Singularities on toric fibrations

Abstract

In this paper we investigate singularities on toric fibrations. In this context we study a conjecture of Shokurov (a special case of which is due to M Kernan which roughly says that if is an -lc Fano-type log Calabi-Yau fibration, then the singularities of the log base are bounded in terms of and where and are the discriminant and moduli divisors of the canonical bundle formula. A corollary of our main result says that if is a toric Fano fibration with being -lc, then the multiplicities of the fibres over codimension one points are bounded depending only on and . Bibliography: 20 titles.

Shokurov's conjecture on conic bundles with canonical singularities

“…Hence Conjecture 1.5 under assumption (2') holds when the coefficients of B| F are bounded from below away from zero as a consequence of the Borisov-Alexeev-Borisov conjecture proved by Birkar [Bir19,Bir21]. Very recently, Birkar and Y. Chen [BC21] proved Conjecture 1.5 under assumption (2') for toric morphisms between toric varieties. We refer the readers to [Bir18, Theorems 1.9 and 2.5] for more related results.…”

“…The bounds in Theorem 1.10 are optimal by Example 4.1. Y. Chen informed us that together with Birkar, they also got the lower bound 1 2 in Theorem 1.10 for toric morphisms between toric varieties in an earlier version of [BC21]. As a related result, when dim X − dim Z = 2, Mori and Prokhorov [MP09] showed that any 3dimensional terminal del Pezzo fibration has no fibers of multiplicity > 6.…”

Shokurov’s conjecture on conic bundles with canonical singularities

“…Hence, under assumption (2’), Conjecture 1.5 holds when the coefficients of are bounded from below away from zero as a consequence of the Borisov–Alexeev–Borisov conjecture proved by Birkar [7, 8]. Very recently, Birkar and Y. Chen [10] proved Conjecture 1.5 under assumption (2’) for toric morphisms between toric varieties. We refer the reader to [6, Theorems 1.9 and 2.5] for more related results. Following ideas in [5], it is indicated by [11, Proposition 7.6] (see [13, Theorem 1.10] for an embryonic form) that Conjecture 1.5 might be a consequence of Shokurov’s -lc complements conjecture.…”

“…Hence, under assumption (2’), Conjecture 1.5 holds when the coefficients of are bounded from below away from zero as a consequence of the Borisov–Alexeev–Borisov conjecture proved by Birkar [7, 8]. Very recently, Birkar and Y. Chen [10] proved Conjecture 1.5 under assumption (2’) for toric morphisms between toric varieties. We refer the reader to [6, Theorems 1.9 and 2.5] for more related results.…”

“…Y. Chen informed us that together with Birkar, they also got the lower bound in Theorem 1.10 for toric morphisms between toric varieties in an earlier version of [10]. As a related result, when , Mori and Prokhorov [32] showed that any 3-dimensional terminal del Pezzo fibration has no fibres of multiplicity .…”