Shokurov's conjecture on conic bundles with canonical singularities. (9th June 2022)

• Record Type: Journal Article
• Authors: Han, Jingjun; Jiang, Chen; Luo, Yujie
• Is Part Of: Forum of mathematics. Volume 10(2022)
• Publication Date: 2022-06-09
• Subjects: Conic bundles -- canonical bundle formula -- log canonical threshold -- minimal model program; 14J17 -- 14E30 -- 14C20; Mathematics -- Periodicals; 510
• DOI: 10.1017/fms.2022.32
• Languages: English
• ISSNs: 2050-5094

Abstract

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.