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.
Highlights
A common approach to understand the geometry of the moduli space of principally polarised abelian g-folds, denoted Ag, is to use ideas coming from the geometry of curves
Proposition 5 is generalised in Theorem 7 which, roughly speaking, says that if the Jacobian of a curve contains an abelian subvariety of half the dimension and the type of the restricted polarisation is twice the principal polarisation, there is a double cover of curves that yields the Jacobian and the subvariety
The following conditions are equivalent: 1. there exists an elliptic curve E ⊂ A such that H |E is of type p; 2. there exists an exact sequence
Summary
A common approach to understand the geometry of the moduli space of principally polarised abelian g-folds, denoted Ag, is to use ideas coming from the geometry of curves. That is possible because of the Torelli theorem, which says that the Jacobian completely characterises the curve. Many geometric constructions from the theory of curves give rise to interesting constructions in the theory of Jacobians. One remarkable construction is the Prym construction, which gives a subvariety of a Jacobian for any finite cover of curves. Every cover of curves f : C −→ C induces a pullback map f ∗ : J C −→ J C. J C is a non-simple abelian variety, as it contains im f ∗ and the complementary abelian subvariety, called the Prym variety of the cover
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