Smoothing of 1-cycles over finite fields

TL;DR

This paper proves that any algebraic 1-cycle on a smooth projective variety over a finite field can be smoothed to a combination of smooth curves, and extends Bertini's theorem to finite fields for large tensor powers.

Contribution

It introduces a method to smooth 1-cycles over finite fields and generalizes Poonen's Bertini theorem to ensure smooth divisors in high tensor powers.

Findings

Any algebraic 1-cycle is rationally equivalent to a smooth 1-cycle.

Established a generalized Bertini theorem over finite fields.

Proved the existence of smooth divisors in high tensor powers of line bundles.

Abstract

Let $X$ be a smooth projective variety defined over a finite field. We show that any algebraic $1$-cycle on $X$ is rationally equivalent to a smooth $1$-cycle, which is a $\mathbb{Z}$-linear combination of smooth curves on $X$. We also prove a generalized version of Poonen's Bertini theorem over finite fields. Given a very ample line bundle $\mathcal{L}$ on $X$ and an arbitrary line bundle $\mathcal{M}$, this version implies the existence of a global section of $\mathcal{M}\otimes \mathcal{L}^{\otimes d}$ for sufficiently large $d$ whose divisor is smooth.