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On abelian varieties the theta divisor of which is singular in codimension 3. (Sur les variétés abéliennes dont le diviseur thêta est singulier en codimension 3.) (French) Zbl 0699.14058

This paper investigates the loci \({\mathcal N}_ k^ g\) in the moduli space \({\mathcal A}_ g\) of principally polarized \(g\)-dimensional abelian varieties \((A,\Theta)\). It consists of the points in \({\mathcal A}_ g\) corresponding to \((A,\Theta)\)’s for which the principal polarization \(\Theta\) satisfies \(\dim(\text{Sing}(\Theta))\geq k\).
Important applications of these loci can e.g. be found in Mumford’s work, who used the divisor \({\mathcal N}^ g_ 0\) consisting of those \((A,\Theta)\) with \(\Theta\) singular to prove that \({\mathcal A}_ g\) is of general type for \(g\geq 7\). Also Andreotti and Mayer used such loci in connection with the Schottky problem; they proved that the locus \(\bar {\mathcal I}_ g\) which is the closure in \({\mathcal A}_ g\) of the set of Jacobians of smooth curves of genus \(g\) is an irreducible component of \({\mathcal N}^ g_{g-4}\) for \(g\geq 4\).
However, it is known that this does not completely characterize Jacobians. Other irreducible components of \({\mathcal N}^ g_{g-4}\) and also irreducible components of some other \({\mathcal N}_ k^ g\)’s are given. For \(g=5\) this allows the author to give a complete description of all irreducible components of \({\mathcal N}^ 5_ 1.\)
In the paper two constructions for describing such components are given: Firstly, one can take special Prym varieties \(P(\tilde C/C)\), which are g-dimensional abelian varieties associated with unramified double covers \(\tilde C\to C\) of curves C of genus \(g-1\). A description for which covers one obtains \(P(\tilde C/C)\) with \(\text{codim(Sing}(\Theta))\leq 4\) is due to Beauville. The necessary information about their image in \({\mathcal A}_ g\) is provided by Donagi.
Secondly, the author uses certain pairs \((A,\Theta)\) in which \(A\) is isogenous to a product of abelian varieties. It turns out that all except two of the components found in terms of Prym varieties are in fact subsets of irreducible components which can be described by such non-simple abelian varieties.

MSC:

14K25 Theta functions and abelian varieties
14K30 Picard schemes, higher Jacobians
14K10 Algebraic moduli of abelian varieties, classification
14H40 Jacobians, Prym varieties
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