Lower bounds for Seshadri constants via successive minima of line bundles

TL;DR

This paper improves the lower bounds for Seshadri constants at very general points on projective varieties by employing the concept of successive minima for line bundles, advancing understanding in algebraic geometry.

Contribution

It introduces a refined lower bound for Seshadri constants using successive minima, surpassing previous bounds and connecting geometric invariants with line bundle properties.

Findings

Lower bound for Seshadri constants is larger than (d+1)^{1/d - 1}.

Utilizes successive minima concept to derive bounds.

Enhances previous lower bounds of 1/d for Seshadri constants.

Abstract

Given a nef and big line bundle $L$ on a projective variety $X$ of dimension $d \geq 2$, we prove that the Seshadri constant of $L$ at a very general point is larger than $(d+1)^{\frac{1}{d}-1}$. This slightly improves the lower bound $1/d$ established by Ein, K\"uchle and Lazarsfeld. The proof relies on the concept of successive minima for line bundles recently introduced by Ambro and Ito.