Semiampleness of nef big line bundles on projective K trivial threefolds

Use the growth definition of bigness: limsup h⁰(X,rL)/r³>0. Choose a fixed very ample Cartier H. On H choose an ample Cartier B such that B−L|H has a section nonvanishing on every associated component, by the ample global-generation criterion F3. Its r-th power embeds $O_H(rL)$ into $O_H(rB)$, so the Hilbert-polynomial supplier and Serre vanishing give h⁰(H,rL|H)=O(r²). For arbitrarily large r the cubic growth of h⁰(X,rL) exceeds that bound. The divisor restriction sequence then gives a nonzero section of rL−H, hence an effective G with rL linearly equivalent to H+G. This is the needed decomposition, proved without semiampleness or an unstated stronger vanishing interface. Choose a sufficiently small positive rational eta so (X,B0), B0=eta G, is klt. To specify that terminology here, on a log resolution write $K_Y+B_{0,Y}=\pi \ast (K_X+B0)$; every coefficient of $B_{0,Y}$ is $<1$. The finitely many inequalities hold for small eta because $K_X$ is smooth and the relative canonical multiplicities of smooth-center blowups are nonnegative. We have B0 equivalent_Q eta rL−eta H, $aL-(K_X+B0)$ equivalent_Q (a−eta r)L+eta H, which is ample for $a\ge \eta$ r by [F4]. All choices are rational.

The closed base sets of |k! L| decrease and therefore stabilize, by noetherianity. Bigness ensures nonempty linear series for sufficiently divisible degrees. Choose m0 so the stable base locus B equals Bs|m0L|. For every positive multiple of m0 its base locus is then exactly B: it is contained in Bs|m0L| by products of sections, and contains the stable base locus by definition. If B is empty the conclusion already holds. Assume B nonempty.

Resolve the base ideal of |m0L| together with G. Denote the fixed divisor on this resolution by F, so pi*|m0L|=|M|+F with |M| free and F effective. Choose a very large integer N and general members D1,...,DN of |m0L|, and put $D=(D1+\mathrm{.}\mathrm{.}\mathrm{.}+DN)\mathrm{/}N$. Their strict moving transforms are smooth and meet each other and the fixed SNC support transversely, after a further smooth-center log resolution if needed. For a divisor E in the fixed/exceptional support with pi(E) in B, put $b_E={\mathrm{coeff}}_E(B_{0,Y})<1$ and $f_E={\mathrm{ord}}_E(F)>0$. The finite minimum $c0={\min }_{f_E>0}(1-b_E)\mathrm{/}{f}_E$ is positive. Choose N so large that $c0\mathrm{/}N<1$ and all intersections outside B remain klt: there the only new components are general transverse moving divisors of coefficients c/N, and the fixed boundary is already klt. Hence the first coefficient reaching one in $B_Y(c)=\pi \ast (B0+cD)-(K_Y-\pi \ast K_X)$ occurs at $c=c0$ on a divisor whose center lies in B. This is a finite coefficient calculation on the chosen SNC resolution, not an assumption of lc-center nonvanishing.

We break ties while making the adjoint difference ample even on exceptional fibres. For a smooth-center blowup its negative exceptional divisor is relatively ample. Inducting through the composition, with sufficiently large integer weights on the pulled-back earlier divisors, gives an effective exceptional Q-divisor P0 with −P0 pi-ample. Add arbitrarily small positive rational independent coefficients on the finitely many other SNC components and perturb the exceptional coefficients slightly. Relative ampleness is open, so a resulting effective Q-divisor P can be chosen with −P pi-ample and generic coefficient ratios on the finitely many candidate components. If pi is an isomorphism, the relative ampleness condition is vacuous and one simply uses generic positive coefficients on those components. For small rational $\delta >0$ the adjusted threshold $c_{\delta }={\min }_{f_E>0}[1-b_E-\delta {\mathrm{coeff}}_E(P)]\mathrm{/}{f}_E$ is positive, close to c0, and is attained on a UNIQUE component E. Coefficients with $f_E=0$ stay $<1$ after taking delta small. The moving components outside B also stay $<1$ by the choice of N and smallness of delta. Thus ${\mathrm{\Delta }}_Y=B_Y(c_{\delta })+\delta$ P has exactly one coefficient equal to one, that of a smooth divisor E with pi(E) contained in B; all other coefficients are $<1$. Strict nonexceptional coefficients are nonnegative. Negative coefficients, if any, occur only on exceptional divisors. Set $A=\mathrm{ceil}(-{\mathrm{\Delta }}_Y^{<1})$, Gamma={Delta_Y}. Then A is an effective integral exceptional divisor not containing E, Gamma is effective SNC with coefficients $<1$ not containing E, and Delta_Y=E+Gamma−A. After possibly decreasing delta, eta pi*H−delta P is ample on Y. Indeed −P is relatively ample; tensor a relatively very ample power with a sufficiently high pullback of H, use its relative projective embedding and Serre's criterion, and divide by the power. This is an ordinary ample-bundle construction, not the sought nef contraction.

