On deformation and classification of V-systems

TL;DR

This paper investigates the deformation theory and classification of V-systems, special covector sets related to WDVV equations, focusing on rank 3 cases and their geometric structures.

Contribution

It derives deformation relations for V-systems and provides a comprehensive catalog of all known irreducible rank 3 V-systems.

Findings

Derived relations for infinitesimal deformations of V-systems.

Classified all known irreducible rank 3 V-systems.

Explored matroidal structures and their connection to projective geometry.

Abstract

The V-systems are special finite sets of covectors which appeared in the theory of the generalized Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. Several families of V-systems are known but their classification is an open problem. We derive the relations describing the infinitesimal deformations of V-systems and use them to study the classification problem for V-systems in dimension 3. We discuss also possible matroidal structures of V-systems in relation with projective geometry and give the catalogue of all known irreducible rank 3 V-systems.