On GIT Compactified Jacobians via Relatively Complete Models and Logarithmic Geometry

Abstract

In this thesis we study modular compactifications of Jacobian varieties attached to nodal curves.<br /> Unlike the case of smooth curves, where the Jacobians are canonical, modular compact objects, these compactifications are not unique.<br /> Starting from a nodal curve <i>C</i>, over an algebraically closed field, we show that some celatively complete models, constructed by Mumford, Faltings and Chai, associated with a smooth degeneration of <i>C</i>, can be interpreted as moduli space for particular logarithmic torsors, on the universal formal covering of the formal completion of the special fiber of this degeneration. We show that these logarithmic torsors can be used to construct torsion free sheaves of rank one on <i>C</i>, which are semistable in the sense of Oda and Seshadri. This provides a "uniformization" for some compactifications of Oda and Seshadri without using methods coming from Geometric Invariant Theory.<br /> Furthermore these torsors have a natural interpretation in terms of the relative logarithmic Picard functor. We give a representability result for this functor and we show that the maximal separated quotient contructed by Raynaud is a subgroup of it.