del Pezzo surfaces with one bad prime over cyclotomic $\mathbb{Z}_\ell$-extensions

TL;DR

This paper demonstrates that over the cyclotomic $ ext{Z}_5$-extension of $ ext{Q}$, there are infinitely many del Pezzo surfaces of degrees 3 and 4 with good reduction outside a single prime, contrasting prior finiteness results.

Contribution

It establishes the existence of infinitely many such del Pezzo surfaces over a cyclotomic extension, specifically for degrees 3 and 4, with only one bad prime.

Findings

Infinitely many degree 3 del Pezzo surfaces with good reduction outside one prime.

Infinitely many degree 4 del Pezzo surfaces with good reduction outside one prime.

Contrasts with finiteness results over number fields.

Abstract

Let $K$ be a number field and $S$ a finite set of primes of $K$. Scholl proved that there are only finitely many $K$-isomorphism classes of del Pezzo surfaces of any degree $1 \le d \le 9$ over $K$ with good reduction away from $S$. Let instead $K$ be the cyclotomic $\mathbb{Z}_5$-extension of $\mathbb{Q}$.In this paper, we show, for $d=3$, $4$, that there are infinitely many $\overline{\mathbb{Q}}$ isomorphism classes of del Pezzo surfaces, defined over $K$, with good reduction away from the unique prime above $5$.