On extra-special Enriques surfaces
We refine Cossec and Dolgachev's classification of extra-special Enriques surfaces, providing a complete and concise proof.
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arXiv subjects
Gebhard Martin
Publications and source records attributed to Gebhard Martin.
Automorphism groups of rational elliptic and quasi-elliptic surfaces in all characteristics
We study the groups of automorphisms of rational algebraic surfaces that admit a relatively minimal pencil of curves of arithmetic genus one over an algebraically closed field of arbitrary characteristic. In particular, we classify such surfaces that admit non-trivial automorphisms that act trivially on the Picard group. As an application, we classify classical Enriques surfaces in characteristic $2$ that admit non-trivial numerically trivial automorphisms.
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Weak del Pezzo surfaces with global vector fields
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 non-reduced automorphism scheme exist over $k$ if and only if $p \in \{2,3\}$.
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Numerically trivial automorphisms of Enriques surfaces in characteristic $2$
An automorphism of an algebraic surface $S$ is called cohomologically (numerically) trivial if it acts identically on the second $l$-adic cohomology group (this group modulo torsion subgroup). Extending the results of S. Mukai and Y. Namikawa to arbitrary characteristic $p > 0$, we prove that the group of cohomologically trivial automorphisms $\rm{Aut}_{\rm{ct}}(S)$ of an Enriques surface $S$ is of order $\leq 2$ if $S$ is not supersingular. If $p = 2$ and $S$ is supersingular, we show that $\rm{Aut}_{\rm{ct}}(S)$ is a cyclic group of odd order $n\in \{1,2,3,5,7,11\}$ or the quaternion group $Q_8$ of order $8$ and we describe explicitly all the exceptional cases. If $K_S \neq 0$, we also prove that the group $\rm{Aut}_{\rm{nt}}(S)$ of numerically trivial automorphisms is a subgroup of a cyclic group of order $\leq 4$ unless $p = 2$, where $\rm{Aut}_{\rm{nt}}(S)$ is a subgroup of a $2$-elementary group of rank $\leq 2$.
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Automorphisms of unnodal Enriques surfaces
It follows from an observation of A. Coble in 1919 that the automorphism group of an unnodal Enriques surface contains the $2$-congruence subgroup of the Weyl group of the $E_{10}$-lattice. In this article, we determine how much bigger the automorphism group of an unnodal Enriques surface can be. Furthermore, we show that the automorphism group is in fact equal to the $2$-congruence subgroup for generic Enriques surfaces in arbitrary characteristic (under the additional assumption that the Enriques surface is ordinary if the characteristic is $2$), improving the corresponding result of W. Barth and C. Peters for very general Enriques surfaces over the complex numbers.
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Classification of Enriques surfaces with finite automorphism group in characteristic 2
We classify supersingular and classical Enriques surfaces with finite automorphism group in characteristic 2 into 8 types according to their dual graphs of all $(-2)$-curves (nonsigular rational curves). We give examples of these Enriques surfaces together with their canonical coverings. It follows that the classification of all Enriques surfaces with finite automorphism group in any characteristics has been finished.
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Enriques surfaces with finite automorphism group in positive characteristic
We classify Enriques surfaces with smooth K3 cover and finite automorphism group in arbitrary positive characteristic. The classification is the same as over the complex numbers except that some types are missing in small characteristics. Moreover, we give a complete description of the moduli of these surfaces. Finally, we realize all types of Enriques surfaces with finite automorphism group over the prime fields $\mathbb{F}_p$ and $\mathbb{Q}$ whenever they exist.
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