Revisiting Hilbert's work on invariants of polynomials to study cubic surfaces

In the late 19th century, David Hilbert introduced the problem of the finite generation of the algebra of invariants. Namely, if a polynomial ring is finitely generated and a group acts on it, is the subring of polynomials invariant by the group finitely generated? Can we describe the resulting subring?

These questions resurfaced in the 1960s with the development of Geometric Invariant Theory, by David Mumford, who emphasised their geometric role. Indeed, the construction of classifying spaces for projective manifolds depends deeply on answering these questions.

We will introduce this topic by focusing in one of the main problems considered by Hilbert: the classification of cubic surfaces. We will further enhance this problem by considering also how to simultaneously classify cubic surfaces and cubic curves inside them. In solving this problem we will bring together old tools from singularity theory and modern computational algebra.

This seminar is hosted by Professor Berthold Lausen of the University of Essex, and will feature guest speaker Dr Jesus Martinez Garcia from the University of Bath.