Setup
Michela Brundu and Alessandro Logar [1] offer a computational study of cubic surfaces via the normal form
f=a1(2x2y−2xy2+xz2−xzw−yw2+yzw)+a2(x−w)(xz+yw)+a3(z+w)(yw−xz)+a4(y−z)(xz+yw)+a5(x−y)(yw−xz).
This amounts to fixing an L-set, which is a special configuration of five lines on V(f).
How to compute the Brundu-Logar form in practise? Can we write a1,a2,a3,a4,a5 as rational functions in c1,c2,a3,…,c20? What does this tell us tropically?
Current progress
Avinash Kulkarni has written a Magma script to compute the Brundu-Logar form for generic cubics over p-adic fields, available here. He is not working on this project, but contact him if you would like to discuss the question!
Inspired by the problem of relative realizability of lines on surfaces, [2] investigates the information we can derive tropically from the Brundu-Logar normal form of smooth cubic surfaces. In particular, it is proven that for a residue field of characteristic $\neq 2$ the tropicalization of the Brundu-Logar normal form is not smooth.