Classification of non-$F$-split del Pezzo surfaces of degree $1$

TL;DR

This work classifies non-$F$-split del Pezzo surfaces of degree $1$ in positive characteristic by applying Fedder's criterion to the anti-canonical sextic model in the weighted projective space $\mathbb{P}(1,1,2,3)$. It reduces $F$-splitting to coefficient conditions on the Weierstrass-type equation for the anti-canonical model and derives explicit criteria for $p=5,3,2$, describing the corresponding double covers of $\mathbb{P}(1,1,2)$ and the associated elliptic fibrations. The results yield precise forms for the branch sextic and discriminant divisor, showing, for example, a unique non-$F$-split surface in characteristic $5$ with $\Delta \cong 2\mathbb{P}^1(\mathbb{F}_5)$ and characterizing root configurations that correspond to supersingular members of $|-K_X|$ in other characteristics. The paper also connects non-$F$-splitting with $2$-quasi-$F$-splitting, moduli dimensions, and the finite automorphism groups that enable explicit classification.

Abstract

Using Fedder's criterion, we classify all non-$F$-split del Pezzo surfaces of degree $1$. We give a necessary and sufficient criterion for the $F$-splitting of such del Pezzo surfaces in terms of their anti-canonical system.

Theorems & Definitions (16)

• theorem 1
• theorem 2
• remark 1
• remark 2
• remark 3
• remark 4
• theorem 3
• lemma 1
• lemma 2
• proof