Abelian threefolds in products of projective spaces (Q1912811)

In contrast to the case of abelian surfaces, the construction of abelian threefolds in low dimensional projective spaces is a hard technical problem. In this paper are studied the possibilities to embed an abelian threefold in a product of two projective spaces \(\mathbb{P}^d \times \mathbb{P}^{6-d}\), \(d=1,2,3\). The main result are: (a) There are no abelian threefolds in \(\mathbb{P}^1\times \mathbb{P}^5\). (b) Every abelian threefold in \(\mathbb{P}^2 \times\mathbb{P}^4\) is a product of a plane cubic and an abelian surface of degree 10 in \(\mathbb{P}^4\). (c) There exist abelian threefolds in \(\mathbb{P}^3\times\mathbb{P}^3\); moreover, it is given an explicit construction of a 3-dimensional family of such threefolds. Besides the specific in the construction of the family from (c), the approach is based on the technical lemma 1.2 which states, in particular, that if an abelian \(g\)-fold \(X\) admits a morphism onto a projective \(n\)-fold with \(n<g\), then \(X\) surjects an abelian \(n\)-fold. This way, the projections \(X\to\mathbb{P}^d\) and \(X\to \mathbb{P}^{6-d}\) impose strong restrictions on the structure of the abelian 3-fold \(X\).