Weak del Pezzo surfaces with global vector fields

Abstract

We classify smooth weak del Pezzo surfaces with global vector fields over an arbitrary algebraically closed field $k$ of arbitrary characteristic $p\ge 0$. We give a complete description of the configuration of $(-1)$- and $(-2)$-curves on these surfaces and calculate the identity component of their automorphism schemes. It turns out that there are $53$ distinct families of such surfaces if $p\ne 2,3$, while there are $61$ such families if $p=3$, and $75$ such families if $p=2$. Each of these families has at most one moduli. As a byproduct of our classification, it follows that weak del Pezzo surfaces with non-reduced automorphism scheme exist over $k$ if and only if $p\in \{2,3\}$.

50 pages, 67 figures, comments welcome, v2: added Case 4F

References in corpus (1)

• Infinitesimal automorphisms of algebraic varieties and vector fields on elliptic surfaces

Cited by in corpus (2)

• Which rational double points occur on del Pezzo surfaces?
• Del Pezzo surfaces with infinite automorphism groups