Proof. Take a divisorial test \(v\) centered at \(y\), of discrepancy \(A>0\) and volume \(\sigma\), and let \(a=A/N\). The Gauss extension to the added parameter \(t\) is \[(v\oplus a)\left(\sum_i f_it^i\right) =\min_i\{v(f_i)+ia\}.\] On a log resolution this is a monomial valuation along the divisor of \(v\) and \(t=0\), so its log discrepancy is \(A+a\). Completion does not change the finite colengths. In the power-series ring the quotient decomposes by powers of \(t\), giving \[\mathop{\mathrm{length}}\bigl(R[[t]]/\mathfrak a_k(v\oplus a)\bigr) =\sum_{0\le i<k/a}\mathop{\mathrm{length}}\bigl(R/\mathfrak a_{k-ia}(v)\bigr).\] The defining asymptotic for \(\sigma\), followed by a Riemann sum, gives \(\mathop{\mathrm{vol}}(v\oplus a)=\sigma/a\). Therefore \[\frac{(A+A/N)^{N+1}}{(N+1)^{N+1}}\frac{N\sigma}{A} =\frac{A^N\sigma}{N^N}.\] Take the infimum over the divisorial tests on \(Y\). ◻

Proof. Choose minimal homogeneous algebra generators \(\bar f_1,\ldots,\bar f_e\) of \(\mathop{\mathrm{gr}}_v R\), all of positive value, and lift them to \(f_1,\ldots,f_e\in\mathfrak m\). To show that their classes span \(\mathfrak m/\mathfrak m^2\), start with \(g\in\mathfrak m\). Its initial form is a homogeneous polynomial in the \(\bar f_i\). Subtracting the corresponding polynomial in the \(f_i\) either annihilates the remainder or strictly increases its value. Every positive value that occurs belongs to the semigroup generated by the positive numbers \(v(f_i)\), so only finitely many such values lie below any fixed bound.

The Izumi inequality for a finite-discrepancy valuation on a klt germ (Li 2018, Theorem 3.1) gives \(v(g)\le c\,\mathop{\mathrm{ord}}_{\mathfrak m}(g)\), with a fixed constant \(c\). In particular, a nonzero remainder of value greater than \(c\) lies in \(\mathfrak m^2\). The cancellation process therefore terminates modulo \(\mathfrak m^2\) after finitely many steps. Every polynomial subtracted has no constant term, and modulo \(\mathfrak m^2\) is a linear combination of the \(f_i\). This proves the assertion. ◻

Proof. First take the exact quadratic cone \[Q_n=\mathop{\mathrm{Spec}}R,\qquad R=\mathbb C[z_0,\ldots,z_n]/(z_0^2+\cdots+z_n^2),\] with its ordinary grading. It is klt. For example, blowing up the vertex resolves it with smooth exceptional quadric, whose log discrepancy is \(n-1>0\); further smooth blowups preserve the klt condition.

Normalize a minimizing valuation \(v\) by its discrepancy. Uniqueness makes it invariant under all automorphisms, in particular under cone scalings and the connected orthogonal group \(\operatorname{SO}(n+1,\mathbb C)\). For each \(j\ge0\), the degree-\(j\) piece \(R_j\) is the irreducible representation of complex harmonic polynomials of degree \(j\) in \(n+1\) variables. To identify the quotient, write \(P_j\) for homogeneous complex polynomials of degree \(j\) and \(\mathcal H_j\) for the subspace killed by \(\sum_i\partial_{z_i}^2\); set \(P_k=0\) for \(k<0\). For \(q=\sum z_i^2\), the harmonic decomposition \(P_j=\mathcal H_j\oplus qP_{j-2}\) identifies \(R_j\) with \(\mathcal H_j\); see (Graham 2015, Example 251 and Theorem 257). The complex representation is irreducible already under the compact group \(\operatorname{SO}(n+1)\), so an invariant subspace under \(\operatorname{SO}(n+1,\mathbb C)\) must be trivial or the whole space. For the classical harmonic-polynomial formulation and its extensions, see also (Luo and Xu 2012, sec. 1). This includes \(n=2\) and \(n=3\). The exception for two ambient variables is outside the range under consideration.

For each cutoff \(t\), the subspace \(\{f\in R_j:v(f)\ge t\}\) is invariant, so it is either zero or all of \(R_j\). Every nonzero element of \(R_j\) consequently has a common value \(a_j\). If \(0\ne f\in R_1\), then \(f^j\ne0\), and valuation multiplicativity gives \(a_j=j a_1\). Invariance under cone scalings separates homogeneous pieces of a cutoff: a finite Vandermonde combination of scalings isolates each piece. Thus, for \(g=\sum_j g_j\ne0\), \[v(g)=a_1\min\{j:g_j\ne0\}.\] After localizing at the vertex the same assertion holds, since denominators nonzero at the vertex have value zero. The minimizer is therefore proportional to the ordinary degree valuation.

That valuation is the order along the exceptional quadric of the vertex blowup. Hypersurface adjunction gives discrepancy \(n-1\), and its volume is the hypersurface multiplicity \(2\). It follows that the infimum equals \(2(n-1)^n\).

Finally, an analytic ordinary double point has the same completed local ring as this exact cone. Completion invariance (Liu 2022, Lemma 2.15) gives the same normalized volume. ◻

Proof. Apply stable degeneration to a minimizer at \(x\), and then take the K-polystable special degeneration. In the first passage the induced Reeb valuation has the same volume and discrepancy as the original minimizer (Li and Xu 2018, Lemma 4.10). In the second, equivariant flatness preserves the dimensions of the character spaces and hence the Reeb volume. The relative canonical character is constant, so its discrepancy is also preserved. Li–Xu’s characterization of K-semistability (Li and Xu 2018, Theorem 1.3) shows that the final Reeb valuation minimizes normalized volume over all centered valuations. Thus the two passages preserve the infimum in (1), not only the number of an arbitrary test.

Lemma 4 gives the embedding-dimension inequality for the first passage. Upper semicontinuity along the vertex section of the special flat degeneration gives it for the second. The final vertex is therefore singular.

Choose any positive integral grading of \(S\), and let \(P\ne o\). Its grading orbit specializes to \(o\), so lower semicontinuity gives \[ d(o,C)\le d(P,C). \tag{5}\] Choose a homogeneous function \(f\), of degree \(d>0\), with \(c=f(P)\ne0\), and put \(Y_P=\{f=c\}\subset C_f\). The action map \[Y_P\times\mathbb G_m\longrightarrow C_f,\qquad (Q,a)\longmapsto a\cdot Q,\] is the finite étale cover obtained by adjoining a root \(a^d=f/c\). Near \((P,1)\), it expresses the germ as the product of the \(N\)-dimensional slice \((Y_P,P)\) and a smooth curve. The slice is klt by this étale description. If \(P\) were singular, the slice would be singular as well. The induction hypothesis, Lemma 3, and étale invariance would then imply \[d(P,C)\le p_N<p_n\le d(o,C),\] contrary to (5). Thus the puncture is smooth.

Let \(q\) be the canonical index at \(o\). The cyclic index cover has degree \(q\), is quasi-étale and crepant, and is klt. It has one point over \(o\). Indeed, its algebra is the direct sum of the reflexive canonical powers modulo \(q\). If a homogeneous element of a nonzero residue degree were not nilpotent in the fiber over \(o\), a power returning to degree zero would be a unit. The corresponding section would trivialize a proper positive canonical power, contradicting minimality of \(q\). Thus the nonzero residue-degree pieces are nilpotent, and the reduced fiber consists of one point. Equation (2) and the upper bound \(d\le1\) give \(d(o,C)\le1/q\). Since \(p_n>1/2\), necessarily \(q=1\).

Klt singularities are Cohen–Macaulay (Kollár 2010, Definition 117 and Theorem 120). The canonical module \(\omega_S\) is invertible at the vertex, and a homogeneous element generating its one-dimensional fiber there generates it after localization. Its graded cokernel has empty support: any nonempty closed support invariant under a positive grading contains the vertex. Since \(\omega_S\) is torsion-free, the resulting map \(S\to\omega_S\) is injective as well. It is therefore an isomorphism, with a grading shift. ◻

Proof. In the quotient of real cocharacter space by its integral lattice, recurrence gives \(t_i\to\infty\) and integral cocharacters \(q_i\) with \(q_i=t_i\xi+e_i\), where \(e_i\to0\). One may obtain such a sequence from simultaneous Dirichlet approximation, or from returns to zero of the one-parameter subgroup in its compact closure. Let \(d_i\) be the gcd of \(q_i\) in a lattice basis and put \(\zeta_i=q_i/d_i\). Then \[\zeta_i-\frac{A(\zeta_i)}n\xi =\frac1{d_i}\left(e_i-\frac{A(e_i)}n\xi\right)\longrightarrow0.\] The cocharacters are positive for large \(i\). If their positive integers \(r_i=A(\zeta_i)\) had a bounded subsequence, (6) would make \(\zeta_i\) bounded on that subsequence. A constant lattice subsequence would then give \(\zeta=(r/n)\xi\), forcing the ray to be rational. The rational case is handled by its primitive generator.

Choose homogeneous algebra generators \(g_1,\ldots,g_a\) with \(\mathbb T\)-characters \(\chi_1,\ldots,\chi_a\). Each \(\langle\chi_j,\xi\rangle\) is positive, so (6) gives the asserted two-sided bounds for their integral degrees. More precisely, the normalized cocharacters \((n/r_i)\zeta_i\) converge to \(\xi\). Uniform comparison on the finite set of generator characters then compares their values on every occurring character: each is a sum of generator characters with nonnegative integer coefficients. It follows that the valuation ideals of the normalized degree valuations are squeezed between those of \(\operatorname{wt}_\xi\) with cutoff factors tending to one. Their volumes therefore converge; their discrepancies are all \(n\).

At a nonvertex point some generator is nonzero. Its stabilizer for a primitive grading is cyclic, and its order divides the degree of every nonvanishing generator. It is therefore at most the largest generator degree, proving the final assertion. ◻

Proof. The smooth puncture has finite stabilizers for the positive grading, so its quotient stack is smooth and Deligne–Mumford. It is the usual stack associated with a positively graded affine ring away from its vertex; its coarse space is \(\mathop{\mathrm{Proj}}S\), and a sufficiently divisible positive power of the weight-one line descends to an ample coarse line bundle. This also proves properness.

The grading is effective because the integral cocharacter is primitive in an effective torus. At a point, the finite stabilizer acts trivially on the tangent line of the orbit. If an element acted trivially also on a transverse tangent space, it would have identity differential on the smooth puncture. A finite-order automorphism in characteristic zero is formally linearizable at a fixed point; hence that element would be the identity on a neighborhood and thus on the irreducible cone, contrary to effectiveness. This proves faithfulness of the transverse representation and effectiveness of the orbifold. The weight-one character of the subgroup of \(\mathbb G_m\) is faithful by construction.

Sections of \(jL\) are the homogeneous functions of degree \(j\) on the puncture. Normality and the codimension of the vertex extend them to \(C\), giving the first equality in (7). Contracting \(\Omega_C\) with the Euler field gives a horizontal top form of character \(r\), so \(K_B=-rL\). ◻

Proof. Homogeneous maps form an affine scheme of finite type: in fixed homogeneous generators of \(S\), each coordinate of the map is a polynomial of a specified weighted degree, and the relations of \(S\) give polynomial equations in their finitely many coefficients. The condition of avoiding the vertex outside the origin is open, as may be checked on the proper weighted line. Near the given map, deformation theory can therefore be performed entirely in the smooth target \(C^\circ\).

Let \(E\) be the rank-\(n\) bundle on \(\mathbb P(1,m)\) obtained by descending \(F^*T_{C^\circ}\). Its determinant is \(\mathcal O(r)\), so \(\deg E=r/m\). We are deforming an equivariant map with its source and its grading torsor fixed; equivalently, we are deforming a section of the associated smooth bundle, with the canonical section obstruction theory of (Webb 2022, Theorem 4.13 and Equation (4.18)). There is no quotient by source automorphisms or by changes of the torsor identification. If \(D_\infty\) denotes the residual gerbe at infinity, the tangent and obstruction spaces for fixing its value are \[H^0(\mathbb P(1,m),E(-D_\infty)),\qquad H^1(\mathbb P(1,m),E(-D_\infty)).\] Thus their Euler characteristic is a lower bound for the dimension of every component through the map.

