×

zbMATH — the first resource for mathematics

Unirationality of Enriques surfaces in characteristic two. (English) Zbl 0549.14019
Necessary and sufficient conditions for the unirationality of an Enriques surface over a field of characteristic 2 are given. The result consists in the fact that a non-classical Enriques surface is unirational if and only if it is supersingular. A classical surface is always unirational. In the course of the proof, a consideration of universal coverings of these surfaces takes place and the test of unirationality for K3 surfaces over a field of characteristic 2 is used. The case of characteristic greater than 2 remains open.

MSC:
14M20 Rational and unirational varieties
14J25 Special surfaces
14G15 Finite ground fields in algebraic geometry
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] E. Bombieri and D. Mumford : Enriques’ classification of surfaces in characteristic p > 0, III . Invent. Math 36 (1976) 197-232. · Zbl 0336.14010 · doi:10.1007/BF01390138 · eudml:142405
[2] R. Crew : Thesis , Princeton University, 1981.
[3] E. Kunz : Differentialformen auf algebraischen varietäten mit singularitäten . I. Manuscripta Math. 15 (1975) 91-108. · Zbl 0299.14013 · doi:10.1007/BF01168881 · eudml:154309
[4] E. Kunz : Differentialformen auf algebraischen varietäten mit singularitäten. II . Abh. Math., Seminar Hamburg, 47 (1978), 43-70. · Zbl 0379.14005 · doi:10.1007/BF02941351
[5] T. Katsura : Surfaces unirationnelles en caractéristique p, C. R. Acad. Sc. Paris, Ser. A, t. 288 (1979) 45-47. (Theorem 5). · Zbl 0429.14011
[6] R. Hartshorne : Algebraic Geometry . Graduate Texts in Mathematics, Springer, 1977. · Zbl 0367.14001
[7] J.S. Milne : Étale Cohomology , Princeton University Press, 1980. · Zbl 0433.14012
[8] M. Artin , A. Grothendieck and J.L. Verdier : Théorie des Topos et Cohomologie Étale des Schémas , vol. 2, Lecture Notes in Mathematics 270, Springer Verlag, 1972. · Zbl 0237.00012 · doi:10.1007/BFb0061319
[9] A.N. Rudakov and I.R. Šafarevič : Supersingular K3 surfaces over fields of characteristic two . Math. USSR-Izv. 13 (1979) 147-165. · Zbl 0424.14008 · doi:10.1070/IM1979v013n01ABEH002016
[10] T. Shioda : An example of unirational surfaces in characteristic p . Math. Ann. 211 (1974) 233-236. · Zbl 0276.14018 · doi:10.1007/BF01350715 · eudml:162649
[11] T. Shioda : Some results on unirationality of algebraic surfaces . Math. Ann. 230 (1977) 153-168. · Zbl 0343.14021 · doi:10.1007/BF01370660 · eudml:163036
[12] O. Zariski : An Introduction to the Theory of Algebraic Surfaces . Lecture Notes in Mathematics 83, Springer Verlag, 1969. · Zbl 0177.49001 · doi:10.1007/BFb0082246
[13] O. Zariski and P. Samuel : Commutative Algebra , vol. 2, Van Nostrand, 1960. · Zbl 0121.27801
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.