Proof. We may replace \(H\) by a very ample multiple. The number \(BNH\) is nonnegative. If it were positive, the identity \((K_X+B)NH=N^2H/a=0\) would give \(K_XNH<0\).

For each sufficiently large integer \(l\), choose positive integers \(u_l,v_l\) such that \(u_lH\) and \(v_l(H+lN)\) are very ample. Their general complete intersection is an integral curve \(C_l\) contained in the smooth locus of \(X\). Indeed, normality gives a singular locus of dimension at most one; the first general member meets this locus in finitely many points, and the second avoids them. Bertini’s theorem on the smooth open gives the desired integral curve. Its intersections satisfy \[ \frac{NC_l}{-K_XC_l} =\frac{NH^2}{-K_XH^2-lK_XHN}\longrightarrow0, \tag{6}\] and \(-K_XC_l>0\) for large \(l\).

The quantitative bend-and-break theorem of Jovinelly, Lehmann, and Riedl [10] applies to an integral curve in the smooth locus of a projective variety over an algebraically closed field, with negative canonical degree, and any nef Cartier test divisor. In dimension three it supplies, through every closed point of \(C_l\), a rational curve \(\Gamma\) satisfying \[0\leq N\Gamma\leq4\frac{NC_l}{-K_XC_l}.\] Fix \(l\) so large that the right-hand side is strictly less than one. Since \(N\) is Cartier, it follows that \(N\Gamma=0\).

For completeness, these complete intersections pass through points to which the very-general-point condition applies. For the two fixed very ample systems, the incidence of a point of \(X\) and a pair of members through it is a product of projective bundles over \(X\), hence is irreducible and dominates \(X\). Restricting to the nonempty open set of pairs giving the curves just considered leaves a dense open incidence and a dominant evaluation map. Its image is constructible and contains a nonempty open subset of \(X\). Over the uncountable field, this open contains a closed point outside the countable union excluded by \(n(N)=3\). The curve \(\Gamma\) through that point is a contradiction. This proves (5).

Write \(B=\sum b_TT\) with \(b_T>0\). All numbers \(NHT\) are nonnegative, so (5) gives \(NHT=0\) for each \(T\). Lemma 3 yields \(N|_T\equiv0\). If \(T\) is not a component of \(N\), the canonical section of \(\mathcal O_X(N)\) restricts to a nonzero section on the integral surface \(T\). Were \(T\cap\operatorname{Supp}N\) nonempty, its zero scheme would be a nonzero effective Cartier divisor on \(T\) and would have positive ample degree. This contradicts \(N|_T\equiv0\), so the two supports are disjoint. ◻

Proof. We first find a coefficient-one boundary component lying over \(\operatorname{Supp}N_0\). On a log resolution of \((X,B+N_0)\), the log canonical threshold is the minimum of the finitely many ratios \[\frac{a(E,X,B)}{\operatorname{ord}_E(N_0)} \qquad\bigl(\operatorname{ord}_E(N_0)>0\bigr),\] where \(a(E,X,B)\) denotes log discrepancy and the divisors considered include the strict transforms of the components of \(N_0\). Thus \(\lambda\) is finite and rational, possibly zero, and \((X,B+\lambda N_0)\) is log canonical. There is a log canonical place \(E\) of this pair for which \(\operatorname{ord}_E(N_0)>0\).

Take a projective crepant \(\mathbb Q\)-factorial dlt modification \[\mu:(W,G)\longrightarrow (X,B+\lambda N_0),\] with every extracted divisor having coefficient one; see [24]. Set \[N=\mu^*N_0,\qquad \tau=m^{-1}+\lambda.\] Then \[ K_W+G\sim_{\mathbb Q}\tau N. \tag{11}\] The divisor \(N\) is nef and numerically trivial on every component of its support: a curve in that support maps to a point or to a curve in \(\operatorname{Supp}N_0\), and the assertion follows by the projection formula.

Every component of \(G\) is either contained in or disjoint from \(\operatorname{Supp}N\). For strict transforms of components of \(B\) this follows from the hypothesis, while those coming from \(\lambda N_0\) are contained in the support. An extracted divisor has center in \(\operatorname{Supp}N_0\cup P_{\rm out}\), since \(X\) is klt and the boundary vanishes outside this union. Its irreducible center lies in one of these two disjoint closed sets. In the first case its order on \(N_0\) is positive, and in the second case the divisor is disjoint from \(\operatorname{Supp}\mu^*N_0\).

