Faisceaux sans torsion et faisceaux quasi localement libres sur les courbes multiples primitives

TL;DR

This paper investigates torsion free and quasi locally free coherent sheaves on non reduced primitive multiple curves, establishing their properties, invariants, and implications for moduli spaces of stable sheaves.

Contribution

It introduces the complete type invariant for quasi locally free sheaves and proves the irreducibility of their moduli spaces on non reduced curves.

Findings

Torsion free sheaves are reflexive on non reduced curves.

The complete type invariant classifies quasi locally free sheaves.

Irreducibility of the moduli space of sheaves of fixed complete type.

Abstract

This paper is devoted to the study of some coherent sheaves on non reduced curves that can be locally embedded in smooth surfaces. If Y is such a curve then there is a filtration by subschemes C_i such that C_1 is the reduced curve associated to Y, and that for every P in C, if z is an equation of C_1 in the local ring of Y at P, then (z^i) is the ideal of C_i. A coherent sheaf on Y is called torsion free if it does not have any non zero subsheaf with finite support. We prove that torsion free sheaves are reflexive. We study then the quasi locally free sheaves, i.e. sheaves which are locally isomorphic to direct sums of the structure sheaves of the C_i. We define an invariant for these sheaves, the complete type, and prove the irreducibility of the set of sheaves of given complete type. We study the generic quasi locally free sheaves, with applications to the moduli spaces of stable sheaves on Y.