On Schoen surfaces

TL;DR

This paper introduces a new construction method for Schoen surfaces, revealing their canonical map properties and geometric structure, including their relation to a specific canonical surface with nodes.

Contribution

It provides a novel construction approach for Schoen surfaces and proves their canonical map is a degree 2 finite morphism onto a specific canonical surface.

Findings

Canonical map of a general Schoen surface is a degree 2 finite morphism.

The canonical surface is a complete intersection of a quadric and a quartic in P^4.

The canonical surface has 40 even nodes.

Abstract

We give a new construction of the irregular, generalized Lagrangian, surfaces of general type with p_g=5, \chi=2, K^2=8, recently discovered by Chad Schoen. Our approach proves that, if S is a general Schoen surface, its canonical map is a finite morphism of degree 2 onto a canonical surface with invariants p_g=5, \chi=6, K^2=8, a complete intersection of a quadric and a quartic hypersurface in P^4, with 40 even nodes.