Proof. Apply the log abundance theorem (OpenAI 2026, Theorem 1.1) to the projective lc pair \((Y,0)\) over \(\mathbb C\), whose \(\mathbb Q\)-Cartier canonical divisor is nef. ◻

Proof. Take a common smooth projective resolution carrying \(F\): \[\begin{tikzcd}[column sep=large,row sep=small] & W \arrow[dl,"p"'] \arrow[dr,"q"] & \\ U \arrow[dr,"h"'] \arrow[rr,dashed] & & U^+ \arrow[dl,"h^+"] \\ & R. & \end{tikzcd}\] Write \(g=h\circ p=h^+\circ q:W\to R\) and set \(D=p^*K_U-q^*K_{U^+}\). This divisor is \(q\)-exceptional, since the step extracts no divisors. On every curve contracted by \(g\), the divisors \(-p^*K_U\) and \(q^*K_{U^+}\) have nonnegative degree; in the divisorial case the second has degree zero. Thus \(-D\) is \(g\)-nef, and hence \(q\)-nef. The negativity lemma gives \(D\geq 0\); see (Kollár and Mori 1998, Lemmas 3.38–3.39).

To verify the support assertion, let \(r\in R\) be a closed point with a positive-dimensional fiber on either side. There is a curve in \(g^{-1}(r)\) mapping onto a curve in that fiber: take a component dominating that curve and cut it by sufficiently general ample divisors. The relative ampleness signs give \(D\cdot C<0\) for this lifted curve. Consequently \(g^{-1}(r)\) meets \(\operatorname{Supp}D\).

We use the fiber-support form of the negativity lemma (Kollár and Mori 1998, Lemma 3.39): the support of an effective divisor antinef over a proper birational morphism to a normal variety either contains or is disjoint from each fiber. In the present setting it can also be seen directly on the reduced irreducible components of the fiber. If such a component \(T\) is not contained in \(\operatorname{Supp}D\) but meets it, a Cartier multiple of \(D\) restricts to a nonzero effective Cartier divisor on the integral projective variety \(T\). Its degree against sufficiently many ample hyperplanes is positive, producing a curve contracted by \(g\) with positive \(D\)-degree. This contradicts antinefness. This argument does not require \(T\) to be normal or the fiber to be reduced: only its support is at issue. Components of dimension zero already lie in the support if they meet it. The fibers of \(g\) are connected because \(R\) is normal and \(g\) is proper and birational (Hartshorne 1977, Corollary III.11.4). Containment therefore propagates through their reduced irreducible components.

It follows that every fiber in question is contained in \(\operatorname{Supp}D\). If the center of \(F\) on \(R\) is contained in this locus, then \(F\subset\operatorname{Supp}D\) on our chosen resolution. Finally, \[a(F,U^+)-a(F,U)=\operatorname{coeff}_F D,\] which proves both assertions. ◻

Proof. The usual MMP properties give preservation of \(\mathbb Q\)-factoriality, invariance of the Picard number under a flip, and a drop of one under an elementary divisorial contraction (Kollár and Mori 1998, Propositions 3.36–3.37). There are therefore only finitely many divisorial contractions. Since the steps extract no divisors, only finitely many prime divisors of \(X\) can disappear; denote this finite set by \(\mathcal D\).

All models are terminal. This follows from (Kollár and Mori 1998, Corollaries 3.42–3.43), or directly from Lemma 4: previously exceptional valuations retain log discrepancy greater than one, and a newly contracted divisor increases strictly from log discrepancy one. Moreover, every valuation exceptional over the smooth \(X\) has integral log discrepancy at least two. Discrepancies never decrease along the sequence.

