Genus 2 curves and generalized theta divisors

Abstract

In this paper we investigate generalized theta divisors Θr in the moduli spaces UC(r, r) of semistable vector bundles on a curve C of genus 2. We provide a desingularization Φ of Θr in terms of a projective bundle π : P(V) → UC (r − 1, r) which parametrizes extensions of stable vector bundles on the base by OC . Then, we study the composition of Φ with the well known theta map θ. We prove that, when it is restricted to the general fiber of π, we obtain a linear embedding.

Nodal curves and polarizations with good properties

“…Remark 3. 8 We point out that, if C is of compact type, the family of connected curves {A j }, defined in Definition 3.6, can be used to obtain the conditions of w-stability in [25] (see also Remark 2.14).…”

“…These spaces are interesting by themselves as higher dimensional varieties but also for important related constructions: just to mention some, one can consider higher-rank Brill-Noether theory, Theta divisors and Theta functions and the moduli spaces of coherent systems. For surveys on these topics see, for example [3,6,7]; for some results by the authors see [5,8,[11][12][13][14]. When the curve is singular, these spaces are not in general complete.…”

On kernel bundles over reducible curves with a node

“…In particular, their stability properties, which have been studied with respect to different point of view (see, for instance, [EL92,Mis08]) are closed related to higher rank Brill-Noether theory and moduli spaces of coherent systems (see [BGMN03,BBN08,BB12] for example). Finally, they have been useful in studying theta divisors and the geometry of moduli space of vector bundles on curves (see [Bea03,Pop07,Bri18,BF19a], for example).…”

Nodal curves and polarizations with good properties

“…These spaces are interesting by themselves as higher dimensional varieties but also for important related constructions: just to mention some, one can consider higher-rank Brill-Noether theory, Theta divisors and Theta functions and the moduli spaces of coherent systems. For surveys on these topics see, for example, [Bra09], [BGPMN03] and [Bea13]; for some results by the authors see [BF19a], [BF20b], [Bri15], [Bri17], [BB12] and [BV12]. When the curve is singular, these spaces are not in general complete.…”