Shokurov’s conjecture on conic bundles with canonical singularities

Abstract

AbstractA 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 thatXhas canonical singularities andZis$\mathbb {Q}$-Gorenstein, thenZis 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.