The flipped loci. Consider a flip whose target has a codimension-two component \(T\) of its flipped locus. A terminal variety is smooth in codimension two (Kollár and Mori 1998, Corollary 5.18). Blowing up \(T\) at its generic smooth point therefore defines a prime valuation \(F\) with \[a(F,U_{i+1})=2.\] Its center on the contraction base is contained in the locus of positive-dimensional fibers of the flipped contraction. Lemma 4 implies \[ a(F,X)\leq a(F,U_i)<2=a(F,U_{i+1}). \tag{4}\] Thus \(F\) is not exceptional over \(X\): it is an original prime divisor. It is exceptional over \(U_{i+1}\), so it belongs to \(\mathcal D\). After this occurrence its discrepancy is at least two forever, and it cannot satisfy (4) at a later step. Each flip with such a target component therefore uses a distinct member of a finite set. After a finite prefix, all steps are flips and every flipped locus has codimension at least three.

The flipping loci. We have removed codimension-two components on the target side. It remains to remove them on the source side; only then can a general surface be transported unchanged through every later finite stage. The argument adapts the homological decrease used in the proof of (Alexeev et al. 2007, Lemma 3.1), keeping only classes of algebraic cycles. Put \(d=n-2\) and define the finite-dimensional real vector space \[\mathcal A_d(V) =\operatorname{span}_{\mathbb R} \{[T]:T\subset V\text{ is a closed integral $d$-fold}\} \subset H_{2d}(V,\mathbb R).\] Cycle classes, Borel–Moore localization, and their compatibility with restriction to an open subset are as in (Fulton 1998, sec. 19.1). The homology of the compact complex algebraic varieties here is finite-dimensional and vanishes above their real dimension.

For a remaining flip \(U\dashrightarrow U^+\) over \(R\), remove the images in \(R\) of both exceptional loci and take inverse images. This gives a common open subset \(\mathcal U\). Indeed, a proper birational morphism to a normal variety is an isomorphism over its quasi-finite locus. Each exceptional locus consists of positive-dimensional fibers; its image has dimension at most one less than its own. Since the source contraction is small and the target exceptional locus has dimension at most \(n-3\), the closed complements \(Z\subset U\) and \(Z^+\subset U^+\) satisfy \[\dim Z\leq d,\qquad \dim Z^+\leq d-1.\] The additional points removed outside the exceptional loci lie in an isomorphism locus and have dimension at most \(n-3\).

Restriction gives maps \[\mathcal A_d(U)\longrightarrow H^{\mathrm{BM}}_{2d}(\mathcal U,\mathbb R) \longleftarrow\mathcal A_d(U^+)\] with the same image: intersect an integral \(d\)-fold with \(\mathcal U\) and take its closure on the other model, while a \(d\)-fold contained in the complement restricts to zero. The right-hand map is injective, by the localization sequence and \(H_{2d}(Z^+,\mathbb R)=0\). Hence \[ \dim\mathcal A_d(U^+)\leq\dim\mathcal A_d(U). \tag{5}\] If the flipping locus has a \(d\)-dimensional component \(T\), its class belongs to the kernel of the left-hand restriction and is nonzero: for an ample Cartier divisor \(A\), \[\langle c_1(A)^d,[T]\rangle=A^d\cdot T>0.\] Then (5) is strict. A nonnegative integer can drop only finitely many times, so eventually the flipping loci also have codimension at least three. The same construction of the common open then gives this codimension bound on both complements. ◻

Proof of Proposition 3. Choose an effective ample rational divisor \(B\) for which \((X,B)\) is klt and \(K_X+B\) is ample. For example, take a sufficiently positive general smooth very ample divisor and multiply it by a rational number strictly between zero and one. Since \(K_X\) is pseudo-effective, \(K_X+tB\) is big for every rational \(t>0\).

We shall repeatedly use a simple consequence of normality. If a birational map \(X\dashrightarrow U\) extracts no divisors, every prime divisor of \(U\) corresponds to a prime divisor of \(X\). For compatible transformed divisors, the divisorial criterion for regularity of a rational section consequently gives \[ H^0(X,kD)\hookrightarrow H^0(U,kD_U) \tag{6}\] whenever the multiples are Cartier. Thus effectiveness and bigness of \(B_U\), and bigness of \(K_U+tB_U\), persist on every model constructed below. In particular, we do not need \(B_U\) to be ample.

