Lines on $p$-adic and real cubic surfaces

TL;DR

This paper investigates the number of lines on smooth cubic surfaces over $p$-adic and real fields, demonstrating all possible counts occur and exploring probabilistic distributions and Galois group properties.

Contribution

It proves that all counts of lines predicted by Segre are realizable and explores probabilistic and Galois group aspects of $p$-adic cubic surfaces.

Findings

All possible line counts are achieved on $p$-adic cubic surfaces.

Probabilistic sampling estimates the likelihood of each line count.

Experimental analysis of Galois groups attached to these surfaces.

Abstract

We study lines on smooth cubic surfaces over the field of $p$-adic numbers, from a theoretical and computational point of view. Segre showed that the possible counts of such lines are $0,1,2,3,5,7,9,15$ or $27$. We show that each of these counts is achieved. Probabilistic aspects are also investigated by sampling both $p$-adic and real cubic surfaces from different distributions and estimating the probability of each count. We link this to recent results on probabilistic enumerative geometry. Some experimental results on the Galois groups attached to $p$-adic cubic surfaces are also discussed.