Prym loci of branched double coverings and generalized Andreotti-Mayer loci

TL;DR

The paper generalizes the Andreotti-Mayer framework to abelian varieties with non-principal polarization and proves that the Prym locus for branched double coverings, with polarization type $\Delta=(\underbrace{1,\dots,1}_{n-1},\underbrace{2,\dots,2}_{g})$ and dimension $d=g+n-1$, forms an irreducible component of the generalized Andreotti-Mayer locus $\mathcal{N}_{d,n-4}^{\Delta}$. It develops a detailed analysis of the higher base locus $S(P,\mathcal{L})$ of the Prym theta divisor, describes the tangent cones $\mathcal{T}_{\Theta,x}$ via quadratic forms $\mathcal{V}_{E,\mathcal{G}}$, and establishes AM-type dimension bounds by studying kernels of the multiplication map and their generating quadrics. The results show that $\mathcal{P}_{g,2n}$ is an irreducible component of $\mathcal{N}_{d,n-4}^{\Delta}$ for $n\ge4$, with dimension matching that of the Prym locus, thereby extending Schottky-type characterizations from principal to non-principal polarizations and connecting branched Prym loci to generalized Andreotti-Mayer loci.

Abstract

The Andreotti-Mayer locus is a subset of the moduli space of principally polarized abelian varieties, defined by a condition on the dimension of the singular locus of the theta divisor. It is known that the Jacobian locus in the moduli space is an irreducible component of the Andreotti-Mayer locus. In this paper, we generalize the Andreotti-Mayer locus to the case of the moduli space of abelian varieties with non-principal polarization and prove that the Prym locus of branched double coverings is an irreducible component of the generalized Andreotti-Mayer locus.

Theorems & Definitions (34)

• Theorem 1.1: Naranjo-Ortega NO2
• Theorem 1.2: Andreotti-Mayer AM
• Theorem 1.3: Debarre D
• Theorem 1.4
• Proposition 2.1: M2 p334
• Proposition 2.2: NO1 Proposition 1.6.
• Proposition 2.3
• Lemma 2.4
• proof
• Lemma 2.5: M2 p338