Random hypersurfaces and embedding curves in surfaces over finite fields

TL;DR

This paper proves that the only obstruction to embedding a curve in a smooth surface over a perfect field is the obvious one, and confirms a conjecture on the asymptotic probability of hypersurfaces with prescribed singularities over finite fields.

Contribution

It establishes the finite field analogue of a Bertini-type theorem and verifies a conjecture on hypersurface singularities, advancing understanding of algebraic geometry over finite fields.

Findings

Obstruction to embedding curves in surfaces is solely the obvious one.

Finite field analogue of Bertini's theorem proven.

Confirmed conjecture on hypersurface singularities probability.

Abstract

We use Poonen's closed point sieve to prove two independent results. First, we show that the obvious obstruction to embedding a curve in a smooth surface is the only obstruction over a perfect field, by proving the finite field analogue of a Bertini-type result of Altman and Kleiman. Second, we prove a conjecture of Vakil and Wood on the asymptotic probability of hypersurface sections having a prescribed number of singularities.