The Abel-Jacobi map of the space of conics for double sextic threefolds

TL;DR

This paper studies the Abel-Jacobi map for a specific class of threefolds, showing it induces an isogeny between the Albanese variety of a surface of certain curves and the intermediate Jacobian of the threefold.

Contribution

It establishes a new relationship between the Albanese variety and the intermediate Jacobian via the Abel-Jacobi map for double sextic threefolds.

Findings

The Abel-Jacobi map induces an isogeny between the Albanese and intermediate Jacobian.

The result applies to a general class of double sextic threefolds.

Provides new insights into the geometry of curves on threefolds.

Abstract

Let $X$ be a double cover of $\mathbb P^3$ branched along a sextic surface $Y$. In this paper, we show that, for general $X$, the Abel-Jacobi map associated to the normalization $\tilde F(X)$ of the surface $F(X)$ of curves contained in $X$ which are preimages of lines bitangent to $Y$, gives an isogeny between the Albanese variety of $\tilde F(X)$ and the intermediate Jacobian of $X$.