An Octanomial Model for Cubic Surfaces

TL;DR

This paper introduces a novel normal form for cubic surfaces tailored for p-adic geometry, highlighting their combinatorial structure and enabling explicit symbolic and p-adic computations.

Contribution

It proposes a new eight-term polynomial normal form for cubic surfaces that connects to E6 hyperplane arrangements and tropical geometry.

Findings

Tropical smoothness implies 27 distinct tropical lines.

The normal form facilitates explicit symbolic and p-adic computations.

Reveals intrinsic combinatorics of cubic surfaces in tropical and p-adic contexts.

Abstract

We present a new normal form for cubic surfaces that is well suited for p-adic geometry, as it reveals the intrinsic del Pezzo combinatorics of the 27 trees in the tropicalization. The new normal form is a polynomial with eight terms, written in moduli from the E6 hyperplane arrangement. If such a surface is tropically smooth then its 27 tropical lines are distinct. We focus on explicit computations, both symbolic and p-adic numerical.