For completeness, orbifold Riemann–Roch here says \(\chi(E)=n+\deg E-\operatorname{age}(E_\infty)\). It follows by taking the coarse pushforward: locally diagonalizing at the stack point subtracts the fractional character weights from the degree, after which ordinary Riemann–Roch on \(\mathbb P^1\) applies. Restriction to \(D_\infty\) has Euler characteristic equal to the dimension of the invariant fiber, because \(\mu_m\) is linearly reductive. Subtracting that invariant dimension is precisely the effect of representing each trivial character by \(m\) instead of zero. The exact sequence for restriction to \(D_\infty\) therefore gives Equation (12).

Since \(\sum b_j'/m\leq n\), we have \(\chi\geq r/m\). The Euler characteristic \(\chi\) is an integer. If \(r/m>N\), it is at least \(N+1=n\). If \(r/m=N\) and some character is nontrivial, the inequality is strict and the same conclusion follows. ◻

Proof. Take \(F_0\in H\) with \(F_0(1,0)=o\). Choose a pointed curve in \(H\) through \(F_0\), with general point in the locus avoiding the vertex, and normalize it. Its generic map gives a twisted stable graph map to \(B\times\mathbb P^1\): the second map is the coarse identification of \(\mathbb P(1,m)\) with \(\mathbb P^1\), and infinity is marked. This graph is stable even before any contraction because its second map has degree one. After a finite base change, properness of twisted stable maps gives a limiting graph map (Abramovich et al. 2011, Theorem 4.3). The target is a tame proper smooth Deligne–Mumford stack with projective coarse space, as required by that theorem.

There is a unique component of the limiting coarse source of second degree one. It maps isomorphically to \(\mathbb P^1\); call it the horizontal component. All other components form rational trees attached to it, since the total arithmetic genus is zero. The first map on the horizontal component is the coarse rational map defined by \(F_0\), after cancellation of its basepoints. Indeed the two maps agree on the open set where the limiting homogeneous coordinates do not vanish simultaneously.

We spell out why a positive-degree tail occurs over the finite basepoint \([1:0]\). Choose a sufficiently divisible integer \(k\) such that \(kL\) descends to a very ample bundle on the coarse space of \(B\), and \(m\mid k\). Its pullback along the original maps is the coarse bundle \(\mathcal O_{\mathbb P^1}(k/m)\). The limiting projective coordinates are binary forms of that degree. They have a common zero at \([1:0]\), because \(F_0(1,0)=o\), and are not all zero, because \(F_0(0,1)=P\ne o\). Cancelling their common divisor loses positive degree at \([1:0]\). A general hyperplane avoiding the value of the canceled map at that point has the lost positive number of inverse-image points specializing there. In the limiting stable map these points cannot lie on the horizontal component. They are therefore accounted for by positive first degree on a tree over \([1:0]\). This also proves the degree assertion directly, without making an assumption about components that might occur over infinity.

Choose a terminal leaf of that tree. It does not contain the infinity marking, since its second image is \([1:0]\). It has one node and no markings. If its first map had degree zero, it would be constant in both factors and unstable: stability requires the induced coarse map to be a Kontsevich stable map (Abramovich et al. 2011, sec. 4), and an unmarked constant terminal leaf has only one special point. Hence its first degree is positive. Its normalization is a twisted rational curve with at most one stack point: the only allowed stack points of a twisted source are nodes and markings. Projection from the product target to \(B\) preserves representability, since \(\mathbb P^1\) has trivial inertia. The leaf consequently gives a representable nonconstant map \(\mathbb P(1,\ell')\to B\) of the type used in the minimum.

Its \(L\)-degree is at least the original minimal degree \(1/m\). Degree is additive on the limiting source and every component has nonnegative \(L\)-degree, while the total degree is \(1/m\). Thus this leaf consumes the full first degree. The horizontal component has first degree zero, and its constant value is the coarse image of \(P\). It follows that the image of the affine map \(F_0\) lies in the closure of the grading orbit of \(P\).

The normalization of that orbit closure is \(\mathbb A^1\), with a parameter of grading weight \(m\), normalized to have value one at \(P\). To see the weight, the gcd of the degrees of the coordinates nonzero at \(P\) is its stabilizer order \(m\). The map from the normal source \(\mathbb A^2\) factors through this normalization: its parameter is integral over the coordinate ring of the orbit curve and hence regular on \(\mathbb A^2\). The parameter pulled back along \(F_0\) is a weighted homogeneous polynomial of degree \(m\), so it has the form \[y+c x^m.\] The coefficient of \(y\) is one because the value at \((0,1)\) is fixed. Evaluation at \((1,0)\) is the vertex only if \(c=0\). Therefore the vertex fiber is supported at the single contracted map.

Finally, let \(R_H\) be the positively graded coordinate ring of \(H\). The preceding fiber statement says that \(R_H/S_+R_H\) is finite dimensional. Lift a homogeneous basis of this quotient. Induction on the positive degree shows that these lifts generate \(R_H\) as an \(S\)-module; this is graded Nakayama. Thus evaluation is finite. Its image is closed and has dimension \(\dim H\geq n\), by Equation (13). Since \(C\) is integral of dimension \(n\), the image is all of \(C\), and \(\dim H=n\). ◻

Proof. The source automorphisms \((x,y)\mapsto(x,y+t x^m)\), for \(t\in\mathbb C\), preserve the homogeneous-map scheme, the fixed value at \((0,1)\), and its open locus avoiding the vertex. The connected additive group therefore preserves the component \(H\). Its vector field is nonzero: a coordinate nonzero at \(P\) has a nonzero pure power of \(y\), whose derivative under this action is nonzero in characteristic zero. Differentiating evaluation gives a regular vector \(V\) along evaluation over \(C^\circ\). It is generically nonzero, since the finite dominant evaluation is generically separable and hence has injective differential on a nonempty smooth open set. For a target coordinate of degree \(d\), its component is \(\partial_yF'(1,0)\), of coefficient-grading degree \(d-m\). Thus \(V\), as a vector along evaluation, has derivation weight \(-m\).

Over the generic point, take the symmetric product of the vectors on all \(b\) sheets. This gives a nonzero element of \(\mathop{\mathrm{Sym}}^bT_{C^\circ}\) over \(\mathbb C(C)\); the product, unlike the ordinary trace, cannot cancel. It is invariant under permutation of the sheets and so descends to that field. It is regular as well. On a smooth affine open set of \(C^\circ\) trivializing the tangent bundle, the components of \(V\) lie in a finite algebra over the base ring. The coefficients of their symmetric product are therefore integral over the base ring and lie in its fraction field. Normality of the base makes them regular. This argument does not require \(H\) to be normal; alternatively one can first use its finite normalization. The construction is equivariant and multiplies weights by \(b\), giving weight \(-bm\). ◻

Proof. The Chern connection induced on \(\mathop{\mathrm{Sym}}^bT_{C^\circ}\) has zero mean curvature: contraction of the curvature on \(T_{C^\circ}\) is Ricci curvature, and contraction commutes with the induced tensor and symmetric power operations. Put \(u=|s|^2\). The holomorphic Bochner formula and Cauchy–Schwarz give, with a consistent positive complex-Laplacian normalization, \[\Delta u=|\nabla^{1,0}s|^2\geq0, \qquad |\partial u|^2\leq u\,\Delta u.\] Consequently \(\log(u+a)\) is subharmonic for each \(a>0\), and so is \((u+a)^{p/2}\) for every \(p>0\), by the chain rule applied to the increasing convex exponential function. Letting \(a\) decrease to zero shows that \(|s|^p\) is subharmonic in the distributional sense, including across the zero set of \(s\).

Let \(\rho\) be the metric radial coordinate and \(M=\{\rho=1\}\) its compact smooth link. A contravariant tensor of rank \(b\) and derivation weight \(\tau\) has norm of radial degree \(\tau+b\). If \(\tau+b<0\), choose \(p>0\) such that \[d=p(\tau+b)\in(2-2n,0).\] Then \(|s|^p=\rho^d\phi\) for a nonnegative continuous function \(\phi\) on \(M\) with positive integral. The cone-Laplacian formula, valid also distributionally, is \[\Delta(\rho^d\phi) =\rho^{d-2}\bigl(d(d+2n-2)\phi+\Delta_M\phi\bigr).\] Integrating the angular term over \(M\) gives zero. Testing subharmonicity against a nonnegative radial function supported in an annulus would therefore imply \(d(d+2n-2)\int_M\phi\geq0\), which is impossible. Thus \(\tau+b\geq0\). ◻

Proof. For the fixed eigen-generators \(f_j\), positivity of \(\xi\) and absolute convergence of the error give constants \(c_1,c_2>0\) such that their integral degrees \(d_j=\alpha_j(\zeta)\) satisfy \[ c_1r\leq d_j\leq c_2r \quad\text{for every }j. \tag{16}\] These bounds include the exact rational grading.

The differentials \(df_j\) span the cotangent bundle of \(C^\circ\). Pairing a symmetric tensor with all ordered \(b\)-tuples of these differentials embeds its space of sections into a finite tuple of regular functions on \(C^\circ\). Normality extends these functions over \(o\). This also shows that the torus action on sections is locally finite, so the tensor has a nonzero torus eigencomponent; every such component retains its specified \(\zeta\)-weight.

Let \(\chi\) be the character of a nonzero component, and choose a nonzero pairing with \(df_{j_1}\otimes\cdots\otimes df_{j_b}\). Its function has character \(\beta=\chi+\alpha_{j_1}+\cdots+\alpha_{j_b}\) and integral degree \[0\leq d=-bm+d_{j_1}+\cdots+d_{j_b}\leq c_2rb.\] It has a polynomial lift in the generators consisting of monomials of character \(\beta\), by torus equivariance of the polynomial presentation of \(S\). Each such monomial contains at most \(d/(c_1r)\leq(c_2/c_1)b\) factors, by Equation (16). As the generator characters are fixed, this proves \(\|\beta\|=O(b)\), and then \(\|\chi\|=O(b)\), uniformly in the tensor, its rank and the grading.

Put \(\tau=\chi(\xi)\). The eigencomponent satisfies Lemma 14, so \(\tau\geq-b\). On the other hand, \[-bm=\chi(\zeta)=\frac rn\tau+\chi(\epsilon) \geq-\frac rn b-cb\|\epsilon\|.\] Dividing by \(b\) proves Equation (15). ◻

Proof of Theorem 10. For each grading choose the minimal curve and its stabilizer order \(m\) as above. If the exceptional case in Lemma 11 occurs, then \(r/m=N\) and the \(\mu_m\)-action on \(T_{C,P}\) is trivial. Faithfulness of the transverse tangent action implies \(m=1\), hence \(r=N\). Equation (9) then forces \(m_{\max}=1\).

For the remaining indices, Lemmas 12 and 13 apply. The character estimate and Equation (11) give \[0\leq nm-r\leq nc\|\epsilon\|.\] The left side is an integer. Since the error tends to zero, we have \(nm=r\) for all sufficiently large indices. For a rational ray with its constant exact grading the error is zero, so this equality holds directly.

Now the finite evaluation has \(\dim H=n\), whereas its deformation dimension satisfies \[n=\dim H\geq n+\frac rm-\sum_{j=1}^n\frac{b_j'}m =2n-\sum_{j=1}^n\frac{b_j'}m\geq n.\] All equalities must hold. In particular every \(b_j'=m\), so the tangent action is trivial. Its faithfulness again gives \(m=1\), and now \(r=n=N+1\). Finally \[m_{\max}\leq\frac{N+1}{N}<2\] forces \(m_{\max}=1\).

The quotient is therefore an ordinary smooth projective Fano \(N\)-fold and \(L\) an actual ample line bundle. The Kobayashi–Ochiai theorem in its ample-line-bundle formulation (Kobayashi and Ochiai 1973), also supplied by (Araujo et al. 2008, Theorems 1.1 and 1.3), applied to \(-K_B=rL\) with \(r=N+1\) or \(N\), gives precisely the two polarized varieties stated in the theorem. The polarization is the standard one, as also follows by substituting into their Picard groups. By the section-ring identity from the cone setup, \[S=\bigoplus_{j\geq0}H^0(B,jL).\] This is the polynomial ring for \((\mathbb P^N,\mathcal O(1))\), and the homogeneous coordinate ring of a nondegenerate quadratic hypersurface for \((Q^N,\mathcal O(1))\). Its affine cone is respectively smooth or the \(n\)-dimensional ordinary double point. ◻

Proof. Leading monomials multiply, and the associated leading coefficient is a scalar. This proves additivity and the one-dimensional-leaf assertion. In particular, each leading index in a fixed degree accounts for one dimension of \(S_j\).