Finite stages. Set \(t_j=2^{-j}\) for every integer \(j\geq0\). Starting with \(V_0=X\), construct \(V_{j+1}\) from \(V_j\) by running the MMP for \[K_{V_j}+\Delta_j,\qquad \Delta_j=t_{j+1}B_{V_j},\] with scaling of \(C_j=(t_j-t_{j+1})B_{V_j}\). The precise input is (Birkar et al. 2010, Corollary 1.4.2): for a projective \(\mathbb Q\)-factorial klt pair \((V,\Delta)\) with big boundary \(\Delta\), and \(C\geq0\) such that \((V,\Delta+C)\) is klt and \(K_V+\Delta+C\) is nef, the MMP with scaling of \(C\) terminates. Here the inductive hypothesis is that \((V_j,t_jB_{V_j})\) is klt and its adjoint is nef. The smaller pair is klt because \(B_{V_j}\) is effective; its boundary is big by (6); and the larger pair is exactly the one in the inductive hypothesis.

The adjoint remains big by (6), so a Mori fiber space cannot occur: an effective representative of a positive multiple cannot have negative degree on curves covering general fibers. Each stage is consequently a finite, possibly empty, sequence ending at a nef adjoint. Standard MMP preservation gives the next klt pair and a projective \(\mathbb Q\)-factorial model. The rational divisor \(K_{V_j}+t_jB_{V_j}\) is nef and big, hence semiample by the klt base point free theorem (Kollár and Mori 1998, Theorem 3.3).

Canonical steps. To apply Lemma 5, we must identify every adjoint step as a canonical MMP step. This requires canonical negativity on the contracted ray and, for a flip, canonical ampleness on the flipped side. On its source \(U\), let \(D=K_U+t_{j+1}B_U\). The contracted ray \(R_0\) satisfies \[D\cdot R_0<0,\qquad (D+\lambda C)\cdot R_0=0,\qquad \lambda>0,\] where \(C=(t_j-t_{j+1})B_U\) is the current scaling divisor. Therefore \(B_U\cdot R_0>0\) and \(K_U\cdot R_0<0\). This is the usual ray calculation for a directed program (Birkar et al. 2010, proof of Corollary 1.3.3), and it already identifies each divisorial step as a \(K\)-negative divisorial contraction.

For a small contraction \(h:U\to R\), the adjoint flip makes \(D^+\) relatively ample. We show that \(K_{U^+}\) is relatively ample as well. Choose the positive rational number \[b=\frac{K_U\cdot R_0}{D\cdot R_0}.\] The divisor \(K_U-bD\) is numerically trivial over \(R\) and descends up to rational linear equivalence. To see this, a Cartier multiple is relatively semiample by the relative klt base point free theorem for the pair \((U,t_{j+1}B_U)\) (Kollár and Mori 1998, Theorem 3.24): subtracting its adjoint \(D\) gives relative ampleness, because \(-D\) is relatively ample. Its semiample morphism is constant on each connected projective fiber, as its degree on every fiber curve is zero. The resulting image is proper and quasi-finite over \(R\), because each connected fiber maps to one point. It is therefore finite and birational over the normal \(R\), hence equals \(R\). Thus \(K_U-bD\sim_{\mathbb Q}h^*A\) for a rational Cartier divisor \(A\) on \(R\).

Both morphisms in the adjoint flip are small, so this relation transforms to \[K_{U^+}-bD^+\sim_{\mathbb Q}(h^+)^*A.\] Since \(D^+\) is relatively ample and \(b>0\), so is \(K_{U^+}\). The adjoint flip is therefore a \(K\)-flip. Thus all steps of the finite stages satisfy the hypotheses of Lemma 5.