The center on \(W\) of the place \(E\) lies in \(\operatorname{Supp}N\) and, by crepancy and the dlt property, in \(\lfloor G\rfloor\). Choose a component \(S\) of \(\lfloor G\rfloor\) containing this center. The preceding separation then implies \(S\subset\operatorname{Supp}N\). This argument also covers \(\lambda=0\); it does not require all components of \(N\) to occur in \(G\).

The modification \(\mu\) is an isomorphism outside \(\operatorname{Supp}N_0\cup P_{\rm out}\). Indeed it has no exceptional divisors there, and a projective small birational morphism to a \(\mathbb Q\)-factorial variety is an isomorphism. To recall the latter fact, push down a relatively ample Cartier divisor. A Cartier multiple of its pushdown pulls back to the same multiple upstairs, because there are no exceptional divisors. It has degree zero on every contracted curve, contradicting relative ampleness if such a curve exists.

Choose a rational \(\epsilon>0\) sufficiently small that \[ G-\epsilon S\geq0,\qquad \tau N-\epsilon S\geq0. \tag{12}\] Apply the partial MMP with lowered boundary \(G-\epsilon S\) and restoration divisor \(\epsilon S\) [24]. Its steps are \((K_W+G-\epsilon S)\)-negative and \((K_W+G)\)-trivial. The lowered pair is dlt, and its log canonical divisor is pseudoeffective by (11) and (12). The termination required here is therefore supplied by [23]. The threefold MMP package used in this application, including in characteristic five, is recalled in [17].

The partial-MMP conclusion is either a Mori fiber contraction for the lowered log divisor, trivial for the restored divisor, or nefness of the restored divisor minus \(\delta\) times the transform of \(S\) for every sufficiently small rational \(\delta>0\). We first track \(S\), the pullbacks of \(N\), and support separation through the steps, and then exclude the fiber-type alternative. Write \(W_i\dashrightarrow W_{i+1}\) for a step, with transforms \(G_i,N_i,S_i\). The equality \(K_{W_i}+G_i\sim_{\mathbb Q}\tau N_i\) persists by pushforward, so its contracted ray \(R_i\) satisfies \[ N_i\cdot R_i=0,\qquad S_i\cdot R_i>0. \tag{13}\] In particular \(S_i\) cannot be the exceptional divisor of a divisorial step: the negativity lemma would give a contracted curve of negative \(S_i\)-degree. Hence \(S\) survives throughout the MMP.

Let \(a:T\to W_i\) and \(b:T\to W_{i+1}\) be a common resolution of a step. The divisor \[F=a^*N_i-b^*N_{i+1}\] is \(b\)-exceptional. If a curve on \(T\) is contracted by \(b\), its image on \(W_i\) is a point or a curve contracted to the common contraction target. Thus \(F\) has degree zero on every such curve by (13). The negativity lemma applied to \(F\) and \(-F\) gives \[ a^*N_i=b^*N_{i+1}. \tag{14}\] This applies to divisorial contractions and flips. The same argument for the full log divisors gives \[a^*(K_{W_i}+G_i)=b^*(K_{W_{i+1}}+G_{i+1}).\] Consequently the restored pair remains log canonical.

Equation (14) also preserves nefness and numerical triviality on the support. For the latter assertion, lift a curve in \(\operatorname{Supp}N_{i+1}\) to a curve on \(T\) dominating it. Its image on \(W_i\) lies in \(\operatorname{Supp}N_i\) or is a point: the inverse image of the support of an effective \(\mathbb Q\)-Cartier divisor is exactly the support of its pullback. The projection formula now gives degree zero. Boundary components contained in \(\operatorname{Supp}N_i\) remain contained in the transformed support. If a boundary component is disjoint from that support, its strict transform on \(T\) is disjoint from \(a^*N_i\), hence from \(b^*N_{i+1}\). Since that strict transform maps onto its image on \(W_{i+1}\), the transformed boundary component is still disjoint from \(\operatorname{Supp}N_{i+1}\).

