Periods of cubic surfaces with the automorphism group of order 54

TL;DR

This paper computes the period map for special cubic surfaces obtained as triple covers of the projective plane branched over smooth elliptic curves, linking their periods to those of the elliptic curves.

Contribution

It explicitly calculates the period map for cubic surfaces with automorphism group of order 54, specifically those arising from triple covers branched over elliptic curves.

Findings

Periods expressed through elliptic curve periods

Explicit calculation for cubic surfaces with automorphism group of order 54

Connection between cubic surface periods and elliptic curve periods

Abstract

To any cubic surface, one can associate a cubic threefold given by a triple cover of $\mathbb P^3$ branched in this cubic surface. D. Allcock, J. Carlson, and D. Toledo used this construction to define the period map for cubic surfaces. It is interesting to calculate this map for some specific cubic surfaces. In this paper, we have calculated it in the case when the cubic surface is given by a triple covering of $\mathbb P^2$ branched in a smooth elliptic curve. In this case, the periods can be expressed through periods of the corresponding elliptic curve.