The fixed model. Concatenate the finite stages and apply Lemma 5. Choose an endpoint \(Y=V_s\) after the resulting finite prefix. If only finitely many nonempty stages occur, choose \(s\) after the last one. For every \(j>s\), the finite composition \(Y\dashrightarrow V_j\) is an isomorphism outside codimension-at-least-three closed subsets on both models. This property is preserved under finite composition: on a common open, remove the next bad set and take its closure on the preceding model; its dimension does not increase. Let \(Z_j\) be the resulting closed subset of \(Y\).

The divisor \(M_j=K_Y+t_jB_Y\) agrees on this common open with \(K_{V_j}+t_jB_{V_j}\). Choose a multiple Cartier on both varieties and base point free on \(V_j\). A line bundle on a normal variety has the same global sections after deleting a subset of codimension at least two, since it is reflexive. The two spaces of sections are therefore identified, and the complete system on \(Y\) is generated outside \(Z_j\). This proves (ii) for each \(j\).

Finally, \(Y\) is projective, \(\mathbb Q\)-factorial and terminal by the preceding construction, and a finite composition of these steps extracts no divisors. On a common smooth projective resolution of the finite chain, the effective differences in Lemma 4 telescope to (3). A divisor mapping onto a prime divisor of \(Y\) also corresponds to a prime divisor of \(X\), so its coefficient in \(E\) is zero. Thus \(E\) is \(q\)-exceptional. For an original prime divisor contracted on \(Y\), terminality gives log discrepancy greater than one on \(Y\), compared with one on \(X\). Its strict transform has positive coefficient in \(E\), proving (iii). ◻

Proof. Put \(r=n-2\). Very general members of \(|H|\) have a smooth complete intersection \(S\) avoiding \(\operatorname{Sing}Y\) and every \(Z_j\). Indeed, each of these countably many closed sets has dimension at most \(n-3\), so the condition that \(r\) hyperplanes meet any one of them is a proper closed incidence condition. Over \(\mathbb C\) these can be avoided simultaneously, together with the closed conditions excluded by Bertini’s theorem. The inverse image of \(S\) is therefore exactly the smooth complete intersection cut out by the \(r\) pulled-back equations; there are no additional components over the singular locus. Each \(M_j|_S\) is semiample, hence nef. Passing to the limit on every curve on \(S\) shows that \(L|_S\) is nef and proves (8).

Write \(A=\pi^*H\), choose an ample Cartier divisor \(A_0\) on \(\widehat Y\), and make the fixed choice \[ P=K_{\widehat Y}+rA+A_0. \tag{11}\] Since \(A\) is nef and ampleness is open (Lazarsfeld 2004a, Corollary 1.4.10 and Theorem 1.4.23), there exists \(\eta>0\) such that \[ A_0+(r-i)A-\epsilon\pi^*B_Y \quad\text{is ample whenever }0\leq i\leq r \text{ and }0\leq\epsilon<\eta. \tag{12}\] For a given positive \(m\) with \(d\mid m\), first choose \(j=j(m)\) so that \(mt_j<\eta\), and then choose a multiple \(k\) of \(d_j\) with \(k>m\). The pullback of \(|kM_j|\) is generated near \(S\). A general member \(D_m\) is smooth there, by Bertini, and \[R_m=\frac{m}{k}D_m\geq 0, \qquad R_m\sim_{\mathbb Q}m\pi^*M_j, \qquad \mathcal J(R_m)=\mathcal O_{\widehat Y} \quad\text{near }S.\] Here \(\mathcal J(R_m)\) is the multiplier ideal; its asserted local triviality follows from \(m/k<1\) and the smoothness of \(D_m\) there.

