Weak del Pezzo surfaces with global vector fields

Martin, Gebhard; Stadlmayr, Claudia

doi:10.2140/gt.2024.28.3565

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Weak del Pezzo surfaces with global vector fields

Gebhard Martin and Claudia Stadlmayr

Geometry & Topology 28 (2024) 3565–3641

DOI: 10.2140/gt.2024.28.3565

Abstract

We classify smooth weak del Pezzo surfaces with global vector fields over an arbitrary algebraically closed field $k$ of arbitrary characteristic $p \geq 0$ . 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 $p \neq 2, 3$ , 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 nonreduced automorphism schemes exist over $k$ if and only if $p \in {2, 3}$ .

Keywords

del Pezzo surfaces, vector fields, automorphisms, group schemes, positive characteristic

Mathematical Subject Classification

Primary: 14E07, 14J26, 14J50, 14L15

References

Publication

Received: 28 April 2021

Revised: 26 April 2022

Accepted: 9 June 2022

Published: 20 December 2024

Proposed: Mark Gross

Seconded: Dan Abramovich, Jim Bryan

Authors

Gebhard Martin
Mathematisches Institut Universität Bonn Bonn Germany

Claudia Stadlmayr
Zentrum Mathematik Technische Universität München Munich Germany

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Volume 28, issue 8 (2024)