On discriminants and incidence resolutions

TL;DR

This paper investigates the incidence complex associated with morphisms of sheaves, establishing conditions under which it forms a resolution of the ideal sheaf, and explores the properties of discriminants in linear systems on the projective line.

Contribution

It introduces the incidence complex as a potential resolution of the ideal sheaf and proves it is a resolution when the scheme is Cohen-Macaulay, linking discriminants to classical polynomial discriminants.

Findings

Incidence complex is a local complete intersection when the sheaf morphism is surjective.

The incidence complex provides a resolution of the ideal sheaf for Cohen-Macaulay schemes.

Relates discriminants of linear systems on the projective line to classical polynomial discriminants.

Abstract

In this paper we study the incidence complex of an arbitrary morphism of locally free sheaves relative to an arbitrary quasi compact morphism of schemes. We prove it is a local complete intersection in the case when the sheaf morphism is surjective. We construct a complex - the incidence complex - which is a candidate for a resolution of the ideal sheaf of the incidence scheme. When the initial scheme is Cohen-Macaulay we prove the incidence complex is a resolution. We also study the rational points of the incidence scheme and discriminant scheme of any linear system on the projective line over any field of characteristic zero. We use this study to relate the discriminant to the classical discriminant of degree d polynomials.