2022
DOI: 10.1017/fms.2022.32
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Shokurov’s conjecture on conic bundles with canonical singularities

Abstract: A conic bundle is a contraction $X\to Z$ between normal varieties of relative dimension $1$ such that $-K_X$ is relatively ample. We prove a conjecture of Shokurov that predicts that if $X\to Z$ is a conic bundle such that X has canonical singularities and Z is $\mathbb {Q}$ -Gorenstein, then Z is always $\frac {1}{2}$ -lc, and the mul… Show more

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Cited by 5 publications
(6 citation statements)
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“…By the above claim, there exists a prime divisor C 0 ⊆ Supp f * π * z but C 0 Supp G. Hence G is very exceptional over Z (see [Bir12 Proof of Theorem 1.6. The case when ε = 1 is solved in [HJL22,Theorem 1.4]. So we may suppose that ε < 1.…”
Section: Proofs Of Main Theoremsmentioning
confidence: 99%
See 3 more Smart Citations
“…By the above claim, there exists a prime divisor C 0 ⊆ Supp f * π * z but C 0 Supp G. Hence G is very exceptional over Z (see [Bir12 Proof of Theorem 1.6. The case when ε = 1 is solved in [HJL22,Theorem 1.4]. So we may suppose that ε < 1.…”
Section: Proofs Of Main Theoremsmentioning
confidence: 99%
“…When d = 3 and X has only terminal singularities, Mori and Prokhorov [MP08] proved that Z is 1-lc. When dim X − dim Z = 1 and ε = 1, Han, Jiang and Luo [HJL22] proved that Z is 1 2 -lc. For toric fibrations, this conjecture was confirmed by Alexeev and Borisov [AB14].…”
Section: Introductionmentioning
confidence: 99%
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“…Thus it is reasonable to assume dim X − dim Z = d rather than dim X = d (although the conjecture was commonly stated assuming the latter). See [32] for other variations.…”
Section: Introductionmentioning
confidence: 99%