Shokurov's conjecture on conic bundles with canonical singularities

Abstract

A conic bundle is a contraction X → Z between normal varieties of relative dimension 1 such that −KX is relatively ample. We prove a conjecture of Shokurov which predicts that, if X → Z is a conic bundle such that X has canonical singularities and Z is Q-Gorenstein, then Z is always 1 2 -lc, and the multiplicities of the fibers over codimension 1 points are bounded from above by 2. Both values 1 2 and 2 are sharp. This is achieved by solving a more general conjecture of Shokurov on singularities of bases of lc-trivial fibrations of relative dimension 1 with canonical singularities.

An effective upper bound for anti-canonical volumes of canonical $\mathbb{Q}$-Fano threefolds

“…In this section, we investigate lower bounds of log canonical thresholds on weak del Pezzo surfaces. First we prove 2 useful lemmas on (local) log canonical thresholds, of which we got the ideas during the preparation of [12] (cf. [12,Theorem 1.11]).…”

“…First we prove 2 useful lemmas on (local) log canonical thresholds, of which we got the ideas during the preparation of [12] (cf. [12,Theorem 1.11]). Here in Lemma 3.1 we do not assume that Supp C is smooth as in [12,Theorem 1.11], so these 2 results do not contain each other.…”

“…[12,Theorem 1.11]). Here in Lemma 3.1 we do not assume that Supp C is smooth as in [12,Theorem 1.11], so these 2 results do not contain each other. Also it appears to us that the assumptions in Lemma 3.1 are quite simple, so the statement might be known to experts, but we could not find any similar statement in the literature (cf.…”

“…So after replacing D by D i c i , we may assume that Supp D is irreducible.If Supp D is a line L on S, then D = 3L. Then by[12, Theorem 1.11],…”