New examples of theta divisors for some syzygy bundles

Abstract

Let C be a smooth complex irreducible projective curve of genus g, 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 . The aim of this paper is twofold. Firstly, to give new examples of stable syzygy bundles admitting a theta divisor over Petri curves. We prove that if $$M_L$$ is strictly semistable then $$M_L$$ admits a theta divisor. Secondly, to study the cohomological semistability of $$M_L$$ , and in this direction we give another proof of cohomological semistability of $$M_L$$ when L induces a birational map. This proof gives us precise conditions for the cohomological semistability of $$M_L$$ where such conditions agree with the semistability conditions for $$M_L$$ . Finally, we relate these two properties by showing that under certain conditions on Petri curves the cohomological semistability of $$M_L$$ implies the existence of reducible theta divisor for $$M_L$$ .