base \(Q\).

If \(\dim Q\leq1\), Proposition 36 gives \(H_Q=0\), while Theorem 48 gives \(s=0\) at both ends.

If \(\dim Q=3\), both endpoint bases have dimension three and are birational to \(Q\). On the generic fibre the link is an isomorphism of smooth projective conics. Thus the endpoint base fields and quaternion classes are identified, including all eligible embedded surface fields and their candidates. Therefore \(\Delta s=0\), which equals \(H_Q\) by Proposition 36.

Suppose \(\dim Q=2\) and put \(k=\mathbb C(Q)\). In case (ii) of Lemma 38, the calculation of Proposition 42 gives \(c_{u,\mathrm{hor}/Q}=\Delta w\). Here \(w\) is evaluated on the structured generic surface over \(k\), including its conic structure when \(\dim B=3\), as in Definition 40. On a del Pezzo side, \(h_Q=0\) and \(s=w\). On a conic side, its generic curve base over \(k\) is geometrically integral of genus zero. Lemma 39 identifies its singular-fibre set and component cover with \(\Delta_k\) and the residue cover. The common del Pezzo surface \(W\) has positive degree, so (34) gives \(r_k=\deg_k\Delta_k=8-K_W^2\leq7\). Hence \(k\) is eligible. The same lemma gives \(h_Q=-u(\Delta_k)\), and by (38) and Theorem 48, \[s=w-h_Q.\] This remains valid if the candidate is zero. Consequently \(H_Q=\Delta(w-h_Q)=\Delta s\).

In case (i) of Lemma 38, the structured generic surfaces are isomorphic after cancelling any identical generic contractions. Their exceptional horizontal contribution and \(H_Q\) are zero by that lemma. At a del Pezzo node the Picard-class sets defining \(w\) identify. At a conic node the isomorphism identifies the embedded base field and its quaternion class, so the intrinsic corrections identify. Thus \(\Delta s=0\) here as well. This last argument requires no bound on the discriminant degree of an isomorphic conic structure.

It follows from (46) that every link satisfies \[c_u(\text{link})=\Delta(v+s).\] Add over the factorization and use the cocycle property of Lemma 8. At a node over a point there are no vertical divisors in the defining difference for \(v\), and \(s=0\) by definition. The sum therefore vanishes. All lower bounds on \(d\) arose from the finitely many fixed families specified after (40); none depends on the length or starting resolution of the factorization. ◻

Proof. We spell out the exceptional sets, including their algebraicity. Pass to Hassett’s marked space and then add finite level structure. Fix a smooth irreducible component that is finite and surjective over a dense open \(U\subset\mathcal C_d\), and denote it by \(U'\). The rank-two marking makes \(K\) constant. Its orthogonal \(\mathbb T=K^\perp\) is an integral polarized variation on \(U'\); we trivialize this complementary local system only on period charts. All exceptional loci below are constructed on this one fixed cover.

The cubic Hodge numbers and middle lattice are \[h^{3,1}=h^{1,3}=1,\quad h^{2,2}=21,\quad b_4=23, \qquad H^4(X,\mathbb Z)\simeq \langle1\rangle^{\oplus21} \oplus\langle-1\rangle^{\oplus2};\] see (Hassett 2000, sec. 2.1 and Proposition 2.1.2). Thus \(\mathbb T\) has rank \(21\) and signature \((19,2)\) for the cubic cup pairing. In a marked period chart its period line varies in an open subset of \[\mathcal D_T= \left\{[\omega]\in\mathbb P(T\otimes\mathbb C): (\omega,\omega)=0,\ (\omega,\overline\omega)<0\right\}.\] Here \(T\) denotes the underlying fixed lattice in that chart. The openness follows equivalently from the associated K3 period description (48). The domain \(\mathcal D_T\) is analytically open in a smooth irreducible projective quadric, and this quadric spans \(\mathbb P(T\otimes\mathbb C)\).

