Preprint C185/2015
On autoduality for treelike curves, Abel maps for stable curves and translations for the compactified Jacobian
Flávio Ferreira da Rocha
Keywords: Autoduality | Abel Map | Compactified Jacobian.

This thesis is about compactified Jacobians of reduced nodal curves, and addresses mainly three issues: the existence of an autoduality theorem for reducible curves, the construction of degree-1 and degree-0 Abel maps in a natural way and the existence of a theory of translations for compactified Jacobians.

Concerning the first issue, we use the autoduality theorem, proved by Arinkin, for integral curves whose singularities are planar in order to show our autoduality theorem for treelike curves whose singularities are planar as well.

Regarding the second issue, the first to construct Abel maps for reducible stable curves were Caporaso and Esteves. However, in order to do it, they used a special type of invertible sheaves called twisters and the compactified Jacobian constructed by Caporaso as target of the Abel maps. In this thesis we show that by putting Simpson's compactified Jacobian as target of the Abel maps, it is possible to construct such maps without to use the twisters.

Finally, relative to the last issue, we exhibit two reducible curves for which one can define, in a non trivial way, an action of the Jacobian parameterizing invertible sheaves on certain compactified Jacobians constructed by Esteves, so that this action induces an isomorphism between any two of these compactified Jacobians.