The divisors \(m\pi^*L+P-iA\) are integral Cartier divisors, and \[m\pi^*L+P-iA-K_{\widehat Y}-R_m \sim_{\mathbb Q} A_0+(r-i)A-mt_j\pi^*B_Y\] is ample for \(0\leq i\leq r\). Nadel vanishing (Lazarsfeld 2004b, Theorem 9.4.8), equivalently (Lazarsfeld 2010, Theorem 2.4), gives \[ H^q\!\left(\widehat Y, \mathcal O_{\widehat Y}(m\pi^*L+P-iA) \otimes\mathcal J(R_m)\right)=0 \qquad(q>0,\ 0\leq i\leq r). \tag{13}\]

Set \(\mathcal F= \mathcal O_{\widehat Y}(m\pi^*L+P)\otimes\mathcal J(R_m)\). Tensoring the Koszul complex of the \(r\) equations with \(\mathcal F\) yields the exact complex \[0\longrightarrow\mathcal F(-rA) \longrightarrow\cdots\longrightarrow \mathcal F(-A)^{\oplus r}\longrightarrow\mathcal F \longrightarrow\mathcal O_S(mL|_S+P|_S) \longrightarrow 0.\] At points of \(S\), the equations form a regular sequence and \(\mathcal F\) is locally free. Outside \(S\), one equation is a unit, so the Koszul complex is contractible even after tensoring with an arbitrary sheaf. This proves exactness everywhere. Splitting the complex into short exact sequences and using (13) gives surjectivity on global sections onto its last term. The inclusion \(\mathcal F\subset\mathcal O_{\widehat Y}(m\pi^*L+P)\) then proves (9).

Adjunction and (11) give \(P|_S-K_S=A_0|_S\). Hence \(mL|_S+P|_S-K_S\) is ample. Kodaira vanishing, the zero-boundary ample case of (Lazarsfeld 2010, Theorem 2.4), and surface Riemann–Roch (Hartshorne 1977, V, Section 1) give \[\begin{split} h^0(S,mL|_S+P|_S) ={}&\frac{m^2}{2}(L|_S)^2 +\frac m2 L|_S\cdot(2P|_S-K_S)\\ &+\frac12 P|_S\cdot(P|_S-K_S) +\chi(S,\mathcal O_S). \end{split}\] If \(S\) is disconnected, the formula is read componentwise and summed. All its coefficients are fixed before \(m,j,k,D_m\) vary. Together with (9), this proves (10). ◻

Proof. Choose a line bundle \(G\) of positive degree on the fixed smooth curve \(N\) such that \(\iota^*\Omega^1_V\) embeds as a subbundle of \(G^{\oplus a}\) for some \(a\). Such a choice follows by global generation of \((\iota^*\Omega^1_V)^*\otimes G\) for a sufficiently ample \(G\), followed by dualization. Set \(g=\deg G>0\).

For a nonzero \(s\), the integer \[b=\min_{p\in\Gamma\text{ closed}}\operatorname{ord}_p(s)\] is finite and attained. The leading ambient jet gives a nonzero section of \[ \operatorname{Sym}^b(\iota^*\Omega^1_V) \otimes\iota^*\mathcal D. \tag{15}\] Here is a justification valid also at singular points of \(\Gamma\). For \(b\geq1\), the bundles of principal parts on the smooth variety \(V\) have the locally split exact sequence \[0\longrightarrow\operatorname{Sym}^b\Omega^1_V\otimes\mathcal D \longrightarrow\mathcal P^b(\mathcal D) \longrightarrow\mathcal P^{b-1}(\mathcal D) \longrightarrow 0.\] The lower jet of \(s\) has zero value at every closed point of the reduced curve \(\Gamma\). A section of a vector bundle on a reduced variety with this property is zero, so that lower jet restricts to the zero section. The principal-parts sequence is locally split as a sequence of \(\mathcal O_V\)-modules, and therefore remains exact after restriction to \(\Gamma\) and pullback to \(N\). The restricted order-\(b\) jet consequently belongs to the left-hand bundle. Its value is nonzero at a point where the minimum \(b\) is attained; the induced map on its fiber at any point of \(N\) above that point is an isomorphism. Its pullback is thus regular and nonzero. For \(b=0\), use \(s|_\Gamma\) directly. This proves (15) without assuming smoothness of \(\Gamma\).