These steps also preserve the open complement of the support. Indeed every contracted curve has negative degree against the effective divisor \(\tau N_i-\epsilon S_i\), so is contained in \(\operatorname{Supp}N_i\). A connected positive-dimensional projective fiber contains a curve through each of its closed points. Connectedness of fibers therefore implies that a fiber meeting the complement of \(\operatorname{Supp}N_i\) consists of a single point and does not meet the support. Properness and normality show that the contraction is an isomorphism from this complement onto an open subset of its target. For a flip, this open in the target is \(\mathbb Q\)-factorial, being isomorphic to the corresponding open of \(W_i\). The new small morphism is therefore also an isomorphism over it.

There is no Mori fiber output. The effective representative \(\tau N_i-\epsilon S_i\) persists on each model, as does \(\tau N_i-tS_i\) for every rational \(0<t\leq\epsilon\). A curve in a fiber through a general point outside this effective support has nonnegative degree, whereas the log divisor driving a Mori fiber contraction has negative degree on every contracted curve. This rules out the fiber-type alternative for the lowered class and for every such smaller perturbation of the restored class.

Let \(Y\) be the resulting model. The partial-MMP conclusion gives (8) for some rational \(0<\delta<\epsilon\). Negative MMP steps preserve the \(\mathbb Q\)-factorial dlt pair with lowered boundary, so \((Y,G_Y-\epsilon S_Y)\) is dlt. In particular its \(\mathbb Q\)-factorial underlying variety \(Y\) is klt: every place of zero log discrepancy for a dlt pair has center in its reduced boundary, and removing that effective \(\mathbb Q\)-Cartier boundary strictly increases its discrepancy. Here dlt preservation is used for the lowered boundary; for the restored boundary \(G_Y\) we use the crepant comparison just proved.

It remains to see why nefness isolates \(S_Y\). If \(T\ne S_Y\) is a component of \(N_Y\), numerical triviality of \(N_Y|_T\) and (8) imply that \(-S_Y|_T\) is nef. On the other hand, \(S_Y|_T\) is an effective \(\mathbb Q\)-Cartier divisor. If it were nonzero, its intersection with an ample Cartier divisor on the integral projective surface \(T\) would be positive, contradicting anti-nefness. Therefore \(S_Y\cap T=\varnothing\). Each other component of \(G_Y\) is either such a component of \(N_Y\), or is disjoint from its support. This proves the asserted separation and the local identities (9). The equality \(N_Y|_{S_Y}\equiv0\) gives \(S_Y|_{S_Y}\equiv0\).

Finally, all models share the stated open \(U_X\). Resolve the graph of the correspondences among all the models, keeping this smooth open unchanged, to obtain \(V_0\). Composing (14) with \(N=\mu^*N_0\) proves (10). Pullback preserves this equality on every further common resolution. ◻

Proof. The pair \((Y,S_Y)\) is log canonical since \((Y,G_Y)\) is log canonical and \(G_Y\geq S_Y\). Take a crepant dlt modification extracting only log canonical places. Its boundary \(\Sigma\) consists of the strict transform of \(S_Y\) and its exceptional divisors, all with coefficient one. Every extracted divisor has center contained in \(S_Y\): outside \(S_Y\) the pair is the klt variety \(Y\). Such a divisor has positive order on \(S_Y\), which proves (15). There are no exceptional divisors off \(S_Y\); since \(Y\) is \(\mathbb Q\)-factorial, the small-morphism argument in the preceding proof shows that \(f\) is an isomorphism there.

Crepancy and (9) give (16). Every curve in \(\Sigma\) maps to a point or to a curve in \(S_Y\), where \(S_Y\) is numerically trivial. The projection formula proves the final numerical assertion. Under the identification in Proposition 6, \(U_X\) is disjoint from \(S_Y\) and hence is unchanged by \(f\). ◻

