On pairs, triples and quadruples of points on a cubic surface

TL;DR

This paper investigates the birational relationships between symmetric powers of a cubic surface, revealing that the product of the fourth symmetric power with the surface is stably birational to the third symmetric power times the surface, contrary to some expectations.

Contribution

It demonstrates that $X^{(4)}\times X$ is stably birational to $X^{(3)}\times X$, providing new insights into the birational geometry of symmetric powers of cubic surfaces.

Findings

$X^{(4)}\times X$ is stably birational to $X^{(3)}\times X$

Examples exist where $X^{(4)}$ is not stably birational to $X^{(3)}$

The result clarifies the birational relationship between different symmetric powers

Abstract

Let $X^{(n)}$ denote $n$-th symmetric power of a cubic surface $X$. We show that $X^{(4)}\times X$ is stably birational to $X^{(3)}\times X$, despite examples when $X^{(4)}$ is not stably birational to $X^{(3)}$.