Non-simple abelian varieties and (1; 3) Theta divisors

Abstract

This thesis studies non-simple Jacobians and non-simple abelian varieties. The motivation of the study is a construction which gives a distinguished genus 4 curve in the linear system of a (1, 3)-polarised surface. The main theorem characterises such curves
as hyperelliptic genus 4 curves whose Jacobian contains a (1, 3)-polarised surface.

This leads to investigating the locus of non-simple principally polarised abelian g-folds. The main theorem of this part shows that the irreducible components of this locus are Is g d, defined as the locus of principally polarised g-folds having an abelian subvariety with induced polarisation of type d = (d1, . . . , dk), where k ≤g/2. Moreover, there are theorems which characterise the Jacobians of curves that are ´etale double covers or double covers branched in two points.

There is also a detailed computation showing that, for p > 1 an odd number, the hyperelliptic locus meets Is4 (1,p) transversely in the Siegel upper half space.

Date of Award	21 Nov 2012
Original language	English
Awarding Institution	University of Bath
Supervisor	Gregory Sankaran (Supervisor)