Results on an extended Torelli map and singularities of degenerate abelian varieties

Abstract

The Torelli map associates to a smooth genus g projective curve a g-dimensional principally polarized abelian variety and is a map on the respective moduli spaces Mg -> Ag. There is a somewhat canonical compacti fication of Mg, the Deligne-Mumford compactifi cation, but there are many natural compacti fications of Ag. In this thesis we consider two toroidal compacti fications, the central cone compacti fication and the second Voronoi compactification. Speci fically, we answer the following two questions: Does the Torelli map extend to a regular map to the central cone compactification in genera 7 and 8? What singularities occur on pairs in the image of the extended Torelli map to the second Voronoi compactification? The fi rst question is important because genera 7 and 8 represented the only remaining cases in which this particular extension question was unknown. The result, which is a product of joint work, is that the map does extend in these cases. The second question seeks to extend the 1995 result of Koll ar which states that principally polarized abelian pairs are log canonical. In this thesis we show that in fact all pairs in the boundary of the second Voronoi compactification are semi-log canonical, the analog of log canonical in the non-normal setting.