Choose an integer \(M>0\) divisible by all stabilizer orders such that \(ML\) descends to an ample line bundle on the coarse space. In a fixed residue class \(j_0\) modulo \(M\), prescribe a finite compatible Taylor jet of a section of \(j_0L\). The kernel of the map to that finite jet space is coherent. Serre vanishing after tensoring by sufficiently large powers of \(ML\), or after exact tame pushdown to the coarse space, makes the map on global sections surjective. Thus the prescribed jet is realized in every sufficiently large degree \(j_0+kM\). Taking all the finitely many admissible residue classes proves the assertion about progressions. In particular, constants occur in all sufficiently large degrees divisible by \(m\), and each monomial \(z^\gamma\) occurs as a leading term in sufficiently large compatible degrees.

Differences of two such constant indices whose degrees differ by \(m\) give \((m,0)\) in the generated group. For each \(i\), a realized linear jet gives an index \((j_i,e_i)\) with \(j_i\equiv b_i\pmod m\). The vectors \((m,0),(j_1,e_1),\ldots,(j_N,e_N)\) generate the full congruence lattice in Equation (19). All indices satisfy that congruence, proving equality of groups. These same indices show that the cone contains the positive constant ray, has full dimension, and projects onto the exponent orthant.

We next prove the uniform estimate. Fix torus eigen-generators of \(S\) and the resulting affine embedding of \(C\). Their \(\zeta_\nu\)-degrees are bounded below by a fixed positive multiple of \(r_\nu\) for all sufficiently large \(\nu\), because their \(\xi\)-weights are positive. A section of degree \(j\) has a homogeneous polynomial lift in these generators, of ordinary polynomial degree at most \(Cj/r\). At a smooth point of the punctured cone, a homogeneous function has the same local vanishing order as its restriction to a transverse slice: in orbit and slice coordinates it is a unit power of the orbit coordinate times its slice expression.

Take a general complete-intersection curve through the smooth point, cut out by affine hyperplanes in the fixed embedding. It is smooth there and is not contained in the zero locus of the nonzero function. The local intersection multiplicity with the function is at least its local vanishing order. Bézout’s inequality bounds this multiplicity by the polynomial degree times the degree of the fixed embedded cone. This proves Equation (22) for \(\Gamma\), and then for \(K\) by homogeneity and closure. In particular, bounded degree cuts are compact.

The positive constant ray makes every nonempty fiber over an exponent \(x\) an upper ray in degree. Every exponent fiber is nonempty, since nonnegative combinations of the indices \((j_i,e_i)\) reach every \(x\ge0\). Closedness and the degree bound make the lowest degree in each fiber attainable. Hence \(K\) is the epigraph of a finite convex positively homogeneous function in the scaled coordinates. Equation (22) becomes Equation (24). A full-dimensional closed convex cone is the closure of its interior.

The original lattice has covolume \(m\). The coordinate change \((j,\gamma)\mapsto(Nj/r,\gamma)\) multiplies covolumes by \(N/r\); the resulting covolume is \(Nm/r=1/s\). For clarity, we give the nonsaturated semigroup counting argument. This is the conductor-and-exhaustion method for value semigroups (Kaveh and Khovanskii 2012, Theorems 1.4 and 1.6), also used for graded linear series in (Lazarsfeld and Mustaţă 2009, Proposition 2.1). Its asymptotic conclusion does not assert exact saturation. Let \(\Gamma_0\) be a finitely generated subsemigroup with full generated group \(\Lambda\), and let \(K_0\) be its cone. There is a vector \(c\in\Gamma_0\) such that \[ c+(K_0\cap\Lambda)\subset\Gamma_0. \tag{27}\] Indeed every lattice point of \(K_0\) lies in a cone generated by a linearly independent subset of the finite generating set. Removing the integer parts of its nonnegative coefficients writes it as a semigroup element plus a lattice point in one of finitely many bounded parallelepipeds. There are only finitely many possible remainders \(v\). Since the group is \(\Lambda\), write each such remainder as \(v=v_+-v_-\) with \(v_\pm\in\Gamma_0\). The sum of all the \(v_-\) is a single \(c\) for which Equation (27) holds. Translating a bounded linear cutoff changes its scale by a constant, so ordinary lattice counting between \(c+(K_0\cap\Lambda)\) and \(K_0\cap\Lambda\) gives its leading lattice volume. After the coordinate change this is Euclidean volume with density \(s\).

Now exhaust \(\mathop{\mathrm{int}}K\) by cones of finitely many indices of \(\Gamma\), always retaining a fixed finite set whose group is \(\Lambda\). Every interior point belongs to the interior of one of these finite cones: choose a small simplex surrounding it in \(\mathop{\mathrm{int}}K\) and approximate the simplex vertices by elements of \(\operatorname{cone}(\Gamma)\). The resulting finite cones give the lower leading count. Counting all lattice points of \(K\) gives the upper count. Compact convex cutoff bodies have boundary of Lebesgue measure zero, and their volumes are exhausted from the interior. This proves Equation (25).

Finally, on the uniformizer the canonical character is the negative of the sum of the coordinate characters, in the convention for equivariant line bundles used here. The identity \(-K_B=rL\) therefore gives \(r\equiv\sum_i b_i\pmod m\), proving Equation (26). ◻

Proof. Use the local line-bundle extraction over the chosen uniformizer. Write \(t\) for its line coordinate, of grading weight one. Its finite quotient presentation has the action \[(t,z_1,\ldots,z_N)\longmapsto (\varepsilon^{-1}t,\varepsilon^{b_1}z_1,\ldots, \varepsilon^{b_N}z_N).\] A homogeneous function of degree \(j\) pulls back to \(t^jF_j(z)\). The monomial valuation with costs \[v(t)=N/r=1/(sm),\qquad v(z_i)=\delta\] therefore restricts to Equation (28). Distinct original degrees have different powers of \(t\), so even when scalar values agree their lowest monomials cannot cancel. Rational positive costs give a divisorial monomial valuation up to rescaling. At \(\delta=0\) the rule is the rescaled ordinary degree valuation. The positive degree cost and the positivity of all algebra-generator degrees imply that its center is the vertex.

Let \(\Omega_C\) be the homogeneous canonical generator of weight \(r\). On this chart it has the form \[t^{r-1}u(z)\,dt\wedge dz_1\wedge\cdots\wedge dz_N, \qquad u(0)\ne0.\] The exponent follows from homogeneity, and the coefficient is a unit because \(\Omega_C\) is nonvanishing on the punctured cone. Computing discrepancy from this pulled-back form, including the finite-chart ramification in the restricted value scaling, gives \[A_C(v_\delta)=\frac Nr+N\delta+(r-1)\frac Nr =N+N\delta.\] Equivalently one can take a weighted monomial extraction on the smooth chart and apply the canonical divisor formula there; the same calculation descends to the quotient. For \(\delta=0\) it is the canonical degree formula.

In each \(S_j\), a leading-term basis is compatible with total Taylor order. Thus the colength below a cutoff \(q\) is the number of indices of \(\Gamma\) satisfying \(D_0+\delta|x|<q\). These finite quotients are supported at the vertex, so computing them in \(S\) or in its local ring gives the same length. Proposition 16 applies to this strictly positive cutoff functional. Since \(K\) has dimension \(n\), homogeneity and integration by layers give \[n!\operatorname{Leb}\{(D_0,x)\in\widetilde K:D_0+\delta|x|<1\} =\int_{\widetilde K}e^{-D_0-\delta|x|}\,dD_0\,dx =\int_{\mathbb R_{\ge0}^N}e^{-f(x)-\delta|x|}\,dx.\] Combining this identity with the lattice density and Equation (29) proves Equation (30). ◻

Proof. Write \(F(\delta)\) for the right side of Equation (30). The volume hypothesis and the test at zero give \[2N^n\le\widehat{\operatorname{vol}}(o,C)\le F(0)=N^n sH,\] which is the first inequality.

Consider the probability measure \(d\mu_f=H^{-1}e^{-f(x)}dx\). Positive homogeneity gives, for every \(t>0\), \[\int e^{-tf(x)}dx=t^{-N}H.\] Differentiating, justified by Equation (24), yields \[ \mathbb E_{\mu_f}f=N,\qquad \mathbb E_{\mu_f}f^2=N(N+1). \tag{33}\] In particular, \[\begin{align*} \frac{F'(0)}{F(0)} &=n-\mathbb E_{\mu_f}|x|\\ &=1+\mathbb E_{\mu_f}(f-|x|) =\frac{\mathbb E[(1+B_0)e^{-B_0}]}{H}. \tag{34}\end{align*}\]

We check the uniformity needed for near-minimality. Equation (24) and Equation (33) give uniform first and second moments of \(|x|\) under \(\mu_f\). Indeed they are at most \(c_C^{-1}N\) and \(c_C^{-2}N(N+1)\), respectively. Since \[\frac{F(\delta)}{F(0)} =(1+\delta)^n\mathbb E_{\mu_f}e^{-\delta|x|},\] twice differentiating shows \[ \sup_{0\le\delta\le1}\frac{|F''(\delta)|}{F(0)}\le C'_C \tag{35}\] for a fixed finite constant. The three terms are bounded by constant multiples of \(1\), \(\mathbb E_{\mu_f}|x|\), and \(\mathbb E_{\mu_f}|x|^2\), because \(e^{-\delta|x|}\le1\).

The test at zero is a rescaling of the current degree valuation, so Lemma 8 gives \(F_\nu(0)\to\widehat{\operatorname{vol}}(o,C)\). If \(F'_\nu(0)/F_\nu(0)\) were bounded above by a fixed negative number along a subsequence, Taylor’s formula and Equation (35) would give a fixed small positive rational \(\delta\) and a fixed \(\rho>0\) such that \(F_\nu(\delta)\le(1-\rho)F_\nu(0)\). This is incompatible with \(F_\nu(\delta)\ge\widehat{\operatorname{vol}}(o,C)>0\) and the preceding convergence. Thus \(F'_\nu(0)/F_\nu(0)\ge-\eta_\nu\) for some \(\eta_\nu\to0\). Equation (34) proves the second inequality. At an exactly minimizing degree valuation, every positive rational test has volume at least \(F(0)\), so its right derivative is nonnegative. ◻

Proof. The preservation of log-concavity under convolution is a special case of the classical theory of Prékopa (Prékopa 1973, Theorems 7–8); we give the elementary exponential-specific argument needed here. A nonzero coefficient vector with sum zero has a positive and a negative coefficient. The difference \(aE-bE'\) for \(a,b>0\) has density \[p(x)=\frac1{a+b} \begin{cases}e^{x/b},&x\le0,\\e^{-x/a},&x\ge0. \end{cases}\] It is positive, continuous, and log-concave.

Let \(r\) be a positive log-concave function for which the indicated one-sided integrals are finite. Set \(R(x)=\int_{-\infty}^x r(y)dy\). For \(t\ge0\), concavity of \(\log r\) implies that \(r(x-t)/r(x)\) is nondecreasing in \(x\). Thus \[\frac{R(x)}{r(x)}=\int_0^\infty\frac{r(x-t)}{r(x)}dt\] is nondecreasing, and \((\log R)'=r/R\) is nonincreasing. Consequently \(R\) is log-concave. The same argument with \(r(x+t)/r(x)\) shows that \(T(x)=\int_x^\infty r(y)dy\) is log-concave: \(T/r\) is nonincreasing, so \((\log T)'=-r/T\) is nonincreasing. These statements can equivalently be read almost everywhere; the positive functions involved are locally absolutely continuous.

If \(p\) is a positive log-concave probability density, adding \(cE\) or \(-cE\), \(c>0\), produces respectively the densities \[\frac{e^{-x/c}}c\int_{-\infty}^x e^{y/c}p(y)dy, \qquad \frac{e^{x/c}}c\int_x^\infty e^{-y/c}p(y)dy.\] The integrands are exponential tilts of \(p\), hence log-concave, and the appropriate one-sided integrals are finite. The preceding argument proves log-concavity of the new densities. Positivity and continuity are immediate from the displayed formulas. Induction starting with \(aE-bE'\) proves the assertions about finite exponential sums, and translation does not affect them.

