Non-simple principally polarised abelian varieties

TL;DR

This paper characterizes the structure of non-simple principally polarised abelian varieties, identifying their irreducible components and providing equations for these loci, with applications to Jacobians of certain algebraic curves.

Contribution

It introduces a detailed description of the irreducible components of the non-simple locus and derives Humbert-like equations for these components for any dimension and polarization type.

Findings

Irreducible components are characterized by subvarieties with specific polarizations.

Provides Humbert-like equations for these components across all dimensions.

Characterizes Jacobians of certain double covers of curves.

Abstract

The paper investigates the locus of non-simple principally polarised abelian $g$-folds. We show 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 $D=(d_1,\ldots,d_k)$, where $k\leq\frac{g}{2}$. The main theorem produces Humbert-like equations for irreducible components of $\Is^g_{D}$ for any $g$ and $D$. Moreover, there are theorems which characterise the Jacobians of curves that are \'etale double covers or double covers branched in two or four points.