For \(0\ne t\in T\otimes\mathbb Q\), consider the locus in \(U'\) where some flat translate of \(t\) is a Hodge class. After clearing denominators and Tate twisting, (Cattani et al. 1995, Theorem 1.1 and Corollary 1.3) makes this locus closed algebraic, with finitely many local branches. Here one fixed denominator works for the whole monodromy orbit, and the flat polarization has the same value on all its translates. The cited corollary obtains this orbit locus from images of connected components of the bounded-norm space of Hodge classes, which the theorem makes finite over the base. In a period chart each branch lies in \((t')^\perp\cap\mathcal D_T\) for a nonzero flat translate \(t'\). Each such section is proper because the period domain spans \(T\otimes\mathbb C\); hence the global locus is proper as well. There are countably many rational \(t\). Since \(H^4(X,\mathbb Q)=K_\mathbb Q\oplus K_\mathbb Q^\perp\), excluding these loci removes every Hodge class outside \(K_\mathbb Q\). Thus \(A_X\otimes\mathbb Q=K\otimes\mathbb Q\); saturation gives \(A_X=K\) and consequently \(\operatorname{rank}T_X=21\).

The middle lattice is unimodular, and both \(A_X\) and \(T_X\) are primitive. To recall the determinant argument, write \(L=H^4(X,\mathbb Z)\). Every homomorphism \(A_X\to\mathbb Z\) extends to \(L\), since \(L/A_X\) is torsion-free. Unimodularity therefore makes the restriction map \(L\to A_X^*\) surjective with kernel \(T_X\). Reducing modulo \(A_X\) gives \[L/(A_X\oplus T_X)\simeq A_X^*/A_X.\] Interchanging \(A_X\) and \(T_X\) gives the same quotient as \(T_X^*/T_X\). The two discriminant groups have the same order; hence \(|\det T_X|=\det A_X=d\).

Next fix a nonscalar \(a\in\operatorname{End}_{\mathbb Q}(T\otimes\mathbb Q)\). If \(a\) is a Hodge endomorphism, then \(\mathbb C\omega\) is an eigenline for \(a\). Hence its Hodge locus is contained in \[\bigcup_{\lambda}\mathbb P\ker(a-\lambda)\cap\mathcal D_T.\] This is a finite union of proper linear sections of the period quadric. Indeed, an eigenspace equal to \(T\otimes\mathbb C\) would make \(a\) scalar; and an irreducible quadric spanning the ambient vector space cannot be contained in a finite union of proper linear subspaces. There are countably many rational endomorphisms \(a\). Apply the same argument to the weight-zero polarized variation \(\operatorname{End}(\mathbb T)\), clearing denominators. The locus where some flat translate of \(a\) is Hodge is closed algebraic with finitely many local branches. Monodromy acts by conjugation and preserves nonscalarity, so the eigenline argument makes every local branch proper. These global loci are therefore proper. Excluding them gives \(\operatorname{End}_{\mathrm{Hdg}}(T_X\otimes\mathbb Q)=\mathbb Q\).

For completeness, simplicity also follows directly from the Hodge numbers. By polarizability, a rational Hodge substructure has a Hodge complement. In any nontrivial direct-sum decomposition of \(T_X\otimes\mathbb Q\), only one summand could contain the one-dimensional \((3,1)\) part, and that summand would also contain its conjugate. Every other summand would consist entirely of rational \((2,2)\) classes, contrary to the definition of \(T_X\).

Finally take the finite images in \(U\) of the exceptional loci in \(U'\). These images are closed and proper, since \(U'\) is irreducible and finite over \(U\). Their closures in \(\mathcal C_d\) are still proper. Adding \(\mathcal C_d\setminus U\) gives a countable union of proper closed algebraic subsets, whose complement is nonempty over \(\mathbb C\). The categorical assertion holds throughout \(\mathcal C_d\) by (49); if one uses only the Addington–Thomas theorem, one additionally removes the complement of its dense open subset. This proves the simultaneous assertion. ◻

Proof of Theorem 1. Let \(d_0\) be as in Proposition 50, and fix any admissible \(d>d_0\). The divisor \(\mathcal C_d\) is nonempty. By Lemma 51, outside a countable union of proper closed algebraic subsets of this divisor, the lattice \(T_X\) has rank \(21\), absolute determinant \(d\), and rational Hodge endomorphism ring \(\mathbb Q\). The same lemma supplies an exact \(\mathbb C\)-linear equivalence \[\operatorname{Ku}(X)\simeq D^b(\operatorname{Coh}(S))\] for a smooth projective K3 surface \(S\). Proposition 50 applies to each such \(X\) and proves that it is irrational.