For completeness, Equation (40) is the one-dimensional log-concave form of Grünbaum’s centroid inequality (Grünbaum 1960); see (Lovász and Vempala 2007, Lemma 5.4) and (Meyer et al. 2018, sec. 3). It follows directly here from the same integral argument. If \(G\) is the distribution function of \(Z\), both \(G\) and \(1-G\) are log-concave. Since \(G(Z)\) is uniform on \((0,1)\), Jensen’s inequality gives \[\log G(\mathbb EZ)\ge\mathbb E\log G(Z) =\int_0^1\log t\,dt=-1.\] Applying the same reasoning to \(1-G\) gives \(\log(1-G(\mathbb EZ))\ge-1\). All the means in the present application are finite, and both logarithmic expectations just used equal \(-1\). These two inequalities give Equation (40). ◻

Proof. Write \(p\) for the density of \(Z\) and first consider \[\psi(t)=\mathbb E k(Z+t) =\int_0^\infty(1-y)e^{-y}p(y-t)dy.\] This is continuous, by continuity of translation in \(L^1\) and boundedness of \(k\). A positive log-concave density has a continuous representative whose logarithm is finite and concave. For \(t_1<t_2\), the ratio \[r(y)=p(y-t_2)/p(y-t_1)\] is nondecreasing in \(y\). If \(\psi(t_1)=0\), subtraction of the constant \(r(1)\) gives \[\psi(t_2) =\int_0^\infty k(y)p(y-t_1)(r(y)-r(1))dy\le0:\] the integrand is nonpositive on both \((0,1)\) and \((1,\infty)\). Equality would force \(r\) to be constant on \((0,\infty)\), by positivity and continuity. Thus \(\log p\) would have constant increments across the fixed interval length \(t_2-t_1\) on a half-line. Concavity forces it to be affine there: its almost-everywhere derivative is nonincreasing, and equality of this derivative across a fixed shift makes it constant on that half-line. Integrability then gives \(p(y-t_1)=A e^{-\lambda y}\) for \(y>0\), with \(A,\lambda>0\). But \[\psi(t_1)=A\int_0^\infty(1-y)e^{-(1+\lambda)y}dy =\frac{A\lambda}{(1+\lambda)^2}>0,\] a contradiction. Hence the inequality is strict. Reversing the likelihood-ratio comparison proves strict positivity to the left of a zero. Continuity shows that \(\psi\) either has one strict sign everywhere or has exactly one zero, with positive sign to its left and negative sign to its right.

Now put \(F(c)=\mathbb E\psi(cE)\). For \(c\ne0\), the density of \(cE\) is \[q_c(t)=|c|^{-1}e^{-t/c}\mathbf1_{\{tc>0\}}.\] If \(c_1<c_2\) have the same sign, then on their common support \[\frac{q_{c_2}(t)}{q_{c_1}(t)} =\frac{|c_1|}{|c_2|} \exp\bigl(t(1/c_1-1/c_2)\bigr)\] is strictly increasing in \(t\). This holds on the negative half-line as well as on the positive half-line. If \(F(c_*)=0\) and \(c_*\ne0\), then \(\psi\) must have its strict crossing inside that half-line. Subtracting the likelihood ratio at this crossing proves strict positivity of \(F(c)\) for \(c<c_*\) and strict negativity for \(c>c_*\) within the same parameter half-line.

The comparisons across zero are also strict. If \(c_*>0\), the crossing of \(\psi\) lies in \((0,\infty)\), so \(\psi>0\) on \((-\infty,0]\) and \(F(c)>0\) for \(c\le0\). If \(c_*<0\), the crossing lies in \((-\infty,0)\), giving \(F(c)<0\) for \(c\ge0\). Finally \(F(0)=\psi(0)\); a zero there gives the two required signs directly from those of \(\psi\). This proves the assertion for all real parameters. ◻

Proof. Equivalently consider a local minimum of \(\mathbb Eh(Y)\), where \(Y=u+\sum_i a_iX_i\). At a nonzero coefficient vector, Lemma 20 supplies a continuous positive log-concave density for \(Y\). First and second differentiation under the expectation are justified by boundedness of \(h'\), its global Lipschitz property, the finite second moments of the variables \(X_i\), and the absence of an atom at \(Y=0\). Stationarity in the direction \(e_i-e_j\) gives \[ 0=\mathbb E[(X_i-X_j)h'(Y)] =(a_i-a_j)\mathbb E[X_iX_jk(Y)]. \tag{42}\] To see the second equality, condition on all other variables and on \(S=X_i+X_j\). Given \(S=s\), \(X_i\) is uniform on \([0,s]\) and \(X_j=s-X_i\). Integration by parts with \(x(s-x)\), whose endpoint values vanish and whose derivative is \(s-2x\), gives exactly Equation (42).

For distinct indices, weighting by \(X_iX_j\) changes those two independent exponential laws into independent Gamma\((2,1)\) laws. Since \(\mathbb E X_iX_j=1\), no normalization factor appears. Thus \[ \mathbb E[X_iX_jk(Y)] =\mathbb E k(Y+a_iE'+a_jE''), \tag{43}\] where the added exponentials are independent of all variables in \(Y\). If three distinct coefficient values occurred, fix an index \(i\) at one value and choose indices at the other two. Equations (42) and (43) would give two distinct zeros of \[c\longmapsto\mathbb E k(Y+a_iE'+cE'').\] Its base variable has a positive full-support log-concave density by Lemma 20, contradicting Lemma 21. Therefore there are at most two coefficient values.

Suppose the two values are \(b<c\) and the higher value occurs at indices \(i\ne j\). A high-low pair gives a zero at \(b\) of \(d\mapsto\mathbb E k(Y+cE'+dE'')\). Strict single crossing makes its value at \(c\) negative. By Equation (43), this says \(\mathbb E[X_iX_jk(Y)]<0\). As \(a_i=a_j\), \(Y\) depends on this pair only through \(S\), and \[\mathbb E[(X_i-X_j)^2\mid S]=S^2/3, \qquad \mathbb E[X_iX_j\mid S]=S^2/6.\] Consequently the Hessian in the permissible splitting direction is \[\mathbb E[(X_i-X_j)^2k(Y)] =2\mathbb E[X_iX_jk(Y)]<0,\] contradicting a minimum. The higher value therefore occurs once. The sum-zero constraint forces it to be positive and gives Equation (41). ◻

Proof of Theorem 19. Put \(\Phi_N(a)=(e^u/u)\mathbb E g(u+a\cdot X)\) and let \(\mathcal P_N\) be its supremum under the sum-zero constraint. First control escape to infinity with \(N\) fixed. From an unbounded sequence \(a_k\), pass to a subsequence with \(a_k/\|a_k\|\to b\ne0\). The vector \(b\) has sum zero, so \(b\cdot X\) has mean zero and a continuous log-concave density. Almost surely away from its zero set, \[g(u+a_k\cdot X)\longrightarrow\mathbf1_{\{b\cdot X<0\}}.\] Bounded convergence and Equation (40) imply that every such subsequential limit is at most \[ B_*:=\frac{e^u}{u}(1-1/e). \tag{44}\] Thus, if \(\mathcal P_N>B_*\), a maximizing sequence is bounded and a finite maximizer exists by continuity. The zero vector has value \(1+1/u=7/4\). Every other finite maximizer has the form in Lemma 22. If \(\mathcal P_N\le B_*\), the boundary estimate already suffices; we make no assertion of attainment in that case.

Appending a zero coefficient shows that \(\mathcal P_N\) is nondecreasing in \(N\), and it is bounded by \(e^u/u\). Consider its limit. If that limit is at most \(\max(B_*,7/4)\), those two values suffice. Otherwise sufficiently large dimensions have finite nonzero maximizers. After permutation write them as in Equation (41), with exceptional coefficient \(t_N>0\). On a common probability space their arguments are \[Y_N=u+t_N\left(X_1-\frac1{N-1}\sum_{i=2}^NX_i\right).\] Pass to a subsequence on which \(t_N\) converges in \([0,\infty]\). If \(t_N\to t<\infty\), the law of large numbers and bounded convergence give \[\Phi_N(a_N)\longrightarrow \Phi_\infty(t):=\frac{e^u}{u}\mathbb E g(u+t(X_1-1)).\] If \(t_N\to\infty\), then \(Y_N/t_N\to X_1-1\) almost surely. The limit has no atom at zero; on its positive and negative sets, respectively, \(Y_N\) tends to \(+\infty\) and \(-\infty\). Hence bounded convergence gives \[\Phi_N(a_N)\longrightarrow \frac{e^u}{u}\mathbb P(X_1<1)=B_*.\] This argument uses no comparison between the growth of \(t_N\) and \(\sqrt N\).

It remains to maximize the scalar expression. For \(0\le t\le u\), the argument \(u+t(X_1-1)\) is nonnegative, and direct integration gives \[ \Phi_\infty(t)= \begin{cases} \displaystyle\frac{e^t}{u} \left(\frac{1+u-t}{1+t}+\frac{t}{(1+t)^2}\right), &0\le t\le u,\\[7pt] \displaystyle\frac{e^u}{u} \left(1-e^{-1+u/t}\frac{t^2}{(1+t)^2}\right), &t\ge u. \end{cases} \tag{45}\] For the second branch, split the exponential integral at \(X_1=1-u/t\); above that point translate the integration variable so that the argument of \(g\) is \(t\) times the new variable. The integral there is \(e^{-1+u/t}((1+t)^{-1}+t(1+t)^{-2})\), which yields the displayed formula.

The derivative of the first branch has the sign of \(t(-t^2+ut+u-1)\). For \(u=4/3\), the second factor is positive throughout \([0,u]\), as its endpoint values are positive and it is concave. On the second branch, the logarithmic derivative of the positive quantity being subtracted is \[-\frac u{t^2}+\frac2t-\frac2{1+t} =\frac{(2-u)t-u}{t^2(1+t)}.\] Thus this branch is maximal at \(t=u/(2-u)=2\), and its value is \[\Phi_\infty(2)=\frac34e^{4/3}-\frac e3=C_*.\] In particular \(C_*>\Phi_\infty(0)=7/4\). Moreover \[C_*-B_*=\frac34e^{1/3}-\frac e3>0,\] because \(e^{1/3}>1+1/3\) and \(e<3\). The preceding compactness and limit argument proves the upper bound \(\sup_N\mathcal P_N\le C_*\). Taking \(t_N=2\) and applying the law of large numbers proves the reverse inequality and the asserted approach to the supremum.

Finally the strict gap has an elementary rational certificate. The exponential series gives \(e>1359/500\) and \(e^{4/3}<1897/500\). For the latter bound one may retain terms through degree eight and bound the remaining tail by its first term divided by \(1-(4/3)/10\). Therefore \[C_*<\frac34\frac{1897}{500}-\frac13\frac{1359}{500} =\frac{3879}{2000}<2.\] ◻

Proof. Suppose that \(f(\mathbf1)\ge N\). At the interior point \(\mathbf1\) of the exponent orthant, choose a subgradient \(p\) of \(f\). Positive homogeneity gives \(p\cdot\mathbf1=f(\mathbf1)\) and \(f(x)\ge p\cdot x\) on the orthant. Therefore \(B_0(x)\ge(p-\mathbf1)\cdot x\), and the coefficient sum of \(p-\mathbf1\) is nonnegative. Subtracting the same nonnegative number from each coefficient gives a vector \(a\) with sum zero and \[ B_0(x)\ge a\cdot x\qquad(x\in\mathbb R_{\ge0}^N). \tag{47}\]

Take \(u=4/3\). Applying Equation (37) pointwise to Equation (47), and using Lemma 18, gives \[\begin{align*} (1-\eta_\nu/u)H &\le\mathbb E\left[\left(1+\frac{1+B_0(X)}u\right) e^{-B_0(X)}\right]\\ &\le\frac{e^u}{u}\mathbb E g(u+a\cdot X) \le C_*<2. \end{align*}\] All expectations are finite by Equation (24). But \(H\ge2/s\ge2\), so the displayed inequality is impossible once \(\eta_\nu\) is sufficiently small. Hence \(f(\mathbf1)<N\) for all sufficiently large gradings. At an exact minimizing rational grading, \(\eta_\nu=0\) gives the contradiction immediately.

