Lefschetz pencils on a certain hypersurface in positive characteristic. (English) Zbl 1388.14127

Katsura, Toshiyuki

Oguiso, Keiji (ed.) et al., Higher dimensional algebraic geometry. In honour of Professor Yujiro Kawamata’s sixtieth birthday. Proceedings of the conference, Tokyo, Japan, January 7–11, 2013. Tokyo: Mathematical Society of Japan (MSJ) (ISBN 978-4-86497-046-4/hbk). Advanced Studies in Pure Mathematics 74, 265-278 (2017).

Summary: We examine Lefschetz pencils of a certain hypersurface in \(\mathbb{P}\) 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 T. Shioda [Math. Ann. 211, 233–236 (1974; Zbl 0276.14018)].
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 \(\mathbb{F}_q\) (resp. over \(\mathbb{F}_{q^2}\)) and the lines over \(\mathbb{F}_q\) (resp. over \(\mathbb{F}_{q^2}\)) with \(q=p^a\).
For the entire collection see [Zbl 1388.14012].

MSC:

14J70	Hypersurfaces and algebraic geometry
14G17	Positive characteristic ground fields in algebraic geometry
14J25	Special surfaces
14C21	Pencils, nets, webs in algebraic geometry

Keywords:

hypersurface; Lefschetz pencil; positive characteristic; configuration

Citations:

Zbl 0276.14018