Aspects of Brill-Noether geometry in moduli theory of algebraic and tropical curves

Abstract

The aim of this thesis is to develop some possible generalizations of the Brill-Noether theory of smooth curves, to singular curves and to tropical curves. We briefly summarize the contents hereafter: 1) let Alpha*d/X be the Abel map of multidegree d of a singular curve X of genus g. We describe the closure of ImAlpha*d/X inside Caporaso’s compactified Jacobian PdX for irreducible curves, curves of compact type and binary curves; 2) we construct the moduli spaces of tropical curves and tropical principally polarized abelian varieties, working in the category of (what we call) stacky fans. We define the tropical Torelli map between these two moduli spaces and we study the fibers (tropical Torelli theorem) and the image of this map (tropical Schottky problem). Finally we determine the image of the planar tropical curves via the tropical Torelli map and we use it to give a positive answer to a question raised by Namikawa on the compactified classical Torelli map; 3) we give some results on quadratic normality of reducible curves canonically embedded and partially extend this study to their projective normality; 4) we study nodal curves with two components investigating about sufficient and necessary conditions in order for them to be k-gonal.