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

TL;DR

This work studies del Pezzo surfaces over the cyclotomic $\mathbb{Z}_5$-extension $K=\mathbb{Q}_{\infty,5}$ and constructs infinite families of surfaces of degrees $3$ and $4 with good reduction away from the unique prime above $5$. The construction uses blow-ups of $\mathbb{P}^2$ at Galois-orbits of points defined from a primitive $5^{r+1}$-th root of unity, yielding surfaces $X_r$ (deg $3$) and $Y_r$ (deg $4$) defined over $\mathbb{Q}_{r,5}$. Nonisomorphism among members is established by invariant theory: Clebsch–Salmon invariants for cubic surfaces and binary quintic invariants for degree-$4$ surfaces show only finitely many parameter choices can yield isomorphic surfaces, producing infinite pairwise nonisomorphic families. The results demonstrate that Shafarevich-type finiteness for del Pezzo surfaces fails over the infinite extension $K=\mathbb{Q}_{\infty,5}$ in degrees $3$ and $4$, informing the arithmetic of del Pezzo surfaces over cyclotomic towers.

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 $5$.

Theorems & Definitions (17)

• Theorem 1
• Theorem 2
• Lemma 3
• Definition 3
• Definition 4
• Example 4
• Example 5
• Proposition 6
• proof
• Remark 7