Pathologies of Hilbert scheme of points of supersingular Enriques surface

TL;DR

This paper investigates the unique properties of Hilbert schemes of points on supersingular Enriques surfaces in characteristic 2, revealing their complex structure and deviations from classical Hodge number formulas.

Contribution

It demonstrates that these Hilbert schemes are simply connected symplectic varieties that are not irreducible symplectic, highlighting new phenomena specific to characteristic 2.

Findings

Hilbert schemes are simply connected symplectic varieties

They are not irreducible symplectic due to high Hodge number h^{2,0}

Examples of varieties with trivial canonical class that are neither irreducible symplectic nor Calabi-Yau

Abstract

We show that Hilbert schemes of points on supersingular Enriques surface in characteristic 2 are simply connected, symplectic varieties but are not irreducible symplectic as the hodge number $h^{2,0} > 1$, even though a supersingular Enriques surface is an irreducible symplectic variety. These are the classes of varieties which appear only in characteristic 2 and they show that the hodge number formula for G\"ottsche-Soergel does not hold over characteristic 2. It also gives examples of varieties with trivial canonical class which are neither irreducible symplectic nor Calabi-Yau, thereby showing that there are strictly more classes of simply connected varieties with trivial canonical class in characteristic 2 than over $\mathbb{C}$ as given by Beauville-Bogolomov decomposition theorem.