Proof. We reproduce the proof because both the finite exceptional set and the nonreduced divisor will occur in our application. Replace \(Z\) by \(Z\cap\operatorname{Supp}E\). At \(z\in Z\), write \(R=\mathcal O_{T,z}\), with maximal ideal \(\mathfrak m\), and let \(t_z\) be a local equation of \(E\). Normality gives \[H^1_{\mathfrak m}(R)=0, \qquad \ell_R\bigl(H^2_{\mathfrak m}(R)\bigr)<\infty.\] Here the second assertion uses the dimension-three hypothesis. Indeed, present \(R=A/I\) with \(A\) regular local of dimension \(n\) and essentially of finite type over the field. Each nonmaximal localization of \(R\) is normal of dimension at most two, hence Cohen–Macaulay. Catenarity gives \(\dim A_{\mathfrak q}-\dim R_{\mathfrak p}=n-3\) for a prime \(\mathfrak q\) above \(\mathfrak p\); the Auslander–Buchsbaum formula therefore gives projective dimension \(n-3\) over \(A_{\mathfrak q}\). Thus \(\operatorname{Ext}^{n-2}_A(R,A)\) is supported only at the closed point. It is a finite module, so it has finite length; local duality gives the assertion for \(H^2_{\mathfrak m}(R)\).

Choose \(j\geq1\) such that \(t_z^{j-1}\) annihilates \(H^2_{\mathfrak m}(R)\) at every point of \(Z\). A single \(j\) works because \(Z\) is finite. For every integer \(u\), the restriction sequences give a commutative diagram \[\begin{tikzcd}[column sep=small] 0\arrow[r]&\mathcal O_T((u-j)E)\arrow[r]\arrow[d] &\mathcal O_T(uE)\arrow[r]\arrow[d,equal] &\mathcal O_{jE}(uE)\arrow[r]\arrow[d]&0\\ 0\arrow[r]&\mathcal O_T((u-1)E)\arrow[r] &\mathcal O_T(uE)\arrow[r] &\mathcal O_E(uE)\arrow[r]&0. \end{tikzcd}\] In local trivializations the left vertical map is multiplication by \(t_z^{j-1}\). Local cohomology with support in the finite set \(Z\) is the direct sum of the corresponding local groups. The vanishing of the ambient \(H^1_Z\) makes the connecting maps from the \(H^1_Z\) of the rightmost terms to the ambient \(H^2_Z\) injective. Naturality and our choice of \(j\) therefore imply \[ H^1_Z\bigl(T,\mathcal O_{jE}(uE)\bigr) \longrightarrow H^1_Z\bigl(T,\mathcal O_E(uE)\bigr) \quad\text{is zero for every }u\in\mathbb Z. \tag{17}\] The same \(j\) works for every twist: locally the invertible sheaves in the diagram are free of rank one, and the annihilating map is still multiplication by \(t_z^{j-1}\).

Now fix this thickening \(jE\). The ideal of \(E_{\mathrm{red}}\setminus Z\) in \(jE\setminus Z\) is nilpotent. Lift a chosen generator of \(\mathcal O_T(E)|_{E_{\mathrm{red}}\setminus Z}\) to local generating sections on a finite open cover of \(jE\setminus Z\). On overlaps the lifts differ by sections in that nilpotent ideal times the line bundle. For \(u=p^e\) sufficiently large, their \(u\)th tensor powers agree: in local frames this is the identity \((a+b)^{p^e}=a^{p^e}+b^{p^e}\), with \(b^{p^e}=0\). They consequently glue to a generator \[s_j\in H^0\bigl(jE\setminus Z,\mathcal O_{jE}(uE)\bigr).\] Let \(s\) be its restriction to \(E\setminus Z\). The obstruction to extending \(s\) over \(E\) is the image of the obstruction to extending \(s_j\) over \(jE\). Equation (17), applied to the exact sequences for sections with support in \(Z\), kills that image. Hence \(s\) extends to a section of \(\mathcal O_E(uE)\).

This extended section has no zeros. On each integral projective surface component \(F\) of \(E_{\mathrm{red}}\), it is nonzero because it is a generator off \(Z\). If its restriction to \(F\) vanished anywhere, its zero scheme would be a nonempty effective Cartier divisor on \(F\), and would have positive intersection with an ample divisor. This contradicts the numerical triviality of \(\mathcal O_F(uE)\). The argument uses only integrality of \(F\), not normality. The section is therefore a generator on the reduction, and then on \(E\), since an element invertible modulo a nilpotent ideal is invertible. Thus \(\mathcal O_E(uE)\simeq\mathcal O_E\). ◻