Write $L_Y=\pi \ast L$. It is curve-nef. For every sufficiently large integer m, and $M_m=m{L}_Y+A$, calculation gives $M_m-E-K_Y-\mathrm{\Gamma }=m{L}_Y-(K_Y+{\mathrm{\Delta }}_Y)$ equivalent_Q $(m-c_{\delta }m0-\eta r)L_Y+\eta {\pi }^{\ast }H-\delta P$. The right side is ample by [F4] and step4.1. Apply the ample-SNC vanishing [F1] to the integral line bundle $M_m-E$ on Y. It gives $H^1(Y,M_m-E)=0$, hence a SURJECTIVE restriction $H0(Y,M_m)$ -> $H0(E,M_m\mathrm{\mid }E)$. Adjunction on the smooth divisor E, and the same equality restricted to E, show $M_m\mathrm{\mid }E-K_E-\mathrm{\Gamma }\mathrm{\mid }E$ is ample. The SNC property makes Gamma|E an effective SNC fractional boundary with coefficients $<1$. Therefore [F1] also gives $H^q(E,M_m\mathrm{\mid }E)=0$ for all $q>0$ and all such m.

We prove $H0(E,M_m\mathrm{\mid }E)$ is nonzero for all sufficiently large m by an explicit surface calculation. Put $D_E=L_Y\mathrm{\mid }E$ and $A_E=A\mathrm{\mid }E$. The latter is effective (the divisor A does not contain E). Choose one large integer a such that $Q=A_E+a{D}_E-K_E-\mathrm{\Gamma }\mathrm{\mid }E$ is ample. Since $D_E$ is nef, its square is nonnegative, either by the attached numerical bridge and taking ample limits or by surface Hodge index. Use the Euler polynomial [F5]. If $D_E^2>0$ its leading term is positive. If $D_E^2=0$ but $D_E$ is not numerically zero, its linear coefficient satisfies $D_E\cdot (A_E-K_E\mathrm{/}2)$ = $(1\mathrm{/}2)D_E\cdot [A_E+(A_E-K_E-\mathrm{\Gamma }\mathrm{\mid }E)+\mathrm{\Gamma }\mathrm{\mid }E]$ = $(1\mathrm{/}2)D_E\cdot [A_E+Q+\mathrm{\Gamma }\mathrm{\mid }E]$ >0. Here subtracting aD_E does not change the product because $D_E^2=0$. The effective divisors contribute nonnegative intersections. The ample Q contributes strictly positively: if $D_E\cdot Q=0$, Hodge index and $D_E^2\ge 0$ force $D_E$ numerically zero. Thus $\chi (M_m\mathrm{\mid }E)>0$ for m large in these two cases, and the higher-cohomology vanishing in step5.1 makes h0 equal to it. If $D_E$ is numerically zero, $A_E-K_E-\mathrm{\Gamma }\mathrm{\mid }E$ is itself ample because it is numerically equivalent to Q. The vanishing [F1] applies already to $A_E$, so $\chi (A_E)=h0(A_E)>0$: the canonical section of the effective divisor $A_E$ is nonzero, including $A_E=0$. The Euler polynomial is constant in m, hence again $h0(M_m\mathrm{\mid }E)=\chi (A_E)>0$ for every sufficiently large m. This covers numerical dimensions two, one and zero without a general lc-center or abundance theorem.

Since A is effective exceptional over the smooth, hence normal, X, $\pi \ast O_Y(A)=O_X$. A rational function regular off the exceptional image has no pole on any codimension-one point of X and is regular by normality. The projection formula therefore identifies H0(Y,mL_Y+A)=H0(X,mL). Choose m sufficiently large AND divisible by m0. Every section of mL vanishes on B, and hence its pullback vanishes along E, since pi(E) is contained in B. Multiplication by the canonical section of A does not remove that vanishing, because E is not a component of A. Thus every section on the left restricts to zero on E. This contradicts the surjectivity in step5.1 and the nonzero target in step6.1. Therefore B was empty. Some multiple of L is globally generated.