Taking symmetric powers of the subbundle inclusion above embeds (15) into a direct sum of copies of \(G^{\otimes b}\otimes\iota^*\mathcal D\). A nonzero component is a nonzero section of this line bundle on \(N\), so \[0\leq bg+\deg_N\iota^*\mathcal D.\] Since \(\operatorname{ord}_z(s)\geq b\) for every \(z\in\Gamma\), (14) follows. ◻

Proof. Every curve in \(\operatorname{Supp}F_j\) maps to a point of \(T\), by the generation assumption. Fix a very ample Cartier divisor \(H_T\) on \(T\) and an ample Cartier divisor \(J\) on \(T_1\), and put \(Q=f^*H_T\). The projection formula gives \[ Q^2=H_T^2\cdot[T]>0, \qquad Q\cdot F_j=f^*D_j\cdot F_j=0. \tag{17}\] The Hodge index theorem on the fixed smooth surface \(T_1\) (Hartshorne 1977, V, Section 1) makes the negative intersection form positive definite on \(Q^\perp\subset N^1(T_1)_{\mathbb R}\). Consequently there is one constant \(K>0\) such that \[(J\cdot F)^2\leq K(-F^2) \qquad\text{for every }F\in Q^\perp.\] Effectiveness of \(F_j\) gives \(J\cdot F_j\geq ck_j(J\cdot e)\). Thus, with the fixed positive number \(\epsilon=c^2(J\cdot e)^2/K\), \[ F_j^2\leq-\epsilon k_j^2. \tag{18}\]

After removal of \(F_j\), the residual linear system has no fixed curve. Its square is nonnegative: choose two members with no common curve and take their effective intersection, or use a nowhere vanishing member if the system is trivial. By (17) and the projection formula, \[0\leq(k_jf^*D_j-F_j)^2 =k_j^2(D_j^2\cdot[T])+F_j^2.\] Now (18) proves (16). ◻

Proof. Suppose that an integral curve \(C_0\subset Y\) satisfies \(L\cdot C_0<0\). Fix a projective resolution \(\pi:\widehat Y\to Y\) which is an isomorphism over the smooth locus. Choose a closed point \(x\in C_0\) outside the images of those irreducible components of \(\pi^{-1}(C_0)\) which do not dominate \(C_0\). There are only finitely many such images, and each is a point. Consequently every point of \(\pi^{-1}(x)\) lies on a component of \(\pi^{-1}(C_0)\) dominating \(C_0\).

A surface through \(x\). Choose \(n-2\) very general members \(H_1,\ldots,H_{n-2}\in|H|\), all passing through \(x\). They meet properly, and their intersection meets each \(Z_j\) and \(\operatorname{Sing}Y\) in at most finitely many points. Indeed, the hyperplanes through \(x\) have no other base point; successive general cuts avoid all positive-dimensional components of the relevant intersections. The countably many conditions can again be imposed simultaneously over \(\mathbb C\). Write their intersection cycle as \[H_1\cdots H_{n-2}\cdot[Y]=\sum_{i=1}^a a_i[T^{(i)}], \qquad a_i>0,\] where each \(T^{(i)}\) is an integral surface. The dimension theorem guarantees a component through \(x\); fix one and denote it by \(T\).

On every \(T^{(i)}\), the system induced by \(|d_jM_j|\) has at most a finite base locus. Its square is nonnegative, including when \(T^{(i)}\) is nonnormal. To check this directly, choose a section nonzero at the generic point, and then choose a second section not vanishing identically on any curve of the first zero divisor. The resulting Cartier intersections with \([T^{(i)}]\) form an effective zero-cycle. If the first section has no zero, the restricted line bundle is trivial and its square is zero. These are the usual Cartier intersection and projection formulas on cycles (Fulton 1998, chap. 2). It follows that \[M_j^2\cdot[T^{(i)}]\geq0, \qquad \sum_{i=1}^a a_i(M_j^2\cdot[T^{(i)}]) =M_j^2\cdot H^{n-2}\longrightarrow 0\] by (19). In particular, \[ M_j^2\cdot[T]\longrightarrow 0. \tag{20}\]