The integer \(d_0\) was fixed before choosing \(d\), so the conclusion holds very generally in every admissible \(\mathcal C_d\) with \(d>d_0\). Its ineffectiveness is inherited from the uniform bound in Theorem 49. Such a cubic satisfies the categorical condition of Kuznetsov’s conjecture and fails its rationality condition; hence the categorical-to-rational implication is false. ◻

Proof of Corollary 2. For admissible \(d\), the classical input (48) supplies the indicated polarized K3 surface for every labelled cubic in \(\mathcal C_d\). Theorem 1 supplies irrationality on the stated very-general set for \(d>d_0\). Since \(K\) contains \(h^2\), the isometry gives \[L^\perp(-1)\simeq K^\perp\hookrightarrow (h^2)^\perp=H^4(X,\mathbb Z)_{\mathrm{pr}}.\] The inclusion is primitive: if a nonzero integer multiple of an integral class in \((h^2)^\perp\) is orthogonal to \(K\), then the class itself is orthogonal to \(K\). This is the primitive polarized K3 Hodge embedding in the cited formulation, with the stated Tate twist and pairing reversal. ◻

Proof of Corollary 3. For \(n\ge2\), the integer \(d=2n^2+2n+2\) is greater than six and is congruent to zero or two modulo six. Taking \(a=1\) realizes Addington’s condition \((***)\), namely \(d=(2n^2+2n+2)/a^2\) for integers \(n,a\) with \(a\ne0\). That condition implies his condition \((**)\), the remaining divisibility restrictions in (1); hence \(d\) is admissible. Addington’s Theorem 2 then supplies the stated birationality for every \(X\in\mathcal C_d\), with a projective K3 surface by his convention (Addington 2016, condition \((***)\), Theorem 2, and the convention in the introduction). Theorem 1 supplies irrationality for a very general such \(X\) when \(d>d_0\). Finally, the values \(2n^2+2n+2\) are strictly increasing and unbounded for \(n\ge2\), so infinitely many exceed \(d_0\). ◻

Proof of Corollary 4. First let \(X\in\mathcal C_d\) be any cubic with a labelling \(K\) of admissible discriminant \(d\). The intersection form satisfies \((h^2,h^2)=\int_Xh^4=3\), so \(h^2\) is primitive in the integral middle lattice: if \(h^2=m\eta\) for an integral class \(\eta\), then \(m^2\) divides \(3\). It is therefore primitive in \(K\) and extends to a basis \((h^2,\sigma)\) of this rank-two lattice.

Voisin’s Theorem 5.6 is stated for the lattice \(P\) generated by the independent integral Hodge classes \(h^2\) and \(\sigma\), which are algebraic on a smooth cubic fourfold as recalled in its proof (Voisin 2017, Theorem 5.6 and its proof). Here \(P=K\), and its discriminant in that theorem is exactly \[D(\sigma)= \det\begin{pmatrix} 3 & h^2\mathbin{\cdot}\sigma\\ h^2\mathbin{\cdot}\sigma & \sigma^2 \end{pmatrix} =\det K=d.\] Admissibility includes \(4\nmid d\). Voisin’s theorem therefore gives universal triviality of \(\operatorname{CH}_0(X)\) for every such labelled cubic, without requiring \(K\) to be the entire lattice of Hodge classes or \(X\) to be very general. The field-extension scope and the equivalence with the displayed integral Chow-theoretic decomposition are given in the introduction and equation (1) of the same paper (Voisin 2017). Theorem 1 supplies irrationality for the stated very general cubics when \(d>d_0\). ◻