Remarks on Ein-Lazarsfeld criterion of spannedness of adjoint bundles of polarized threefolds

TL;DR

This paper refines conditions under which the adjoint bundle K+B of a polarized threefold is globally generated at a point, improving upon previous bounds established by Ein-Lazarsfeld.

Contribution

It provides a weaker set of numerical criteria ensuring the spannedness of K+B on threefolds, extending prior results with less restrictive assumptions.

Findings

K+B is spanned at a point under new numerical conditions

Improves previous bound from B^3 ≥ 92 to B^3 ≥ 51

Corollary: K+3L is spanned if L is ample with L^3 > 1

Abstract

Let B be a nef and big line bundle on a smooth complex threefold X with canonical bundle K. Let x be a point on X and suppose that BC\ge3 for any curve C passing x, B^2S\ge7 for any surface S containing x, and B^3\ge51. Then K+B is spanned at x. (Ein-Lazarsfeld proved the assertion assuming B^3\ge92.) Corollary: K+3L is spanned if L is an ample line bundle with L^3>1.