Vector bundles with Theta divisors I: Bundles on Castelnuovo curves

TL;DR

This paper proves that semistable vector bundles on Castelnuovo curves of genus at least 2 possess theta divisors, and extends this result to certain semistable bundles on smooth curves via degenerations.

Contribution

It establishes the existence of theta divisors for semistable vector bundles on Castelnuovo curves and extends this to general curves through degenerations.

Findings

Semistable bundles on Castelnuovo curves have theta divisors.

Extension of theta divisor existence to general curves via degenerations.

Results applicable for genus g >= 2.

Abstract

In this paper we show that semistable vector bundles on a Castelnuovo curve of genus g >= 2 have theta divisors. As a corollary, we deduce that semistable vector bundles on a smooth, general curve of genus g >= 2 which extend to semistable vector bundles on any Castelnuovo degeneration of the general curve admit a theta divisor.