Extracting Invariants of Isolated Hypersurface Singularities from their Moduli Algebras

TL;DR

This paper develops classical invariant theory methods to derive numerical invariants from moduli algebras of isolated hypersurface singularities, aiding their classification and extending previous results.

Contribution

It introduces a new approach to extract invariants from moduli algebras of hypersurface singularities, advancing the classification and equivalence problem.

Findings

Constructed invariants for complex graded Gorenstein algebras

Extended invariants to quasi-homogeneous hypersurface singularities

Verified the conjecture for binary quartics, ternary cubics, quintics, and sextics

Abstract

We use classical invariant theory to construct invariants of complex graded Gorenstein algebras of finite vector space dimension. As a consequence, we obtain a way of extracting certain numerical invariants of quasi-homogeneous isolated hypersurface singularities from their moduli algebras, which extends an earlier result due to the first author. Furthermore, we conjecture that the invariants so constructed solve the biholomorphic equivalence problem in the homogeneous case. The conjecture is easily verified for binary quartics and ternary cubics. We show that it also holds for binary quintics and sextics. In the latter cases the proofs are much more involved. In particular, we provide a complete list of canonical forms of binary sextics, which is a result of independent interest.