Proof. The relations furnished by isolation are \[ K_{Y_d}+\Sigma\sim_{\mathbb Q}\tau s f^*S_Y \quad\text{on a neighborhood of }\Sigma, \qquad f^*S_Y|_{\Sigma_i}\equiv0. \tag{18}\] Let \(\sigma:\Sigma'\to\Sigma\) be the finite \(S_2\)-ification. It is unchanged away from a finite set \(Z\subset\Sigma\) [24]. Reduced-boundary adjunction gives an effective rational divisor \(\Delta'\) such that \((\Sigma',\Delta')\) is a projective semi-log-canonical surface pair and \[ K_{\Sigma'}+\Delta' \sim_{\mathbb Q}\sigma^*((K_{Y_d}+\Sigma)|_\Sigma) \sim_{\mathbb Q}\tau s\,\sigma^*(f^*S_Y|_\Sigma). \tag{19}\] We use adjunction in the form of [24]; see also [17]. The \(S_2\)-ification is needed because the reduced boundary itself need not satisfy \(S_2\). Its codimension-one singularities are nodes, and on its normalization the adjunction boundary includes the conductor with coefficient one; the normalized pair is log canonical. These are precisely the semi-log-canonical conditions needed for surface abundance.

The surface \(\Sigma'\) is projective because it is finite over \(\Sigma\). The last class in (19) is numerically trivial, by (18) and the projection formula for curves. Posva’s abundance theorem for projective semi-log-canonical surfaces in positive characteristic therefore makes \(K_{\Sigma'}+\Delta'\) semiample [18]. A semiample numerically trivial rational line bundle is \(\mathbb Q\)-linearly trivial: a globally generated Cartier multiple defines a morphism contracting every curve, hence constant on each connected component. Its line bundle is trivial there.

Since \(\tau s\) is positive and rational, clearing denominators in (19) yields a positive Cartier multiple \(M\) of \(f^*S_Y\) with \(\sigma^*\mathcal O_\Sigma(M)\) trivial. In particular, \[\mathcal O_{Y_d}(M)|_{\Sigma\setminus Z} \simeq\mathcal O_{\Sigma\setminus Z}.\] The divisor \(M\) is effective, its reduction is \(\Sigma\), and it is numerically trivial on every component of its support. Applying Lemma 8 on the normal projective threefold \(Y_d\) proves torsion on the full Cartier scheme \(M\). ◻

Proof. Equation (3) and Proposition 5 give the hypotheses of Proposition 6. Choose its isolated model \((Y,G_Y,N_Y,S_Y)\) and the reduced dlt modification \(f:(Y_d,\Sigma)\to(Y,S_Y)\) of Corollary 7. Let \(M\) be the positive Cartier multiple of \(f^*S_Y\) supplied by Proposition 9.

The canonical section and its support. Let \(V_0\) be a smooth projective common resolution of \(X\) and \(Y_d\), chosen to dominate the intervening models. Write \[q:V_0\longrightarrow X,\qquad h_0:V_0\longrightarrow Y_d, \qquad r=f\circ h_0:V_0\longrightarrow Y.\] These models share the open set \[U_X=X_{\mathrm{reg}}\setminus (\operatorname{Supp}N_0\cup P_{\mathrm{out}}),\] where \(P_{\mathrm{out}}\) is the union of the boundary components disjoint from \(\operatorname{Supp}N_0\). We choose the common resolution to be an isomorphism over this open. Projective resolution and resolution of divisor supports in dimension three permit this choice [6, 4]. The pullback equalities through the partial MMP give \[ q^*N_0=r^*N_Y. \tag{20}\]

The integer \(m\) was chosen so that \(N_0\sim m(K_X+B)\) and both \(mK_X\) and \(mB\) are Cartier. It gives a rational section \(\theta\) of \(\omega_{V_0}^{\otimes m}\) with \[ \operatorname{div}(\theta) =q^*N_0+m(K_{V_0}-q^*K_X)-m q^*B. \tag{21}\] Indeed, the right side is an integral divisor linearly equivalent to \(mK_{V_0}\). Put \(E_0=\operatorname{Supp}q^*N_0\). Every prime component in (21) is either contained in \(E_0\) or disjoint from it. For \(q^*B\) this follows from boundary separation on \(X\). For a \(q\)-exceptional divisor, its irreducible center lies in \[X\setminus U_X =\operatorname{Supp}N_0\cup P_{\mathrm{out}} \cup\operatorname{Sing}X.\] A terminal threefold is regular in codimension two, so \(\operatorname{Sing}X\) is finite [17]. The center is therefore contained in \(\operatorname{Supp}N_0\), or is disjoint from it. Taking inverse images gives the assertion for the exceptional divisor.

