14 - Coherent Systems on a Nodal Curve

Summary

Abstract

Let E denote a torsionfree coherent sheaf of rank n, degree d and V ⊂ H⁰(E) be a subspace of dimension k on a nodal curve X. We show that for k ≤ n the moduli space of coherent systems (E, V) which are stable for sufficiently large values of a real parameter stabilizes. We study the nonemptiness and properties like irreducibility, smoothness, seminormality for this moduli space G_L.

Introduction

Coherent systems on smooth curves have been studied and are being studied extensively ([BG], [BOMN], [KN], [LN1], [LN2], [He], to name a few). A brief survey of coherent systems on smooth curves appears in this volume [Br]. In this paper, we initiate the study of coherent systems on a nodal curve. A coherent system on a nodal curve X of arithmetic genus g is a pair (E, V) where E denotes a torsionfree coherent sheaf of rank n, degree d on X and V ⊂ H⁰(E) is a subspace of dimension k. The (semi) stability condition for coherent systems depends on a real parameter α > 0. It is easy to see that if (E, V) is α-semistable then d ≥ 0. If (E, V) is α-stable, then for k ≥ n one has d > 0 and for k ≥ n one has d > 0 except in case (E, V) = (O, H⁰(O)).