## Tokyo Journal of Mathematics

### Characterization of Non-F-split del Pezzo Surfaces of Degree 2

Natsuo SAITO

#### Abstract

We investigate a smooth del Pezzo surface of degree 2 which is not Frobenius split. We give a characterization of non-F-split del Pezzo surfaces of degree 2 which exist only if the characteristic of the ground field is 2 or 3. Moreover, we prove that the set of centers of the blow-ups on $\mathbb{P}^{2}$ which gives a non-F-split del Pezzo surface is projectively equivalent to the only complete 7-arc over $\mathbb{F}_{9}$ if the characteristic is 3.

#### Article information

Source
Tokyo J. Math., Volume 40, Number 1 (2017), 247-253.

Dates
First available in Project Euclid: 8 August 2017

https://projecteuclid.org/euclid.tjm/1502179226

Digital Object Identifier
doi:10.3836/tjm/1502179226

Mathematical Reviews number (MathSciNet)
MR3689989

Zentralblatt MATH identifier
06787098

Subjects
Primary: 14J26: Rational and ruled surfaces

#### Citation

SAITO, Natsuo. Characterization of Non-F-split del Pezzo Surfaces of Degree 2. Tokyo J. Math. 40 (2017), no. 1, 247--253. doi:10.3836/tjm/1502179226. https://projecteuclid.org/euclid.tjm/1502179226

#### References

• P. Cragnolini and P. A. Oliverio, On the proof of Castelnuovo's rationality criterion in positive characteristic, J. Pure and Applied Algebra 68 (1990), 297–323.
• M. Demazure, Surfaces de del Pezzo II-V, Lecture Notes in Mathematics 777, Springer, 1980, 23–69.
• I. Dolgachev, Finite subgroups of the plane Cremona group, Algebraic Geometry in East Asia-Seoul 2008, Adv. St. in Pure Math. 60 (2010), 1–49.
• I. Dolgachev, Classical Algebraic Geometry: A Modern View, Cambridge University Press, 2012.
• N. Hara, A characterization of rational singularities in terms of injectivity of Frobenius maps, Amer. J. Math. 120 (1998), 981–996.
• J. W. P. Hirschefeld, Projective geometries over finite fields, Oxford Science Publications, 1998.
• M. Homma, A combinatorial characterization of the Fermat cubic surface in characteristic 2, Geometriae Dedicata 64 (1997), 311–318.
• V. B. Mehta and A. Ramanathan, Frobenius Splitting and Cohomology Vanishing for Schubert Varieties, Annals of Mathematics 122 (1985), 27–40.
• K. Schwede and K. E. Smith, Globally F-regular and log Fano varieties, Adv. Math. 224 (2010), 863–894.
• T. Shioda, Arithmetic and Geometry of Fermat Curves, Proc. Algebraic Geometry Seminar, Singapore, World Scientific, 1987, 95–102.
• K. E. Smith, Globally F-regular varieties: applications to vanishing theorems for quotients of Fano varieties, Michigan Math. J. 48 (2000), 553–572.
• K.-O. Stöhr and J. F. Voloch, Weierstrass points and curves over finite fields, Proc. London Math. Soc. 52 (1986), 1–19.