## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Higher Dimensional Algebraic Geometry: In honour of Professor Yujiro Kawamata's sixtieth birthday, K. Oguiso, C. Birkar, S. Ishii and S. Takayama, eds. (Tokyo: Mathematical Society of Japan, 2017), 265 - 278

### Lefschetz pencils on a certain hypersurface in positive characteristic

#### Abstract

We examine Lefschetz pencils of a certain hypersurface in ${\bf P}^{3}$ over an algebraically closed field of characteristic $p > 2$, and determine the group structure of sections of the fiber spaces derived from the pencils. Using the structure of a Lefschetz pencil, we give a geometric proof of the unirationality of Fermat surfaces of degree $p^a + 1$ with a positive integer $a$ which was first poved by Shioda [10]. As byproducts, we also see that on the hypersurface there exists a $(q^{3} + q^{2} + q + 1)_{q + 1}$-symmetric configuration (resp. a $((q^{3} + 1)(q^{2} + 1)_{q + 1}, (q^{3} + 1)(q + 1)_{q^{2} + 1}$)-configuration) made up of the rational points over ${\bf F}_{q}$ (resp. over ${\bf F}_{q^{2}}$) and the lines over ${\bf F}_{q}$ (resp. over ${\bf F}_{q^{2}}$) with $q = p^{a}$.

#### Article information

**Dates**

Received: 3 May 2013

Revised: 26 October 2013

First available in Project Euclid:
23 October 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1540319491

**Digital Object Identifier**

doi:10.2969/aspm/07410265

**Mathematical Reviews number (MathSciNet)**

MR3791217

**Zentralblatt MATH identifier**

1388.14127

**Subjects**

Primary: 14J70: Hypersurfaces

Secondary: 14Q10: Surfaces, hypersurfaces

**Keywords**

hypersurface Lefschetz pencil positive characteristic configuration

#### Citation

Katsura, Toshiyuki. Lefschetz pencils on a certain hypersurface in positive characteristic. Higher Dimensional Algebraic Geometry: In honour of Professor Yujiro Kawamata's sixtieth birthday, 265--278, Mathematical Society of Japan, Tokyo, Japan, 2017. doi:10.2969/aspm/07410265. https://projecteuclid.org/euclid.aspm/1540319491