The scaled coordinates of \(c_0\) are \((N,\mathbf1)\), and \(\mathbf1\) is interior to the exponent orthant. The strict epigraph inequality is therefore precisely Equation (46). Its membership in the lattice was already proved in Proposition 16. ◻

Proof. Let \(q:\mathcal X\to X\) be the coarse morphism. For a prime divisor \(B\subset X\), let \(e_B\) be the stabilizer order at the generic point of its inverse image. A rational divisor \(A\) on \(\mathcal X\) has a coarse rational divisor \(A_c\), characterized in codimension one by \(q^*A_c=A\). Locally the coarse map is given in the normal direction by \(t=z^{e_B}\), so \[K_{\mathcal X}=q^*(K_X+R), \qquad R=\sum_B(1-1/e_B)B.\] Only divisors with \(e_B>1\) contribute to \(R\).

The coefficient of \(D_c\) at \(B\) is an integral multiple of \(1/e_B\). Consequently \[ \Delta=R+\Theta_c-\{D_c\} \tag{48}\] is effective: the fractional coefficient being subtracted is at most \((e_B-1)/e_B\). The pair \((X,\Delta)\) is klt. Indeed, on a smooth finite quotient uniformizer \(U\to X\), the log pullback of \(\Delta\) is the pullback of \(\Theta\) minus the effective pullback of \(\{D_c\}\). This boundary is bounded above by the simple normal crossing boundary \(\Theta|_U\) with coefficients less than one. All its log discrepancies are therefore positive. The finite-map discrepancy formula, with ramification included in the displayed canonical identity, gives the same positivity downstairs.

Normal extension from codimension one identifies \[q_*\mathcal O_{\mathcal X}(D)=\mathcal O_X(\lfloor D_c\rfloor).\] For completeness, on a finite quotient chart the pushforward is the invariant part of an invertible module on a normal smooth scheme. Sections extend across sets of codimension at least two there and then remain invariant, so this pushforward is the rank-one reflexive sheaf prescribed by its codimension-one orders. Those orders are exactly \(\lfloor D_c\rfloor\). Taking invariants is exact in characteristic zero, which also gives \[H^q(\mathcal X,\mathcal O_{\mathcal X}(D)) =H^q(X,\mathcal O_X(\lfloor D_c\rfloor)).\]

The coarse space has quotient singularities and is locally \(\mathbb Q\)-factorial. Thus \(\lfloor D_c\rfloor\) is an integral \(\mathbb Q\)-Cartier Weil divisor. By Equation (48), \[\lfloor D_c\rfloor-(K_X+\Delta) =D_c-(K_X+R+\Theta_c),\] which is the nef and big class descended from \(D-(K_{\mathcal X}+\Theta)\). Kawamata–Viehweg vanishing for a klt pair and an integral \(\mathbb Q\)-Cartier Weil divisor now applies (Fujino 2023). It gives the claimed vanishing. This argument requires neither flatness of the coarse morphism nor ampleness in place of nefness and bigness. ◻

Proof. The coordinate inequalities of \(K\) imply \(\alpha_i+1>0\). Since \(\alpha_i\) is integral, \(\alpha_i\ge0\). The bounded-slice property and the positivity of the degree coordinate on \(K\setminus\{0\}\) give \(j+r>0\). Choose a rational number \[0<j_0<j+r \quad\hbox{such that}\quad p=(j_0,\alpha+\mathbf1)\in\mathop{\mathrm{int}}K.\] We construct a section of \(jL\) with the required Taylor initial.

Proof. Fix such jet data, with \(c_0\in\mathop{\mathrm{int}}K\cap\Lambda\). Put \(\varphi(x)=(r/N)f(x)\), so the epigraph of \(\varphi\) uses the original degree coordinate \(j\), rather than \(D_0=Nj/r\). On the open positive orthant, \(\varphi\) is a finite convex function and is locally Lipschitz.

Proof. We first transfer the structure of the monomial algebra to \(\overline S\). By Equation (63), \(\Gamma\) is a finitely generated normal affine semigroup. Hochster’s theorem makes \(\mathbb C[\Gamma]\) Cohen–Macaulay, and the canonical-module formula for normal affine semigroup rings identifies its canonical module with the monomial ideal of interior lattice points; see (Hochster 1972) and (Stanley 1978, Theorem 6.7 and its following normal-monoid reformulation). The second identity in Equation (63) makes this ideal principal, generated in multidegree \((r,1,\ldots,1)\). Thus its bidegree is \((r,N)\). The interior-translation identity, rather than saturation alone, is essential here.

Choose a finite set of generators of \(\Gamma\) and bihomogeneous lifts in \(\overline S\) of their monomials. We lift generation by degreewise subduction, in the terminology of Khovanskii bases (Kaveh and Manon 2019, Algorithm 2.11 and Proposition 2.13). Subtracting products of the lifts cancels any selected initial monomial. Within a fixed original degree there are only finitely many Taylor orders, and within each order only finitely many monomials. Consequently this subtraction terminates and the chosen lifts generate \(\overline S\).

The same finite list gives a one-parameter flat monomial degeneration. Choose positive integral costs on the \(z_i\) that preserve the selected monomial initial of every one of these finitely many lifts. Such costs exist by approximating the specified lexicographic refinement on this finite list. The scalar initial algebra contains \(\mathbb C[\Gamma]\). In each original bidegree its dimension is the dimension of \(\overline S\) in that bidegree, which is also the number of the corresponding points of \(\Gamma\). The inclusion is therefore an equality in every bidegree. Lifting the generators again, with the same degreewise terminating subtraction, shows that the extended Rees algebra for this scalar filtration is finite type. As a subalgebra of a Laurent-polynomial extension it is torsion-free, hence flat, over its parameter line. Its zero fiber is \(\mathbb C[\Gamma]\) and its nonzero fibers are \(\overline S\).

Write \(R\) for this extended Rees algebra and \(\lambda\) for its parameter. It retains original degree and \(T\)-degree, and has the additional scalar filtration grading, with scalar weight \(-1\) on \(\lambda\). A sufficiently large multiple of original degree minus scalar weight gives a positive grading on \(R\), including positive degree for \(\lambda\). At its vertex, the local ring \(A\) has a normal Gorenstein quotient \(A/(\lambda)\). Lifting through the nonzerodivisor \(\lambda\) makes \(A\) Cohen–Macaulay and Gorenstein; for the latter equivalence see (The Stacks Project Authors 2026, Tag 0BJJ).

The local ring \(A\) is also normal. Here is a direct verification using \(R_1+S_2\). It is already Cohen–Macaulay. Let \(P\) be a height-one prime of \(A\). If \(\lambda\in P\), reducedness of \(A/(\lambda)\) makes \(A_P/(\lambda)\) a field, so \(A_P\) is regular. Otherwise choose \(Q\) minimal over \((P,\lambda)\). Since \(A\) is a local domain essentially of finite type over \(\mathbb C\), the dimension formula and the principal ideal theorem give \(\operatorname{ht}Q\leq2\). The normal ring \(A_Q/(\lambda)\) has dimension at most one and is regular. Lifting a regular system of parameters together with \(\lambda\) shows that \(A_Q\) is regular, and then so is its localization \(A_P\).

The normal and Gorenstein loci of \(\mathop{\mathrm{Spec}}R\) are open, contain its vertex, and are invariant under the positive grading action. Every orbit contracts to that vertex; hence \(R\) is normal and Gorenstein everywhere. Its generic fiber, a localization of \(R\), is \(\overline S\otimes_{\mathbb C}\mathbb C(\lambda)\) and is normal. Normality descends under this faithfully flat field extension, proving normality of \(\overline S\). Quotienting \(R\) by \(\lambda-1\) proves the Cohen–Macaulay and Gorenstein assertions for \(\overline S\) as well.

Adjunction tracks the canonical bidegree. The parameter \(\lambda\) has original degree and \(T\)-degree both zero, so it contributes no shift in these two degrees. Choose a lift of the central canonical generator homogeneous also for the scalar grading, with its adjunction shift. Its cokernel in the canonical module of \(R\) is equal to its product by \(\lambda\). Graded Nakayama for the positive combined grading therefore makes the cokernel zero. Restriction to \(\lambda=1\) forgets the scalar grading but preserves original degree and \(T\)-degree. Thus the canonical generator of \(\overline S\) has bidegree \((r,N)\).

We next use the constant Taylor direction to describe \(\overline S\) over a polynomial parameter. The vector \((m,0)\) lies on the constant ray of \(K\) and belongs to \(\Lambda\). Equation (63) provides a section with this leading index. Rescale its value to obtain \(\overline v=\tau^m\). Choose integer representatives \(b_i\) of the stabilizer characters. Eventual compatible jet separation shows that, after inverting \(\tau^m\), every monomial \(\tau^jz^\gamma\) with \[j-\sum_i b_i\gamma_i\in m\mathbb Z\] occurs. Set \(u_i=\tau^{b_i}z_i\) in that localization. The displayed congruence expresses each allowed monomial uniquely as a Laurent power of \(\overline v\) times a monomial in the \(u_i\). This proves Equation (65), including its \(T\)-grading. It also shows directly that there is no residual finite quotient in a nonzero \(\overline v\) fiber.

For each \(q\geq0\), consider the \(T\)-degree-\(q\) piece of \(\overline S\) as a module over \(\mathbb C[\overline v]\). It is torsion-free. It is also finite: in each original-degree residue class modulo \(m\), multiplication by \(\overline v\) identifies its pieces with an increasing sequence of subspaces of the finite-dimensional space of compatible degree-\(q\) Taylor forms. These subspaces are eventually the full compatible space by jet separation. Thus there are only finitely many necessary module generators. This module is consequently free over the principal ideal domain \(\mathbb C[\overline v]\). Equation (65) shows that its rank is \[\binom{q+N-1}{N-1}.\] Reduction modulo \(\overline v\) consequently has exactly this dimension in \(T\)-degree \(q\), proving the Hilbert series in Equation (66).

Since \(\overline v\) is a nonzerodivisor in the normal Cohen–Macaulay domain \(\overline S\), its quotient is equidimensional of dimension \(N\) and Cohen–Macaulay. Adjunction makes the quotient Gorenstein and subtracts \((m,0)\) from the canonical degree. The Hilbert series shows that the \(T\)-degree-zero piece of \(D\) is \(\mathbb C\).

It remains to obtain a basis of \(S\) compatible with the filtration. Lift a bihomogeneous basis of \(D\) to \(\overline S\). These lifts are a \(\mathbb C[\overline v]\)-basis: generation follows by subtracting modulo \(\overline v\) and descending in original degree, and independence follows by reducing a putative relation modulo \(\overline v\) and successively removing its smallest power. Now lift them through the \(T\)-filtration to original-homogeneous sections \(e_i\) of the indicated orders \(T_i\). Their products with powers of \(v\) have the just-constructed basis as initials. Terminating cancellation in original degree proves Equation (67). Choose \(e_i=1\) for the sole basis vector of \(T\)-degree zero; all other \(T_i\) are positive. ◻

Proof. Lift homogeneous generators of \(\mathop{\mathrm{gr}}_F S\) to \(s_i\in S\), of filtration orders \(p_i\). Cancellation of initial forms, terminating within each original degree, shows that \(R_F\) is generated by \(\lambda\) and \(\lambda^{-p_i}s_i\). Indeed, an element of \(F_pS\) has an expression as a sum of monomials in the \(s_i\) whose filtration orders are at least \(p\); multiplication by \(\lambda^{-p}\) then introduces only nonnegative extra powers of \(\lambda\). Taking a sufficiently large multiple of original degree minus filtration degree makes every one of these generators positive, with \(\deg\lambda=1\).

Cartier adjunction (The Stacks Project Authors 2026, Tag 0B4A) for the nonzerodivisor \(\lambda\) identifies the reduction of the canonical module of \(R_F\) with that of \(\mathop{\mathrm{gr}}_FS\), with the shift coming from \(d\lambda\). Since \(\lambda\) has filtration weight \(-1\), a lift of the specified central generator has bidegree \((r,Q-1)\). This lift generates the whole canonical module: its cokernel is equal to its product by \(\lambda\), and a finite module bounded below in the positive combined grading cannot have that property unless it is zero. As the canonical module is torsion-free of rank one over the domain \(R_F\), the resulting generator is free.

