On rational maps from the product of two general curves

TL;DR

This paper investigates the existence of dominant rational maps from the product of two very general curves to surfaces, proving non-existence of such maps of degree greater than one under certain conditions.

Contribution

It extends previous results by combining known theorems to show that certain products of general curves do not admit non-trivial dominant rational maps to non-ruled surfaces.

Findings

No dominant rational maps of degree > 1 from product of two very general curves to non-ruled surfaces for specified genera.

The result applies to the 2-symmetric product of a curve.

Provides conditions under which such rational maps cannot exist.

Abstract

This paper treats the dominant rational maps from the product of two very general curves to nonsingular projective surfaces. Combining the result by Bastianelli and Pirola, we prove that the product of two very general curves of genus $g\geq 7$ and $g'\geq 3$ does not admit dominant rational maps of degree $> 1$ if the image surface is non-ruled. We also treat the case of the 2-symmetric product of a curve.