Weak del Pezzo surfaces with global vector fields

Martin, Gebhard¹ ; Stadlmayr, Claudia²

¹ Mathematisches Institut, Universität Bonn, Bonn, Germany
² Zentrum Mathematik, Technische Universität München, Munich, Germany

Geometry & topology, Tome 28 (2024) no. 8, pp. 3565-3641.

Résumé

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 nonreduced automorphism schemes exist over $k$ if and only if $p\in \{2,3\}$.

Détail

DOI : 10.2140/gt.2024.28.3565

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

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