On \(\lambda\ne0\), both this generator and \(\Omega_C\wedge d\lambda\) generate the same canonical module. Their ratio is a unit of \(S[\lambda,\lambda^{-1}]\). The units of the positively graded domain \(S\) are the nonzero constants, so such a unit is \(c\lambda^a\). The filtration weight of \(\Omega_C\wedge d\lambda\) is \(-1\), and the weight of the lifted generator is \(Q-1\). It follows that \(a=-Q\), proving Equation (69). Computing on a common smooth birational model gives \[ A_{C\times\mathbb A^1}(V) =A_{\mathfrak X}(V)+Q\ell. \tag{71}\] This computation uses rational top forms and does not require the Rees morphism itself to be proper.

On the other hand, \[ A_{C\times\mathbb A^1}(V)\geq A_C(w)+\ell. \tag{72}\] To see this directly, take any smooth proper birational log model of \(C\) used to retract \(w\), and take its product with the parameter line. Include \(\lambda=0\) among the simple-normal-crossing divisors. The monomial retraction of \(V\) on this product has discrepancy equal to the discrepancy of the corresponding retraction of \(w\) plus \(\ell\). Log discrepancy dominates its value on each such retraction (Jonsson and Mustaţă 2012, Corollary 5.4 and Remark 5.6); taking the supremum over the models of \(C\) proves Equation (72). Combining it with Equation (71) proves the lemma. ◻

Proof. The \(T\)-Rees algebra is finite type by Lemma 27 and the generator-lifting argument in Lemma 28. Its nonzero fibers are \(S\), and its central fiber \(\overline S\) is normal, Cohen–Macaulay, and Gorenstein. The regular parameter and these fiber properties imply that the total space is Cohen–Macaulay and Gorenstein. It is normal as well: away from the central fiber it is \(C\times\mathbb G_m\), and at every codimension-one point of the central fiber the quotient by \(\lambda\) is a field. Thus it is regular in codimension one and satisfies Serre’s condition \(S_2\).

Let \(H\) be an irreducible component of the divisor \(\overline v=0\) in \(\mathop{\mathrm{Spec}}\overline S\), and write \(\nu=\mathop{\mathrm{ord}}_H(\overline v)\geq1\). At its generic point the normal central fiber is a discrete valuation ring. The total space is therefore regular there, of local dimension two, with parameters \(\lambda\) and a lift \(t\) of a central uniformizer. In this local ring \[v=a t^{\nu}+\lambda b\] for a unit \(a\) and a regular element \(b\). Take the monomial valuation with costs \(V(\lambda)=1\) and \(V(t)=1/\nu\). Its discrepancy on the total space is \(1+1/\nu\). For its restriction \(w\) to \(\mathbb C(C)\) one has \[w(v)\geq1,\qquad w(e_i)\geq T_i,\] because \(\lambda^{-T_i}e_i\) is regular on the Rees space. Hence \(w(v^ae_i)\geq a+T_i\).

By Lemma 27, every nonconstant original-homogeneous basis vector has a strictly positive lower bound \(a+T_i\). Thus \(w\) is centered exactly at \(o\). The series counting the basis with these costs is \[\sum_{a,i}z^{a+T_i}=(1-z)^{-1}\mathop{\mathrm{Hilb}}_TD(z)=(1-z)^{-n}.\] The vectors of cost below \(k\) consequently number \(k^n/n!+O(k^{n-1})\). They span the valuation quotient, so \[ \mathop{\mathrm{vol}}(w)\leq1. \tag{73}\] These finite-colength quotients of \(S\) are supported at \(o\); their lengths agree with those after localization at the vertex.

Apply Lemma 28 with \(Q=N\) and \(\ell=1\). It gives \[A_C(w)\leq N+1/\nu.\] If \(\nu\geq2\), Equation (73) yields \[\widehat{\operatorname{vol}}(o,C)\leq (N+1/2)^{N+1}<2N^{N+1}.\] For the strict inequality, use \[(N+1)\log\left(1+\frac1{2N}\right) <\frac{N+1}{2N}\leq\frac58<\log2, \qquad N\geq4.\] This contradicts Equation (64). All the multiplicities of \(\overline v\) are therefore one. Its quotient \(D\) is generically reduced, and it is Cohen–Macaulay, hence satisfies \(S_1\). The reducedness criterion \(R_0+S_1\) proves the assertion; see (The Stacks Project Authors 2026, Tag 033P). ◻

Proof. Expand a product \(e_ie_j\) in the basis \(v^ae_k\). Compatibility of that basis with total Taylor initials shows that every nonzero term has \(T_k\geq T_i+T_j\). Its \(W\)-cost \(a+T_k\) is therefore at least \(T_i+T_j\). Terms with equality have precisely \(a=0\) and \(T_k=T_i+T_j\); their images are the product of the images of \(e_i,e_j\) in \(D=\overline S/(\overline v)\). Factoring out the displayed powers of \(v\) proves multiplicativity and the algebra isomorphism in Equation (76). In particular the variable \(\mathtt v\) is regular even if \(D\) has several components.

The Hilbert series follows from Equation (66). Adjoining the polynomial variable of bidegree \((m,1)\) changes the canonical degree \((r-m,N)\) to \((r,n)\). Thus \(G\) is reduced, Cohen–Macaulay, and Gorenstein. The \(W\)-filtration has finite length in fixed original degree, as is clear from its basis description; lifting finitely many generators of \(G\) proves finite generation of its Rees algebra.

The nonzero fibers of this Rees family are \(S\), and its central fiber is \(G\). Consequently the total space is Cohen–Macaulay and Gorenstein. For normality, away from \(\lambda=0\) it is \(C\times\mathbb G_m\). A codimension-one point on \(\lambda=0\) is the generic point of a central component. Since \(G\) is reduced, its local ring there is a field, and the total local ring has maximal ideal generated by the nonzerodivisor \(\lambda\). It is regular. The total space satisfies \(R_1\) and \(S_2\), hence is normal. Notice that normality of \(D\) or of \(G\) is not required in this argument. Finally Lemma 28, with \(Q=n\), gives the canonical generator and top-form identity. ◻

Proof. We first justify that weight one defines an honest line bundle on \(Z=\mathop{\mathrm{Proj}}E\). Choose positive homogeneous algebra generators \(a_j\), with degrees \(w_j\). At a nonvertex closed point \(q\), its grading stabilizer is \[\mu_{\gcd\{w_j:a_j(q)\ne0\}}.\] By hypothesis this gcd is one. Invert the product \(f\) of those active generators. Bézout’s identity supplies integers \(c_j\) with \(\sum_j c_jw_j=1\), so \(u=\prod_j a_j^{c_j}\) is an invertible homogeneous element of degree one in \(E_f\). Consequently \[ E_f=(E_f)_0[u,u^{-1}]. \tag{78}\] These open sets cover the puncture: they contain all its closed points, and a finite-type scheme over \(\mathbb C\) is Jacobson. Their degree-zero charts cover \(Z\). Equation (78) gives a Zariski \(\mathbb G_m\)-torsor over \(Z\), and makes \(L_Z=\mathcal O_Z(1)\) invertible, with \(\mathcal O_Z(k)=L_Z^{\otimes k}\). A positive power of this line bundle is ample by the construction of \(\mathop{\mathrm{Proj}}\), so \(L_Z\) is ample.

The same charts show that \(Z\) is reduced, equidimensional, and Cohen–Macaulay, of dimension \(N-1\). The Hilbert leading coefficient gives \(L_Z^{N-1}=1\). Each irreducible projective component has a positive integral degree for this actual ample line bundle. There is therefore only one component, and reducedness makes \(Z\) integral. The image \(V_0\) of \(E_1\) in \(H^0(Z,L_Z)\) has dimension \(N\): restriction is injective, since a nonzero homogeneous element in a reduced positive-dimensional equidimensional ring cannot vanish on its whole puncture.

We construct successive Cartier sections while tracking the images of \(V_0\), not assuming surjectivity onto full spaces of sections. Suppose \(Z_i\) is a positive-dimensional integral projective Cohen–Macaulay scheme obtained after \(i\) cuts, and that the image \(V_i\) of \(V_0\) on it has dimension \(N-i\). Choose a nonzero section \(s_i\in V_i\). Its zero scheme \(Z_{i+1}\) is an effective Cartier divisor. It is nonempty, because \(L_Z|_{Z_i}\) is ample of positive degree, and it is pure and Cohen–Macaulay. Its degree is again one. The component-degree formula forces a single component of generic multiplicity one; the \(S_1\) property then makes this divisor reduced, hence integral. The kernel of restriction \(V_i\to H^0(Z_{i+1},L_Z|_{Z_{i+1}})\) is exactly \(\mathbb Cs_i\): division by \(s_i\) identifies that kernel with regular global functions on the proper integral scheme \(Z_i\), and those functions are constant. Thus \(\dim V_{i+1}=N-i-1\).

After \(N-1\) cuts, the resulting scheme is a single reduced point and the remaining image space has dimension one. Choose a final section nonzero at that point. Lifting all these sections to \(E_1\) gives \(N\) elements whose common zero on \(\mathop{\mathrm{Proj}}E\) is empty. Their ideal is therefore irrelevant-primary and they form a homogeneous system of parameters. Cohen–Macaulayness makes the sequence regular. The quotient has Hilbert series \[(1-z)^N\mathop{\mathrm{Hilb}}E(z)=1.\] It is \(\mathbb C\), so induction on degree, or graded Nakayama, shows that these \(N\) linear elements generate \(E\). The resulting surjection from a polynomial ring on \(N\) degree-one variables is an isomorphism by the same Hilbert series. ◻

Proof. Otherwise Lemma 31 would make \(D\) a polynomial ring on \(N\) degree-one generators, and Proposition 30 would make \(G\) a polynomial ring on \(n=N+1\) generators. Choose these generators homogeneous also for original degree and lift them to \(S\). Terminating cancellation in the \(W\)-filtration shows that the lifts generate \(S\). A surjection from a polynomial ring on \(n\) variables to the \(n\)-dimensional domain \(S\) has zero kernel. Thus \(C\) would be smooth, contrary to our standing hypothesis. Stabilizers at nonvertex points of a positive grading are finite, so the asserted order \(h\geq2\) exists. ◻

Proof. Choose a positive-\(W\)-degree homogeneous function \(f\) on \(\mathfrak X\) with \(f(y)\neq0\). Such a function exists because \(y_D\) is not the vertex of the positively graded ring \(D\); lift a nonvanishing homogeneous function from the central fiber. Write \(d=W(f)>0\) and \(c=f(y)\). On the open set \(U=\{f\neq0\}\subset\mathfrak X\), take the level slice \(Y_0=\{f=c\}\). The action map \[\theta:Y_0\times\mathbb G_m\longrightarrow U, \qquad (q,a)\longmapsto a\cdot q,\] is finite étale: it is the cover obtained by adjoining a root \(a^d=f/c\) of an invertible function. We take the germ of \(Y_0\) at \(y\) and denote it by \(Y\). Normality and the Gorenstein property descend from the étale product description, and \(\dim Y=\dim\mathfrak X-1=n\). The subgroup \(\mu_h\) fixes \(y\) and preserves the level slice, since \(h\) divides \(d\).

The restriction \(\lambda_Y\) of \(\lambda\) has character \(-1\). It is a nonzero divisor on \(Y\): its pullback to \(Y\times\mathbb G_m\) differs from the pullback of \(\lambda\) only by the invertible group character \(a^{-1}\). It vanishes at \(y\). Since \(\mathfrak X_{\lambda\neq0}\simeq C\times\mathbb G_m\) is klt, the étale product description also shows that \(Y\) is klt away from \(\lambda_Y=0\).

We record the function-field relation to \(C\). In \(K(C)(\lambda)\) the action fixes \(K(C)\) and gives \(\lambda\) weight \(-1\), so \[f=\lambda^{-d}s\quad\text{for some }s\in S, \qquad \lambda_Y^d=s/c\quad\text{on }Y.\] After inverting \(\lambda_Y\), each component of the slice is finite over the open set \(\{s\neq0\}\subset C\). It dominates this open set: the equation is monic in \(\lambda_Y\), and the resulting finite algebra is flat over \(S[s^{-1}]\). Moreover \(\lambda_Y\) does not vanish identically on any component through \(y\). Thus \(K(Y)\) is a finite extension of \(K(C)\), and \(S\) injects into \(K(Y)\).

