Deligne-Lusztig varieties whose canonical divisors have negativity

TL;DR

This paper studies special Deligne-Lusztig varieties with negative canonical divisors, providing explicit descriptions in dimension two and developing a new framework that includes Suzuki-Ree cases.

Contribution

It introduces a novel general framework for Deligne-Lusztig varieties that works over prime fields and encompasses Suzuki-Ree cases, with explicit geometric descriptions.

Findings

In dimension two, describes supersingular K3 surfaces with Artin invariant 1 in characteristic two.

Identifies particular ruled surfaces associated with supersingular elliptic curves and the Ree curve.

Develops a new framework relying on the Isogeny Theorem, applicable over prime fields and including Suzuki-Ree cases.

Abstract

We investigate compactified Deligne-Lusztig varieties whose canonical divisor, when expressed as a linear combination of boundary divisors, has all coefficients strictly negative or zero. In dimension two we obtain explicit descriptions: The case $C_2$ gives the supersingular K3 surface with Artin invariant $\sigma=1$ in characteristic two. The arguments rely on properties of the Tutte-Coxeter graph; in this connection we also gain some insight into the arithmetic of quasi-elliptic Weierstrass equations and rational double points. The twisted cases ${}^2A_2$ and ${}^2C_2$ and ${}^2G_2$ yield particular ruled surfaces, attached in a canonical way to supersingular elliptic curves in characteristic two, or the Ree curve in characteristic three. The latter is a curve of genus fifteen with outstanding symmetries. In the former cases, the surfaces can also be expressed as symmetric squares. To obtain such results, we develop a new general framework for Deligne-Lusztig varieties that relies on the Isogeny Theorem, and works over prime fields rather than their algebraic closures, and includes the notorious Suzuki-Ree cases without effort.