Fixed geometry above \(x\). The surface \(T\) meets \(\operatorname{Sing}Y\) only in finitely many points. Its strict transform \(T_{\widehat Y}\) is thus birational to \(T\) and maps onto it. Choose any \(z\in T_{\widehat Y}\) above \(x\). There is an integral curve \(\Gamma\subset\widehat Y\) through \(z\) dominating \(C_0\). To see this, take an irreducible component \(A\subset\pi^{-1}(C_0)\) through \(z\). By the choice of \(x\), it dominates \(C_0\). If \(\dim A=d\), its fiber \(A_x\) is a proper closed subset of dimension at most \(d-1\). General very ample cuts through \(z\), in number \(d-1\), cut \(A_x\) to dimension at most zero and leave a curve component through \(z\) on \(A\). That component cannot lie in \(A_x\), so it dominates \(C_0\). For \(d=1\), take \(\Gamma=A\).

Let \(\iota:N\to\widehat Y\) denote the normalization map of \(\Gamma\), followed by inclusion. Resolve \(T_{\widehat Y}\) by a smooth projective surface \(T_0\to T_{\widehat Y}\), choose a point \(w\in T_0\) above \(z\), and blow it up. Write \(T_1\) for the resulting surface, \(e\subset T_1\) for the exceptional curve of this blowup, and \(f:T_1\to T\) for the composite map: \[\begin{tikzcd}[column sep=large,row sep=large] T_1 \arrow[r,"\mathrm{Bl}_w"] \arrow[d,"f"'] & T_0 \arrow[r] & \widehat Y \arrow[d,"\pi"] & N \arrow[l,"\iota"'] \arrow[d] \\ T \arrow[rr,hook] & & Y & C_0 \arrow[l,hook] \end{tikzcd}\] The images of \(T_0\) and \(N\) in \(\widehat Y\) meet at \(z\). The exceptional curve \(e\), all these varieties, and all these maps are fixed throughout the remaining argument.

The contradiction on the fixed surface. The projection formula and \(M_j\to L\) give, for one fixed \(\delta>0\), \[ \deg_N\iota^*\pi^*M_j\leq-\delta \qquad\text{for all sufficiently large }j. \tag{21}\] In fact this degree equals the positive degree of \(N\to C_0\) times \(M_j\cdot C_0\). For each such \(j\), choose any positive multiple \(k_j\) of \(d_j\). Lemma 7, applied on the fixed \(\widehat Y\) and \(\Gamma\), shows that every nonzero pullback section \(s'\in H^0(\widehat Y,k_j\pi^*M_j)\) of a section on \(Y\) satisfies \[ \operatorname{ord}_z(s')\geq ck_j, \qquad c=\delta/g>0. \tag{22}\] The constant \(c\) depends only on the fixed normalized curve and the negative-degree margin in (21).

The sections on \(Y\) induce a nonzero linear system of \(k_jf^*(M_j|_T)\), generated outside \(f^{-1}(T\cap Z_j)\). Nonzeroness follows from generation at points of \(T\setminus Z_j\).

Every nonzero section in this induced system has order at least \(ck_j\) at \(w\) before the last blowup. Indeed, the local ring map \(\mathcal O_{\widehat Y,z}\to\mathcal O_{T_0,w}\) sends \(\mathfrak m_z^b\) into \(\mathfrak m_w^b\); apply (22) to a section inducing it. Sections whose restriction is identically zero do not contribute to the induced system. Thus its fixed divisor \(F_j\) satisfies \(\operatorname{coeff}_eF_j\geq ck_j\). All the hypotheses of Lemma 8 hold with \(D_j=M_j|_T\). Lemma 8 gives a fixed \(\epsilon>0\) such that \[M_j^2\cdot[T]\geq\epsilon \qquad\text{for all sufficiently large }j,\] contrary to (20). Therefore no curve \(C_0\) of negative \(L\)-degree exists, and \(L\) is nef. ◻