Extend \(u\) to \(K(Y)(a)\) by the Gauss rule with \(a\) of value zero and residue transcendental over the residue field of \(u\). This is the extension trivial on the group factor. Denote by \(V\) its restriction through \(\theta\) to \(K(\mathfrak X)\). Its center is the generic point of the orbit through \(y\), and \[ A_{\mathfrak X}(V)=A_Y(u), \qquad V(\lambda)=\ell. \tag{82}\] Indeed, on the product a log smooth model defining \(u\) may simply be multiplied by \(\mathbb G_m\); the group parameter contributes no discrepancy. Pulling back log smooth models by the finite étale chart preserves the canonical divisor and discrepancy in the restricted-value scaling. These assertions do not require \(u\) to be \(\mu_h\)-invariant.

Let \(w=V|_{K(C)}\). Since elements of \(S\) are invariant under the Rees action, their pullbacks to \(Y\times\mathbb G_m\) are independent of \(a\), and \[ w(s')=u(s'|_Y)\qquad(s'\in S). \tag{83}\] Equivalently, \(w\) is the restriction of \(u\) along the finite extension \(K(Y)/K(C)\). The Abhyankar equality is preserved by finite extension and restriction, so \(w\) is quasi-monomial (Jonsson and Mustaţă 2012, Proposition 3.7).

Write \(F^pS=\{W\geq p\}\) for the second filtration. If \(s'\in F^pS\), the Rees function \(\lambda^{-p}s'\) is regular on \(\mathfrak X\), hence its restriction is regular at \(y\). Therefore \[ w(s')\geq p\ell. \tag{84}\] Since \(F^1S\) is the vertex maximal ideal, this proves that \(w\) is centered exactly at \(o\). Applying Lemma 28 with canonical shift \(Q=n\), and then Equation (82), gives the discrepancy inequality in Equation (81).

Finally, Equation (79) implies \[ \dim_\mathbb CS/F^pS=\frac{p^n}{n!}+O(p^{n-1}). \tag{85}\] For \(p=\lceil k/\ell\rceil\), Equation (84) gives \(F^pS\subseteq\mathfrak a_k(w)\). The valuation ideal has support only at \(o\), so its colength is the same before and after localizing \(S\) at the vertex. Taking leading coefficients in Equation (85) proves \(\mathop{\mathrm{vol}}(w)\leq\ell^{-n}\). ◻

Proof. If it were not klt, there would be a prime divisor \(E\) on a log resolution of \(Y\) with \(A_Y(E)\leq0\) and center containing \(y\). By Lemma 33, its center is contained in \(\lambda_Y=0\), so \(a=\mathop{\mathrm{ord}}_E(\lambda_Y)>0\). Choose a log resolution also resolving this function and the ideal of \(y\), and choose a closed point \(q\) of \(E\) over \(y\). In local regular parameters at \(q\), give a parameter defining \(E\) cost \(1\) and the other parameters positive costs tending to zero. The resulting quasi-monomial valuations \(u_\varepsilon\), centered exactly at \(y\), satisfy \[A_Y(u_\varepsilon)=A_Y(E)+O(\varepsilon), \qquad \ell_\varepsilon=u_\varepsilon(\lambda_Y) =a+O(\varepsilon).\] Their restrictions \(w_\varepsilon\) are centered at \(o\). Since \(C\) is klt, \(A_C(w_\varepsilon)>0\); hence Equation (81) either already gives a contradiction through a nonpositive upper bound for this discrepancy, or gives \[2N^n\leq\widehat{\operatorname{vol}}(o,C) \leq A_C(w_\varepsilon)^n\mathop{\mathrm{vol}}(w_\varepsilon) \leq \left(N+ \frac{A_Y(u_\varepsilon)}{\ell_\varepsilon}\right)^n.\] The quantity inside the last power is positive whenever this chain is applicable, and its limit superior is at most \(N\). Letting \(\varepsilon\) tend to zero contradicts \(2N^n>N^n\). ◻

Proof. Suppose that \(Y\) were smooth at \(y\). The étale product chart then implies that \(\mathfrak X\) is smooth at the corresponding point. In its regular local ring, \(\lambda\) is an eigenfunction of character \(-1\) for \(\mu_h\). Lift the product parameter \(\mathtt v\in G=D[\mathtt v]\) equivariantly to a function \(\widetilde v\) of character \(1\). Its differential is nonzero: the class of \(\mathtt v\) is nonzero in the cotangent space of \(\mathop{\mathrm{Spec}}G\) at \((y_D,0)\), which is a quotient of the cotangent space of \(\mathfrak X\).

Consequently \(H=\{\widetilde v=0\}\subset\mathfrak X\) is smooth at \(y\), and the germ of \(\mathop{\mathrm{Spec}}D\) at \(y_D\) is the hypersurface cut out in \(H\) by \(\lambda|_H\). This is a nonzero equation: \(\lambda\) and \(\widetilde v\) form a regular sequence because \(\mathtt v\) is regular in \(G\). If \(\lambda|_H\) has a nonzero linear part, \(D\) is smooth at \(y_D\). Otherwise it is a minimal hypersurface equation whose character is \(-1\pmod h\), and this character is nontrivial.

We now use the other deformation \(\bar S\) over \(t=\bar v\). Its special fiber is \(D\), and \(t\) is invariant under the \(T\)-action and therefore under \(\mu_h\). Complete its local ring at the point \(y_D\) of the special fiber. The completion is flat over \(\mathbb C[[t]]\). Choose a minimal system of eigen-coordinates of the completed local ring of \(D\) and lift them equivariantly. Complete Nakayama gives a surjection \[P=\mathbb C[[t,x_1,\ldots,x_e]]\longrightarrow \widehat{\mathcal O}_{\mathop{\mathrm{Spec}}\bar S,y_D}\] with kernel \(I\), where \(t\) is fixed and every \(x_i\) is an eigen-coordinate. Flatness identifies \(I/tI\) with the kernel of the special-fiber presentation.

If \(D\) is smooth at \(y_D\), this kernel is zero and Nakayama gives \(I=0\). In the other case it is generated by the minimal hypersurface relation of character \(-1\). Lift that generator to \(I\) and average it to its character to obtain a relation \(\widetilde F\) of character \(-1\). Nakayama gives \(I=(\widetilde F)\). Because the base parameter is invariant and \(-1\) is a nontrivial character, one has \[\widetilde F(t,0,\ldots,0)=0.\] In both cases, substituting \(x_1=\cdots=x_e=0\) defines a \(\mu_h\)-fixed formal section over \(\mathbb C[[t]]\) through \(y_D\).

Since \(y_D\) is not the vertex, there is a positive-\(T\)-degree homogeneous element of \(D\) nonzero at \(y_D\). Lift it homogeneously to \(\bar S\). Its value on the formal section is a unit, so remains nonzero after passing to \(\mathbb C((t))\). On the other hand, Equation (80) identifies this generic fiber, with its \(\mu_h\)-action, with standard affine \(N\)-space: each coordinate has character \(1\). For \(h\geq2\) its fixed-point scheme is exactly the origin, where every positive-\(T\)-degree function vanishes. This is the required contradiction. ◻

Proof. The valuation ideals are \(\mu_h\)-invariant, so these character quotients are defined. Multiplication by \(\lambda_Y^j\) gives an injection \[(R/\mathfrak a_{t-j\ell})_\chi \lhook\joinrel\longrightarrow (R/\mathfrak a_t)_{\chi-j} \qquad(j\geq0).\] Indeed, the value of a product is the sum of the values, so the kernel is exactly the indicated valuation cutoff. For any two characters one may choose \(0\leq j\leq h-1\) connecting them. With \(H_0=(h-1)\ell\), this gives, for each fixed \(\chi\), \[\mathop{\mathrm{length}}(R/\mathfrak a_{t-H_0}) \leq h\dim_\mathbb C(R/\mathfrak a_t)_\chi \leq \mathop{\mathrm{length}}(R/\mathfrak a_{t+H_0}).\] The total length has asymptotic \(\sigma t^n/n!+o(t^n)\), proving the assertion. The argument uses the primitive character of the nonzero function \(\lambda_Y\), and requires no assumption on the tangent representation. ◻

Proof. Set \(\mathfrak a_k=\{s\in S:w(s)\geq k\}\). For a positive integer \(p\), the colength splits exactly as \[ \dim_\mathbb CS/\mathfrak a_k =\dim_\mathbb CS/(F^pS+\mathfrak a_k) +\dim_\mathbb CF^pS/(F^pS\cap\mathfrak a_k). \tag{87}\] The first term is bounded by \(\dim S/F^pS\). For the second, restriction and division define a linear map \[F^pS\longrightarrow R, \qquad s\longmapsto \lambda_Y^{-p}(s|_Y).\] It is regular because \(\lambda^{-p}s\) belongs to the Rees algebra. Its image has character \(p\pmod h\), since \(s\) is invariant and \(\lambda_Y\) has character \(-1\). By Equation (83), it induces the injection \[ \frac{F^pS}{F^pS\cap\mathfrak a_k} \lhook\joinrel\longrightarrow \left(R/\mathfrak a_{k-p\ell}\right)_{p\bmod h}. \tag{88}\] In particular, this step is a comparison of one character quotient, not a comparison with the full length upstairs.

Fix \(0<x<1\) and take \(p=\lfloor kx/\ell\rfloor\). Apply Equation (85) to the first term in Equation (87), and Lemma 36 to Equation (88). There are only finitely many characters for the fixed slice, so the error is uniform as \(p\bmod h\) varies. Multiplication by \(n!/k^n\) and passage to the limit give \[\mathop{\mathrm{vol}}(w)\leq \frac{x^n}{\ell^n} +(1-x)^n\frac{\sigma}{h}.\] Continuity extends this bound to the endpoints. Taking its minimum proves Equation (86). ◻

Proof. For positive \(\ell\) and \(\sigma\), direct differentiation gives \[\min_{0\leq x\leq1} \left\{\ell^{-n}x^n+\frac{\sigma}{h}(1-x)^n\right\} =\bigl(\ell^{n/N}+(h/\sigma)^{1/N}\bigr)^{-N}.\] The minimum is attained at the unique interior point \[ x=\frac{\ell^{n/N}} {\ell^{n/N}+(h/\sigma)^{1/N}}. \tag{92}\] Together with the discrepancy bound in Equation (81), this proves Equation (89).

Apply Hölder with conjugate exponents \(n/N\) and \(n\) to the vectors \[\bigl(\ell,(h/\sigma)^{1/n}\bigr), \qquad \bigl(N,A'(\sigma/h)^{1/n}\bigr).\] Their scalar product is \(N\ell+A'\). This gives \[(N\ell+A')^n \leq \bigl(\ell^{n/N}+(h/\sigma)^{1/N}\bigr)^N \bigl(N^n+(A')^n\sigma/h\bigr),\] with equality exactly as in Equation (91). Finally, Lemmas 34 and 35 show that \(y\in Y\) is a singular boundary-zero klt germ of dimension \(n\). Since \(u_*\) computes its minimum, \[(A'/n)^n\sigma=d(y,Y)\leq M_n.\] Using \(h\geq2\) proves Equation (90). ◻

Proof. Suppose \(M_n>p_n\). Choose singular klt germs whose densities \(d_i>p_n\) tend to \(M_n\). Proposition 7 gives for each of them a singular Gorenstein K-polystable cone \(o_i\in C_i\), smooth off the vertex, with the same density \(d_i\).

Fix \(i\) and choose the fast gradings of Lemma 8 for this cone. If \(m_{\max}\leq r/N\) holds along an infinite subsequence, Theorem 10 implies that \(C_i\) is smooth or the standard ODP cone. The first possibility contradicts its singularity. The second contradicts \(d_i>p_n\), since the normalized volume at an ODP is \(2(n-1)^n\) by Lemma 5. Thus the large-stabilizer condition holds for all sufficiently far terms. Applying Lemma 38 to one such term gives \[d_i\leq p_n/2+M_n/2.\] Letting \(i\) tend to infinity yields \(M_n\leq p_n/2+M_n/2\), contrary to \(M_n>p_n\).

