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 multiplicities of the fibres over codimension $1$ points are bounded from above by $2$ . Both values $\frac {1}{2}$ 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.

Explicit bound for singularities on the base of a Fano type fibration of relative dimension one

“…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.…”

“…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].…”

“…(5) When dim X − dim Z = 1 and ε 1, Han, Jiang and Luo [HJL22] proved Conjecture 1.2 and gived the optimal value of δ = ε − 1/2.…”

“…Following the ideals in [HJL22], we may reduce Theorem 1.4 to the case that π : X → Z is a P 1 bundle over a smooth curve, and finally reduce to a local problem on estimating the log canonical threshold of a smooth curve with respect to a germ of smooth surface pair. We prove the following theorem, using Newton diagrams.…”

Singularities on toric fibrations

“…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.…”