A characterization of the symmetric square of a curve

TL;DR

This paper provides a new intrinsic geometric characterization of the symmetric square of a curve and the product of two curves, linking surface properties to these classical constructions.

Contribution

It introduces a novel geometric criterion that identifies when a surface is birational to a product or a symmetric square of a curve.

Findings

Surfaces with certain divisor properties are birational to products or symmetric squares.

The characterization applies to surfaces of general type with irregularity q.

Provides a new perspective on the geometry of surfaces related to curves.

Abstract

In this paper a new intrinsic geometric characterization of the symmetric square of a curve and of the ordinary product of two curves is given. More precisely it is shown that the existence on a surface of general type S of irregularity q of an effective divisor D having self-intersection D^2>0 and arithmetic genus q implies that S is either birational to a product of curves or to the second symmetric product of a curve.