Invariants and rigidity of projective hypersurfaces

TL;DR

This paper investigates invariants and rigidity properties of projective hypersurfaces by computing Hilbert-Poincaré series-based invariants for classical examples, providing criteria and computational tools for rigidity and singularity analysis.

Contribution

It introduces new invariants derived from Milnor algebras for classical hypersurfaces and offers a computational approach to assess projective rigidity and singularity types.

Findings

Examples of hypersurfaces that are projectively rigid or not.

A computational tool to determine nodal and rigid hypersurfaces.

Extension of previous research on invariants of hypersurfaces.

Abstract

This paper continues our researches \cite{DS1, DS2, DS3} by computing some invariants based on Hilbert-Poincar\'{e} series associated to Milnor algebras. Our computations are for some of the classical surfaces and 3-folds with different configurations of isolated singularities. As a by-product of a recent result of E. Sernesi, we give examples of classical hypersurfaces which are (or are not) projectively rigid. We also include a Singular program to compute the invariants and to decide if a singular projective hypersurface is nodal and projectively rigid.