Rational maps from products of curves to surfaces with $p_g = q = 0$

TL;DR

This paper investigates the degree of irrationality of products of curves mapping to surfaces with zero geometric genus and irregularity, establishing exact values under certain conditions.

Contribution

It provides new results on the degree of irrationality for products of curves, especially hyperelliptic ones, under genericity and genus assumptions.

Findings

Degree of irrationality equals product of gonalities for certain curve products.

Degree of irrationality of product of two hyperelliptic curves is 4.

Results apply to surfaces with $p_g = q = 0$ under mild assumptions.

Abstract

We study dominant rational maps from a product of two curves to surfaces with $p_{g} = q = 0$. Given two curves which satisfy a mild genericity assumption and have large genus relative to their gonality, we show that the degree of irrationality of their product is equal to the product of their gonalities. Moreover, we prove that the degree of irrationality of a product of two hyperelliptic curves is 4.