Seshadri constants of the anticanonical divisors of Fano manifolds with large index

TL;DR

This paper calculates the Seshadri constants of anticanonical divisors on certain Fano manifolds, revealing conditions under which these constants are bounded by 1, thus advancing understanding of local positivity in algebraic geometry.

Contribution

It provides explicit computations of Seshadri constants for Fano manifolds with coindex at most 3 and characterizes when these constants are at most 1 for very general Fano threefolds.

Findings

Seshadri constants computed for Fano manifolds with coindex ≤ 3

Boundedness of Seshadri constants linked to base point freeness of |-K_X|

Characterization of Fano threefolds with Seshadri constant ≤ 1

Abstract

Seshadri constants, introduced by Demailly, measure the local positivity of a nef divisor at a point. In this paper, we compute the Seshadri constants of the anticanonical divisors of Fano manifolds with coindex at most $3$ at a very general point. As a consequence, if $X$ is a nonsingular Fano threefold which is very general in its deformation family, then $\varepsilon(X,-K_X;x)\leq 1$ for all points $x\in X$ if and only if $\vert-K_X\vert$ is not base point free.