Let \(F_0\) be the union of the components of \(\operatorname{div}(\theta)\) outside \(E_0\). Thus \[ Z_0:=E_0\cup\operatorname{Supp}\operatorname{div}(\theta) =E_0\sqcup F_0. \tag{22}\] The disjointness is the point of this description. Components of \(q^*N_0\) can cancel in (21); we retain all of \(E_0\) when choosing the open set preserved by the subsequent resolution.

Removing the prime-to-\(p\) exponent. Write \(m=p^b m'\) with \(b\geq0\) and \((m',p)=1\). Choose a nonzero rational generator \(\eta\) of \(\omega_{V_0}^{\otimes p^b}\) and write \(\theta=u\eta^{m'}\), where \(u\in k(V_0)^\times\). Adjoin one root \(v\) of \(v^{m'}=u\) and normalize \(V_0\) in the resulting field. This is a finite separable cover, possibly of degree smaller than \(m'\). Over \(V_0\setminus\operatorname{Supp}\operatorname{div}(\theta)\), it is an integral component of the torsor of \(m'\)th roots of the invertible section \(\theta\). This description is independent of the rational generator \(\eta\). Since \(m'\) is invertible in \(k\), that torsor is finite étale.

Resolve the normalization and the total inverse image of \(Z_0\), preserving the regular étale open over \(V_0\setminus Z_0\). We obtain a smooth projective threefold and a generically finite separable morphism \(\pi:V\to V_0\) for which the total pullback support of \(N_0\) is simple normal crossings. The morphisms used to compare divisors are \[\begin{tikzcd}[column sep=large,row sep=large] V\arrow[r,"\pi"] &V_0\arrow[r,"q"]\arrow[d,"h_0"']&X\\ &Y_d\arrow[r,"f"']&Y. \end{tikzcd}\] Here \(q,h_0,f\) are birational; \(\pi\) is generically finite and separable. Set \(\rho=q\circ\pi\) and \(h=h_0\circ\pi\). By (20), \(\rho^*N_0=(f\circ h)^*N_Y\), while the Cartier divisor supplied by Proposition 9 pulls back as \(h^*M\).

The rational section \(v\eta\) is a section of \(\pi^*\omega_{V_0}^{\otimes p^b}\). The determinant of the differential is a nonzero section of the rational line \(\omega_V\otimes\pi^*\omega_{V_0}^{-1}\); denote it by \(J_\pi\). Its nonvanishing at the generic point follows from separability. Consequently \[ \psi=(v\eta)J_\pi^{p^b} \quad\text{is a rational section of } \omega_V^{\otimes p^b}. \tag{23}\] Both factors are invertible on \(\pi^{-1}(V_0\setminus Z_0)\): the first is a root of an invertible section, and the second is the Jacobian of an étale morphism. Thus the divisor of \(\psi\) is supported on \(\pi^{-1}(Z_0)\). By (22), each of its prime components lies in \(\operatorname{Supp}\rho^*N_0\) or is disjoint from that support.

A connected primitive Cartier divisor. Put \(M_V=h^*M\) and choose a connected component \(C\) of \(\operatorname{Supp}M_V\). Since \(M\) is a positive multiple of \(f^*S_Y\), and \(S_Y\) is disjoint from the other components of \(N_Y\), the set \(C\) is also a connected component of \(\operatorname{Supp}\rho^*N_0\). Write the part of \(M_V\) supported on \(C\) as \[ aD,\qquad D=\sum_\alpha m_\alpha D_\alpha, \qquad a=\gcd_{D_\alpha\subset C} \operatorname{mult}_{D_\alpha}(M_V). \tag{24}\] The divisor \(D\) is Cartier because \(V\) is smooth, and its multiplicities are positive with gcd one. Its support is connected and simple normal crossings.

