New examples of reducible theta divisors for some Syzygy bundles

TL;DR

This paper constructs new examples of reducible theta divisors for certain stable syzygy bundles on general curves and investigates their cohomological semistability, providing conditions for semistability related to the linear series.

Contribution

It introduces new reducible theta divisor examples for stable syzygy bundles and analyzes their cohomological semistability under specific geometric conditions.

Findings

Strictly semistable syzygy bundles admit reducible theta divisors.

Cohomological semistability of $M_L$ when $L$ induces a birational map.

Conditions for semistability of $M_L$ align with geometric properties of $L$.

Abstract

Let $C$ be a smooth complex irreducible projective curve of genus $g$ with general moduli, and let $(L,H^0(L))$ be a generated complete linear series of type $(d,r+1)$ over $C$. The syzygy bundle, denoted by $M_L$, is the kernel of the evaluation map $H^0(L)\otimes\mathcal O_C\to L$. In this work we have a double purpose. The first one is to give new examples of stable syzygy bundles admitting theta divisor over general curves. We prove that if $M_L$ is strictly semistable then $M_L$ admits reducible theta divisor. The second purpose is to study the cohomological semistability of $M_L$, and in this direction we show that when $L$ induces a birational map, the syzygy bundle $M_L$ is cohomologically semistable, and we obtain precise conditions for the cohomological semistability of $M_L$ where such conditions agree with the semistability conditions for $M_L$.