Seshadri constants and surfaces of minimal degree

TL;DR

This paper characterizes polarized surfaces where Seshadri constants meet equality conditions, identifying the projective plane and minimal degree surfaces as special cases not fibred by Seshadri curves.

Contribution

It provides a complete classification of surfaces with equality in Seshadri constants, extending previous inequalities to a precise geometric description.

Findings

The projective plane is characterized by equality in Seshadri constants.

Surfaces of minimal degree also satisfy the equality condition.

These surfaces are not fibred by Seshadri curves, unlike the general case.

Abstract

In "Seshadri fibrations of algebraic surfaces" [arXiv:0709.2592v1] we showed that if the multiple point Seshadri constants of an ample line bundle on a smooth projective surface in very general points satisfy certain inequality then the surface is fibred by curves computing these constants. Here we characterize the border case of polarized surfaces whose Seshadri constants in general points fulfill the equality instead of inequality and which are not fibred by Seshadri curves. It turns out that these surfaces are the projective plane and surfaces of minimal degree.