Minimal models in numerical dimension one

1. Stabilize positive perturbations in codimension two

Choose an effective ample rational divisor $B$ such that $(X,B)$ is klt and $K_X+B$ is ample. Put $t_j=2^{-j}$. Starting from the preceding nef endpoint, run the $(K+t_{j+1}B)$-MMP with scaling of $(t_j-t_{j+1})B$. The transformed boundary remains big and effective; ampleness need not persist. BCHM supplies a finite scaling MMP. Bigness of the adjoint excludes a Mori fibre space, and the base point free theorem applies to the nef and big klt adjoint at each endpoint.

[BCHM10, Corollary 1.4.2, p. 10]

On a scaling ray, $(K+t_{j+1}B)\cdot R<0$ and $(K+t_{j+1}B+\lambda C)\cdot R=0$ imply $B\cdot R>0$ and $K\cdot R<0$. For a small contraction, the relative base point free theorem descends $K-b(K+t_{j+1}B)\sim_{\mathbb Q}h^*A$. With $b>0$, relative ampleness of the flipped adjoint gives relative ampleness of the flipped canonical divisor. Every step is therefore a canonical MMP step.

Lemma 2.3 does not assert termination of the concatenation. Finiteness of divisorial contractions and strict increase of discrepancies first remove codimension-two components of the flipped loci. A decrease in the dimension of the finite-dimensional space spanned by algebraic $(n-2)$-cycle classes then handles the flipping loci. Choose one endpoint $Y$ after this finite prefix. Each later finite composition is an isomorphism outside sets of codimension at least three on both sides. Normality transfers sections, giving generation of $|d_jM_j|$ on $Y\setminus Z_j$. Both $d_j$ and $Z_j$ may depend on $j$.

Proposition 2.1; Lemmas 2.2–2.3 · pp. 4–8

2. Force zero square while keeping the ample twist fixed

Put $L=K_Y$. Terminality makes $Y$ smooth in codimension two. Choose a very general complete intersection surface $S\subset Y$ avoiding every $Z_j$ and $\operatorname{Sing}Y$ simultaneously. Each $M_j|_S$ is semiample, so the limit $L|_S$ is nef and $L^2\cdot H^{n-2}\geq0$.

On a resolution $\pi:\widehat Y\to Y$, put $r=n-2$, $A=\pi^*H$, and fix $P=K_{\widehat Y}+rA+A_0$ with $A_0$ ample Cartier. For each $m$, first choose $j=j(m)$ with $mt_j$ sufficiently small, and then choose a multiple $k$ of $d_j$ with $k>m$. A general $D_m\in|k\pi^*M_j|$ gives $R_m=(m/k)D_m$ whose multiplier ideal is trivial near $S$. The difference $m\pi^*L+P-iA-K_{\widehat Y}-R_m$ is ample for $0\leq i\leq r$. Nadel vanishing and the Koszul complex give surjectivity of restriction to $S$.

[Laz10, Theorem 2.4, p. 11]

Surface Riemann–Roch then gives, along multiples of one integer $d$ with $dL$ Cartier,

The twist $P$ and the coefficients of the error term are independent of $m,j,k$. On a common resolution, the effective difference $u^*K_X-r^*\pi^*L\geq0$ transfers sections to $X$. The fixed Cartier divisor $P_X=u_*r^*P$ is bounded above by one ample divisor $A_X$. A positive square would yield $h^0(X,mK_X+A_X)\geq cm^2$, contrary to (1.1). Hence $L^2\cdot H^{n-2}=0$. No uniform $O(m)$ upper bound is inferred from $\kappa_\sigma=1$.

Lemma 3.1 · pp. 8–10 Section 4; equation (4.1) · pp. 13–14

3. Use a surface through a negative curve to establish nefness

Suppose $L\cdot C_0<0$. Fix a surface component $T$ of a different complete intersection through a suitable point $x\in C_0$. The original very general $S$ can miss $C_0$, which necessitates this second surface. The squares of $M_j$ on all components are nonnegative, and their multiplicity-weighted sum is $M_j^2\cdot H^{n-2}\to0$. Thus $M_j^2\cdot[T]\to0$.

On a resolution, fix a curve through a point $z$ above $x$ that dominates $C_0$. The ambient jet estimate in Lemma 3.2 forces every pulled-back section of $k_jM_j$ to vanish to order at least $ck_j$ at $z$, with $c>0$ fixed. Blow up once on a fixed resolution of $T$. Its exceptional curve $e$ occurs in the fixed divisor $F_j$ with $\operatorname{coeff}_eF_j\geq ck_j$. The Hodge index theorem gives a uniform bound $F_j^2\leq-\epsilon k_j^2$. Nonnegative square of the residual moving system yields

contradicting the zero limit. The construction also covers curves contained in the singular locus. This is the application of Proposition 3.4 to $L=K_Y$.

Lemmas 3.2–3.3; Proposition 3.4 · pp. 10–13

4. Recover all birational conclusions from the finite prefix

Projectivity, $\mathbb Q$-factoriality, terminality, and non-extraction of divisors come from the finite canonical prefix in Proposition 2.1. Sum the effective canonical differences of Lemma 2.2 on a common resolution to obtain $p^*K_X=q^*K_Y+E$. Components dominating prime divisors of $Y$ have coefficient zero, so $E$ is $q$-exceptional. Every prime divisor on $X$ contracted by the map changes from log discrepancy one to a value greater than one on terminal $Y$. Its strict transform therefore has positive coefficient in $E$. This accounts for the remaining conclusions of Theorem 1.1.

Proposition 2.1(iii); Lemma 2.2 · pp. 4–5, 8

1. From a nef to a semiample canonical divisor

The variety $Y$ is normal projective terminal, and $K_Y$ is nef and $\mathbb Q$-Cartier. Thus the ordinary pair $(Y,0)$ is lc with effective rational boundary zero. Log abundance in characteristic zero makes a positive Cartier multiple of $K_Y$ itself globally generated; it does not merely supply a numerically equivalent divisor. The map and canonical comparison constructed in Theorem 1.1 are retained, and the same endpoint is a good minimal model.

[OpenAI26LA, Theorem 1.1, p. 2] Corollary 1.2 · p. 2