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We study the cone of moving divisors on the moduli space |${\mathcal{A}}_{g}$| of principally polarized abelian varieties. Partly motivated by the generalized Rankin–Cohen bracket, we construct a non-linear holomorphic differential operator that sends Siegel modular forms to Siegel modular forms, and we apply it to produce new modular forms. Our construction recovers the known divisors of minimal moving slope on |${\mathcal{A}}_{g}$| for |$g\leq 4$|⁠ , and gives an explicit upper bound for the moving slope of |${\mathcal{A}}_{5}$| and a conjectural upper bound for the moving slope of |${\mathcal{A}}_{6}$|⁠. [ABSTRACT FROM AUTHOR]

In this note we prove a sharp bound for the number of 2-torsion points on a theta divisor and show that this is achieved only in the case of products of elliptic curves. This settles in the affirmative a conjecture of Marcucci and Pirola. [ABSTRACT FROM AUTHOR]

In this paper we give a structure theorem for the module of vector valued modular forms in the case of a three dimensional ball with the action of the Picard modular group G3[ v -3]. The corresponding modular variety of dimension 3 is a copy of the Segre cubic. [ABSTRACT FROM AUTHOR]

Extending previous methods, we prove three structure theorems for vector-valued modular forms, where two correspond to 4-dimensional cases (two Hermitian modular groups, one belonging to the field of Eisenstein numbers and the other to the field of Gaussian numbers) and one to a 6-dimensional case (a quaternionic modular group). [ABSTRACT FROM AUTHOR]

In a previous paper, we described some Siegel modular threefolds which admit a Calabi-Yau model. Using a different method we give in this paper an enlarged list of such varieties. Basic for this method is a paper of van Geemen and Nygaard. They study a variety X that is the complete intersection of four quadrics in ℙ7 (ℂ). This is biholomorphic equivalent to the Satake compactification of H2/Γ' for a certain subgroup Γ' ⊂ Sp(2, ℤ) and it will be the starting point of our investigation. It has been pointed out that a (projective) small resolution of this variety is a rigid Calabi-Yau manifold .... Then we will consider the action of quotients of modular groups on X and study possible resolutions that admits a Calabi-Yau model in the category of complex spaces. [ABSTRACT FROM AUTHOR]

We study the cone of moving divisors on the moduli space |${\mathcal{A}}_{g}$| of principally polarized abelian varieties. Partly motivated by the generalized Rankin–Cohen bracket, we construct a non-linear holomorphic differential operator that sends Siegel modular forms to Siegel modular forms, and we apply it to produce new modular forms. Our construction recovers the known divisors of minimal moving slope on |${\mathcal{A}}_{g}$| for |$g\leq 4$|⁠ , and gives an explicit upper bound for the moving slope of |${\mathcal{A}}_{5}$| and a conjectural upper bound for the moving slope of |${\mathcal{A}}_{6}$|⁠. [ABSTRACT FROM AUTHOR]

In this note we study the geometry of principally polarized abelian varieties (ppavs) with a vanishing theta-null (i.e. with a singular point of order two and even multiplicity lying on the theta divisor)—denote by θnull the locus of such ppavs. We describe the locus θnullg−1⊂θnull where this singularity is not an ordinary double point. By using theta function methods we first show θnullg−1⊊θnull (this was shown in [4], see below for a discussion). We then show that θnullg−1 is contained in the intersection θnull∩N0′ of the two irreducible components of the Andreotti-Mayer N0=θnull+2N0′, and describe by using the geometry of the universal scheme of singularities of the theta divisor which components of this intersection are in θnullg−1. [ABSTRACT FROM PUBLISHER]