Every curve in \(\operatorname{Supp}M_V\) maps under \(h\) to a point or to a curve in \(\Sigma\). The projection formula and Proposition 9 therefore show that \(M_V\) is numerically trivial on every component of its support. Near \(C\) we have \(M_V=aD\), so \(D|_{D_\alpha}\equiv0\) for every \(\alpha\). For a curve not contained in \(\operatorname{Supp}D\), effectivity gives nonnegative intersection with \(D\). For a curve in its support, the intersection is zero. Thus \(D\) is nef, and every very ample divisor \(H_V\) satisfies \[ D H_V^2>0,\qquad D^2H_V=0. \tag{25}\]

Torsion also survives on the entire Cartier scheme. Choose \(t>0\) with \(\mathcal O_M(tM)\simeq\mathcal O_M\). The pullback divisor \(M_V\) is scheme-theoretically \(V\times_{Y_d}M\): its local equation is the pullback of a local equation of \(M\), which is a nonzero divisor since \(V\) is integral and \(h\) is dominant. Pulling back the trivialization gives \(\mathcal O_{M_V}(tM_V)\simeq\mathcal O_{M_V}\); no flatness assumption on \(h\) is needed. Restricting along \(D\subset M_V\), and using \(M_V=aD\) near \(C\), gives \[ \mathcal O_D(taD)\simeq\mathcal O_D. \tag{26}\] Hence \(\mathcal O_D(D)\) is torsion.

The section \(\psi\) of (23) has, on a neighborhood \(W\) of \(C\), divisor supported on \(D\). Indeed, its other components are disjoint from \(C\) and can be removed from that neighborhood. Thus there are integers \(b_\alpha\) such that \[ \omega_V^{\otimes p^b}|_W \simeq\mathcal O_W\Bigl(\sum_\alpha b_\alpha D_\alpha\Bigr). \tag{27}\] Numerical proportionality and very general points. It remains to show that no \(D\)-trivial curve passes through a very general point of \(V\). Put \(N_V=\rho^*N_0\). It is a nonzero nef divisor, numerically trivial on every component of its support by the projection formula. Since \(\operatorname{Supp}D\) is contained in that support, we have \[N_VH_V^2>0,\qquad N_V^2H_V=N_VDH_V=0.\] Together with (25), the Hodge index theorem gives \[ D\equiv cN_V,\qquad c=\frac{DH_V^2}{N_VH_V^2}>0. \tag{28}\] We verify that this is numerical equivalence on every curve, rather than only on a fixed hyperplane section, following [17].

Let \(\Gamma\subset V\) be any integral curve. For \(l\gg0\), there is an integral surface \(T\in|lH_V|\) containing \(\Gamma\). To see this, twist the ideal of \(\Gamma\) sufficiently far and then multiply its generators by very ample sections. The resulting linear system has no fixed divisorial component and gives a birational map onto its image off \(\Gamma\). Bertini irreducibility gives an irreducible general member, which is generically reduced on that birational open. As a Cartier divisor on smooth \(V\), it has no embedded components, and is therefore integral. On a smooth projective resolution \(\widetilde T\to T\), let \(d,n,e\) be the pullbacks of \(D|_T,N_V|_T,H_V|_T\), respectively. Their intersections are \[d^2=n^2=dn=0,\qquad e^2>0,\qquad de=lDH_V^2,\quad ne=lN_VH_V^2.\] Hence \(d-cn\) has square zero and is orthogonal to \(e\). The Hodge index form is negative definite on the orthogonal complement of the positive-square class \(e\), so \(d-cn\) is numerically zero. The projective surjection \(\widetilde T\to T\) supplies an integral curve dominating \(\Gamma\): take the closure of a closed point of the fiber over its generic point. The projection formula then gives \((D-cN_V)\cdot\Gamma=0\). This proves (28).

Finally let \(Z_i\subset X\) be the countably many proper closed subsets excluded by the nef reduction of \(N_0\). Remove from \(V\) the non-quasi-finite locus of \(\rho\) and all the sets \(\rho^{-1}(Z_i)\). These are proper closed subsets, since \(\rho\) is dominant and generically finite. Their complement contains very general points because \(k\) is uncountable. Every curve through any point of this complement maps to a curve on \(X\). If it were \(D\)-trivial, then (28) and the projection formula would make its image \(N_0\)-trivial, contradicting maximal nef dimension. All properties in Theorem 2 have now been established. ◻

Proof of Theorem 1. Theorem 2 contradicts Proposition 10. The maximal-nef-dimension case is therefore impossible. By Proposition 4, it remains to apply abundance in nef dimension at most two [23]. This gives semiampleness, and faithful descent along the algebraically closed field extension gives it over the original field. ◻