Linear families of smooth hypersurfaces over finitely generated fields

TL;DR

This paper constructs linear systems of hypersurfaces over finitely generated fields ensuring smooth members under certain characteristic conditions, and provides counterexamples when those conditions are not met.

Contribution

It establishes the existence of smooth hypersurface families over finitely generated fields with specific characteristic restrictions, and presents counterexamples in the divisibility case.

Findings

Existence of smooth hypersurface families over finitely generated fields under certain conditions.

Counterexamples when the characteristic divides gcd(d, n+1).

No restriction in characteristic zero case.

Abstract

Let $K$ be a finitely generated field. We construct an $n$-dimensional linear system $\mathcal{L}$ of hypersurfaces of degree $d$ in $\mathbb{P}^n$ defined over $K$ such that each member of $\mathcal{L}$ defined over $K$ is smooth, under the hypothesis that the characteristic $p$ does not divide $\gcd(d, n+1)$ (in particular, there is no restriction when $K$ has characteristic $0$). Moreover, we exhibit a counterexample when $p$ divides $\gcd(d, n+1)$.