All choices of sufficiently fast gradings in this argument are made separately for each fixed cone \(C_i\). Neither an attained supremum nor approximation estimates uniform over different cones are required. The slice bound used only the definition of \(M_n\), so the argument assumes no same-dimensional ODP estimate. ◻

Proof. The left endpoint of Equations (89)–(90) is now \(p_n\), and the right endpoint is at most \(p_n\). Every inequality is therefore an equality. In particular, \[h=2, \qquad (A')^n\sigma=2N^n.\] Combining this with Equation (91) gives \[N^N=(A')^N\ell\,\sigma/2 =N^n\ell/A',\] so \(A'=N\ell\). Rescaling \(u_*\) by \(\ell^{-1}\) makes \(\ell=1\), hence \(A'=N\) and \(\sigma=2\). Equation (92) then gives \(x=1/2\) and the refined volume bound becomes \(\mathop{\mathrm{vol}}(w)\leq2^{-N}\). The discrepancy bound becomes \(A_C(w)\leq2N\).

Since \(w\) is an admissible centered valuation, \[2N^n=\widehat{\operatorname{vol}}(o,C) \leq A_C(w)^n\mathop{\mathrm{vol}}(w) \leq (2N)^n2^{-N}=2N^n.\] Both factors are positive. Thus both displayed upper bounds are attained, and \(w\) computes the normalized volume minimum. Uniqueness up to scaling (Xu and Zhuang 2021, Theorem 1.1), together with \(A_C(\operatorname{wt}_\xi)=n\), gives the last identity in Equation (93). ◻

Proof. Suppose \(w(v)>2\). Choose a fixed real number \(t\) with \[\frac1{w(v)}<t<\frac12,\] and set \(a_k=\lceil tk\rceil\) and \(p_k=\lfloor k/2\rfloor\). For large \(k\) one has \(p_k-a_k>0\). Use the vector-space basis \(v^b e_i\) supplied by Lemma 27; it is adapted to \(W\), with \(W(v^b e_i)=b+T_i\). Let \(E_k\) be the span of its basis vectors with level less than \(p_k-a_k\).

Multiplication by \(v^{a_k}\) maps \(E_k\) injectively into \(S/F^{p_k}S\). To see injectivity modulo this filtration, the initial form of \(v\) is the regular polynomial variable \(\mathtt v\) in \(G=D[\mathtt v]\). Hence multiplication by \(v^{a_k}\) raises every nonzero initial level by exactly \(a_k\) and kills no initial form. Moreover, every element of \(v^{a_k}E_k\) has \(w\)-value greater than \(k\): \(w\) is nonnegative on \(S\) and \(a_kw(v)>k\).

Thus the kernel of \[S/F^{p_k}S\longrightarrow S/(F^{p_k}S+\mathfrak a_k)\] has dimension at least \[\dim E_k =\dim S/F^{p_k-a_k}S =\frac{(1/2-t)^n}{n!}k^n+o(k^n).\] This is a strict positive leading-order loss in the first term of the exact splitting Equation (87). The second term still satisfies the character estimate in Equation (88). At \(x=1/2\), \(\ell=1\), and \(\sigma/h=1\), these estimates give \[\mathop{\mathrm{vol}}(w) \leq 2^{-n}+2^{-n}-(1/2-t)^n <2^{-N},\] contradicting Lemma 40. Only the single exponent \(a_k\) is used at each cutoff, so there is no overlap between collections of powers to count. ◻

Proof. Choose fixed torus eigen-generators \(g_1,\ldots,g_q\) of \(S\), with characters \(\chi_1,\ldots,\chi_q\). Write \[\zeta=\frac rn\xi+\varepsilon, \qquad \|\varepsilon\|\longrightarrow0.\] The \(\xi\)-weights of the generators are positive. Consequently there are fixed constants \(c_1,c_2>0\) such that their \(\zeta\)-degrees lie between \(c_1r\) and \(c_2r\) for all sufficiently far gradings. Also \(m=m_{\max}\leq\max_i\deg_\zeta g_i=O(r)\), by Lemma 8.

The section \(v\in S_m\) has a polynomial lift in these generators homogeneous for \(\zeta\). Every monomial in such a lift uses a bounded number of generators, because its degree is \(m=O(r)\) and each generator degree is at least \(c_1r\). Its possible torus characters therefore belong to a fixed finite set, independent of the grading and of the chosen section \(v\). If \(\chi\) is any character occurring nontrivially in \(v\), then \[m=\langle\zeta,\chi\rangle =\frac rn\langle\xi,\chi\rangle +\langle\varepsilon,\chi\rangle,\] and hence, uniformly for these characters, \[\langle\xi,\chi\rangle =\frac{nm}{r}+O(\|\varepsilon\|/r).\] Taking the minimum over the nonzero character components of \(v\) is precisely \(\operatorname{wt}_\xi(v)\); the same estimate therefore holds even when \(v\) is not itself a torus eigenfunction. Lemma 40 gives \[ w(v)=\frac{2Nm}{r} +O(\|\varepsilon\|/r) =\frac{2Nm}{r}+o(1/r). \tag{94}\] By Lemma 41, \(w(v)\leq2\). Thus \(Nm-r\leq O(\|\varepsilon\|)=o(1)\). The left side is an integer, so it is nonpositive for all sufficiently far terms. On an exact rational ray the error is zero, and the same conclusion is immediate. All constants here concern the fixed cone \(C\) only. ◻

Proof. Take the fast primitive gradings of Lemma 8. A term either already has \(Nm_{\max}\leq r\), or lies in the large-stabilizer range. In the latter range, all constructions above are available for sufficiently far terms, and Lemma 42 again gives \(Nm_{\max}\leq r\). The entire sequence therefore eventually satisfies the small-stabilizer hypothesis. Theorem 10 identifies \(C\) with the smooth affine cone or the standard quadratic cone. Since \(o\in C\) is singular, only the ordinary double point remains. ◻

Proof. Both sides of (95) are unchanged by a common scaling of the weights. We may therefore assume that they are positive integers. Choose a positive common multiple \(D_1\) of the \(w_i\), and in \(\mathcal O_{X,x}\) put \[I=(z_i^{D_1/w_i}:0\le i\le n).\] This ideal is \(\mathfrak m_x\)-primary.

The ambient weighted valuation has log discrepancy \(\sum_i w_i\), gives value \(d_F\) to \(F\), and value \(D_1\) to the ambient ideal defining \(I\). Hypersurface inversion of adjunction (Ein and Mustaţă 2004, Theorem 1.1), applied to the smooth ambient space with \(F\) of coefficient one, gives \[ \mathop{\mathrm{lct}}_X(I)\le \frac{\sum_i w_i-d_F}{D_1}. \tag{96}\] In particular the numerator on the right is positive, because \(X\) is klt and \(I\) is primary.

Let \(J_t\) be the quotient weighted-order filtration on the hypersurface. The elementary monomial floor inequality gives \[J_{(k+n+1)D_1}\subset I^k\subset J_{kD_1}.\] Indeed, for a monomial of weight at least \((k+n+1)D_1\), the sum of the integers \(\lfloor \alpha_i w_i/D_1\rfloor\) is at least \(k\). The associated graded of the quotient filtration is \[\mathbb C[z_0,\ldots,z_n]/(\operatorname{in}_w F),\qquad H(t)=\frac{1-t^{d_F}}{\prod_i(1-t^{w_i})}.\] This formula holds even if \(\operatorname{in}_w F\) is reducible or nonreduced: it is a nonzero equation in a polynomial ring. The two filtration inclusions and the leading Hilbert coefficient give \[ e(I)=\frac{D_1^n d_F}{\prod_i w_i}. \tag{97}\]

Choose a divisor \(E\) computing \(\mathop{\mathrm{lct}}_X(I)\), and put \(q=\mathop{\mathrm{ord}}_E(I)>0\). Its center is \(x\), since \(I\) is primary. The inclusion \(I^k\subset\mathfrak a_{kq}(E)\) implies \[q^n\mathop{\mathrm{vol}}(E)\le e(I).\] Testing the normalized-volume infimum with \(E\), then using (96) and (97), yields \[\widehat{\operatorname{vol}}(x,X)\le A_X(E)^n\mathop{\mathrm{vol}}(E) \le \mathop{\mathrm{lct}}_X(I)^n e(I),\] which is (95). ◻

Proof. Let \(q=\mathop{\mathrm{mult}}_x(F)\ge2\). Weights all equal to one in Lemma 44 give \[\widehat{\operatorname{vol}}(x,X)\le q(n+1-q)^n,\] and klt implies \(q<n+1\). The function \(b(t)=t(n+1-t)^n\) satisfies \[b'(t)=(n+1)(1-t)(n+1-t)^{n-1}<0 \quad(1<t<n+1).\] Thus \(q>2\) gives a strict bound below \(2(n-1)^n\).

Suppose \(q=2\) and the quadratic part is degenerate. After a linear change of coordinates, choose a direction absent from the quadratic part. Give it weight \(1-\epsilon\), and give every other direction weight one, where \(0<\epsilon<1/3\) is rational. Every term of ordinary degree at least three has weight greater than two, so \(d_F=2\). The bound becomes \[\frac{2(n-1-\epsilon)^n}{1-\epsilon}.\] Its logarithmic derivative at \(\epsilon=0\) is \(-n/(n-1)+1=-1/(n-1)<0\). It is therefore strictly smaller than \(2(n-1)^n\) for sufficiently small \(\epsilon>0\). Equality forces a nondegenerate quadratic part, and the holomorphic Morse Lemma gives the asserted analytic germ. ◻

Proof of Theorem 1. The cases \(n=2,3,4\) were established in Section 2, the last by Theorem 6. Suppose the theorem is known in lower dimensions and let \(n=N+1\ge5\). The supremum argument of Theorem 39 gives \(M_n\le p_n\), hence the asserted inequality for every singular germ.

Now suppose \(d(x,X)=p_n\). Apply Proposition 7 to obtain a K-polystable cone \(C\) with the same density and \(\mathop{\mathrm{edim}}(X,x)\le\mathop{\mathrm{edim}}(C,o)\). The equality argument of Theorem 43 makes \(C\) an ordinary double point. Therefore \(\mathop{\mathrm{edim}}(X,x)\le n+1\).

Since \(x\) is singular, its embedding dimension is exactly \(n+1\), and it is a hypersurface germ. To see this algebraically, start with an embedding in a smooth ambient variety and use equations with independent linear parts to cut the ambient germ down to dimension \(n+1\). The remaining ideal is a height-one prime in a regular local ring, so is principal. Proposition 45 shows that its equation has nondegenerate quadratic part and hence that the analytic germ is ODP.

Conversely, Lemma 5 gives equality at every analytic ODP. This proves both directions of the equality statement and completes the induction. ◻

Proof of Corollary 2. For every closed point \(x\) of a K-semistable \(\mathbb Q\)-Fano \(n\)-fold \(V\), the local-to-global comparison gives \[(-K_V)^n \le\left(\frac{n+1}{n}\right)^n \widehat{\operatorname{vol}}(x,V).\] This follows from (Liu 2018, Theorem 21); K-semistability and Ding semistability agree in this setting by (Liu 2018, Theorem 13(2)). If \(V\) is singular, apply Theorem 1 at a singular closed point: \[(-K_V)^n\le \left(\frac{n+1}{n}\right)^n2(n-1)^n =2\left(\frac{n^2-1}{n}\right)^n<2n^n.\]

If \(V\) is smooth and \(V\not\cong\mathbb P^n\), the theorem of Li–Miao gives \((-K_V)^n\le2n^n\); its equality cases are a smooth quadric and \(\mathbb P^1\times\mathbb P^{n-1}\) (Li and Miao 2026, Theorem 1.1). Both have Kähler–Einstein metrics, given by the homogeneous quadric metric and the suitably scaled Fubini–Study product metric, respectively. They are therefore K-polystable (Berman 2016, Theorem 1.1). Their degrees are \[(-K_{Q^n})^n=(nH)^n=2n^n,\qquad (-K_{\mathbb P^1\times\mathbb P^{n-1}})^n =(2H_1+nH_2)^n=2n^n.\] For \(n=2\), a smooth quadric surface is \(\mathbb P^1\times\mathbb P^1\). The strict singular bound excludes every other equality case. ◻