Automorphismen von Enriques Flächen
Zusammenfassung der Projektergebnisse
A central topic of algebraic geometry is the automorphism group Aut(X) of an algebraic variety X, that is, the set of algebraic and bijective self-maps of X. In particular, it is already interesting to know in which situations these groups are infinite and when they are finite. If X is a curve, then the automorphism group is controlled by its Kodaira dimension κ(X): If κ(X) = −∞, then X is the projective line P1 and then, Aut(X) is infinite. If κ(X) = 0, then this group is infinite (when considered as a genus one curve) or finite (when considered as a pointed genus one curve, that is, an elliptic curve). If κ(X) = 1, that is, if X is a curve of general type, then this group is finite and explicit bounds are known, such as Hurwitz’ bound in characteristic zero. If X is a surface, then much less is known. For example, Aut(X) is finite if κ(X) = 2, that is, if X is a surface of general type. If κ(X) = −∞, then X is birationally ruled over a curve, and the automorphism group depends on the curve and the birational ruling over this curve. One of the most interesting classes of surfaces are minimal surfaces with κ(X) = 0. By the Enriques-Kodaira classification, these fall into four classes: Abelian surfaces; K3 surfaces; Bielliptic surfaces; Enriques surfaces. The objective of the DFG project Automorphisms of Enriques surfaces was the classification of Enriques surfaces with finite automorphism groups. Over the complex numbers, this was already achieved by Shigeyuki Kond¯ in 1986: he o showed that there exist seven classes of such surfaces, two of which come in onedimensional families and five of which correspond to unique surfaces. This project dealt with the classification of Enriques surfaces with finite automorphism groups over fields of positive characteristic. For this, new methods had to be developed, since one of the main tools in the complex case, namely the Torelli theorem, is not (yet) available in positive characteristic. All these obstacles have been overcome by my Ph.D. student Gebhard Martin, who was funded by this DFG Sachbeihilfe, and his coworkers, who achieved a full classification, determined the occurring automorphism groups, and computed the moduli dimensions of these classes. The resulting classification is similar to Kondō’s, especially in large characteristic p. More precisely, (1) if p ≥ 7, then there again exist seven types, two of which come in onedimensional families and five of which correspond to unique Enriques surfaces. (2) if 5 ≥ p ≥ 3, then some of the seven types do not exist, but whenever they do, the situation is an in p ≥ 7 or p = 0. (3) if p = 2, then there is still a finite list of classes: three classes in the singular case, eight classes in the classical case, and five classes in the supersingular case. The list here is very different from all the other characteristics.
Projektbezogene Publikationen (Auswahl)
- Automorphisms of Enriques surfaces. Dissertation, München, Technische Universität München, 2018, 160 S.
G. Martin
- Enriques surfaces with finite automorphism group in positive characteristic.
Algebraic Geometry, Vol. 6. 2019, Issue 5, pp. 592–649.
G. Martin
(Siehe online unter https://doi.org/10.14231/AG-2019-027) - Numerically trivial automorphisms of Enriques surfaces in characteristic 2. Journal of the Mathematical Society of Japan, Vol. 71. 2019, Number 4, pp. 1181-1200.
I. Dolgachev, G. Martin
(Siehe online unter https://doi.org/10.2969/jmsj/78867886) - Classification of Enriques surfaces with finite automorphism group in characteristic 2. Algebraic Geometry, Vol. 7. 2020, Issue 4, pp. 390–459.
Katsura, Toshiyuki; Kondō, Shigeyuki; Martin, Gebhard
(Siehe online unter https://doi.org/10.14231/AG-2020-012)
