Geometry of genus 1 fine compactified Jacobians
A classical construction in algebraic geometry associates with every nonsingular complex projective curve its Jacobian, a complex projective variety of dimension equal to the genus of the curve. A similar construction is available for singular curves, but the resulting Jacobian variety fails in general to be compact. In this talk we introduce a general abstract notion of fine compactified Jacobian for nodal curves of arbitrary genus.
We focus on genus 1 and discuss combinatorial classification results for fine compactified Jacobians in the case of a single stable curve, and in the case of the universal family over the moduli space of stable pointed curves. In the former case, our abstract notion finds back objects that had already been constructed by Oda-Seshadri and others. In the latter case our complete classification exhibits new examples. We then discuss how to calculate the cohomology of these compactified Jacobians.
A joint work with Orsola Tommasi.