A primer on Seshadri constants

TL;DR

This paper provides an overview of Seshadri constants, exploring their development, recent progress, open questions, and examples in the context of local positivity of line bundles on projective varieties.

Contribution

It offers a comprehensive summary of recent advances and open problems in the study of Seshadri constants, highlighting their significance in algebraic geometry.

Findings

Summarizes recent progress in Seshadri constants

Discusses open questions in local positivity

Provides illustrative examples of Seshadri constants

Abstract

Seshadri constants express the so called local positivity of a line bundle on a projective variety. They were introduced by Demailly. The original idea of using them towards a proof of the Fujita conjecture failed but they quickly became a subject of intensive study quite in their own right. Lazarsfeld's book "Positivity in Algebraic Geometry" contains a whole chapter devoted to local positivity and serves as a very enjoyable introduction to Seshadri constants. Since this book has appeared, the subject witnessed quite a bit of development. It is the aim of these notes to give an account of recent progress as well as to discuss many open questions and provide some examples.