Singularities of theta divisors and the geometry of \(\mathcal A_5\). (English) Zbl 1373.14043

Farkas, G.; Grushevsky, S.; Salvati Manni, R.; Verra, Alessandro

J. Eur. Math. Soc. (JEMS) 16, No. 9, 1817-1848 (2014).

Inside the moduli space \(\mathcal{A}_g\) of principally polarized abelian varieties (ppav’s), the condition that the theta divisor of a ppav be singular is of codimension \(1\). The locus where this holds is the well-known Andreotti-Mayer divisor \(N_0\). It consists of two irreducible components \(N_0 = \theta_{\mathrm{null}} \cup N_0'\), depending on whether the singularity occurs at a \(2\)-torsion point or not. The subloci of these components where the singularity is not an ordinary double point are denoted by \(\theta_{\mathrm{null}}^{g-1}\) and \({N_0'}^{g-1} =: H\), respectively. The authors explicitly compute the class \([H] \in CH^2(\mathcal{A}_g)\), and furthermore show that \(\theta_{\mathrm{null}}^{g-1} \subsetneq H\) for \(g \geq 5\), while \(\theta_{\mathrm{null}}^3 = H\) if \(g = 4\). It was previously known that \(\theta_{\mathrm{null}}^{g-1} \subset \theta_{\mathrm{null}} \cap N_0'\).
The authors then proceed to investigate the situation more concretely in genus \(g = 5\). Here it is shown that \(H\) decomposes into two irreducible components \(H = \theta_{\mathrm{null}}^4 \cup H_1\), both of which are characterized as images under the Prym map \(P: \mathcal{R}_6 \to \mathcal{A}_5\) of loci of Prym curves satisfying certain explicit conditions. This description enables the authors to conclude that in fact both components are unirational.
As an application of these results, the authors are able to determine the slope \(s(\overline{\mathcal{A}}_5) = 54/7\) of the perfect cone compactification of \(\mathcal{A}_5\) (in fact the slope is the same for all toroidal compactifications). The slope is attained by \(\overline{N_0'}\), and this divisor is shown to be rigid ({i.e.}, its Kodaira-Iitaka dimension is \(0\)).

Reviewer: Fabian Müller (Berlin)

MSC:

14K10	Algebraic moduli of abelian varieties, classification
14J10	Families, moduli, classification: algebraic theory
14K12	Subvarieties of abelian varieties
14H40	Jacobians, Prym varieties
14K25	Theta functions and abelian varieties
14H10	Families, moduli of curves (algebraic)

Keywords:

theta divisor; singularities; moduli space of prinicipally polarized abelian varieties; Prym variety; Andreotti-Mayer divisor; slope

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