Discriminants of morphisms of sheaves

TL;DR

This paper introduces a unified geometric framework for defining and analyzing discriminants of morphisms of sheaves in algebraic geometry, generalizing classical concepts and establishing irreducibility results.

Contribution

It provides a comprehensive geometric definition of discriminants for morphisms of locally free sheaves, extending classical discriminants to broader contexts including flag varieties.

Findings

Discriminant of any linear system on a flag variety is irreducible.

Unified geometric definition encompasses various classical discriminants.

Analysis of discriminants in multiple geometric situations.

Abstract

The aim of this paper is to give a unified definition of a large class of discriminants arising in algebraic geometry using the discriminant of a morphism of locally free sheaves. The discriminant of a morphism of locally free sheaves has a geometric definition in terms of grassmannian bundles, tautological sequences and projections and is a simultaneous generalization of the discriminant of a morphism of schemes, the discriminant of a linear system on a smooth projective scheme and the classical discriminant of degree $d$ polynomials. We study the discriminant of a morphism in various situations: The discriminant of a finite morphism of schemes, the discriminant of a linear system on the projective line and the discriminant of a linear system on a flag variety. The main result of the paper is that the discrimiant of any linear system on any flag variety is irreducible.