Foliations on smooth algebraic surfaces over positive characteristic

TL;DR

This paper studies the properties of $p$-divisors associated with foliations on smooth algebraic surfaces over fields of positive characteristic, providing a structure theorem for specific cases like the projective plane and Hirzebruch surfaces.

Contribution

It introduces a structure theorem for $p$-divisors of foliations on certain algebraic surfaces and shows conditions under which these divisors are reduced.

Findings

Structure theorem for $p$-divisors on the projective plane.

Reduced $p$-divisors under certain conditions on Hirzebruch surfaces.

Properties of $p$-divisors in positive characteristic settings.

Abstract

We investigate the notion of the $p$-divisor for foliations on a smooth algebraic surface defined over a field of positive characteristic $p$ and we study some of their properties. We present a structure theorem for the $p$-divisor of foliations in the projective plane and the Hirzebruch surfaces where we show that, under certain conditions, such $p$-divisors are reduced.