Triangulations and Severi varieties

TL;DR

This paper explores minimal vertex triangulations of projective planes over Hurwitz algebras, linking face counts to automorphism group representations of Severi varieties, and proposes a geometric complex as a triangulation analogue.

Contribution

It introduces a novel approach connecting triangulations with automorphism representations of Severi varieties, providing a geometric framework for these structures.

Findings

Face counts match dimensions of automorphism group representations

Constructed a complex representing the triangulation structure

Proposed a geometric analogue of triangulations for Severi varieties

Abstract

We consider the problem of constructing triangulations of projective planes over Hurwitz algebras with minimal numbers of vertices. We observe that the numbers of faces of each dimension must be equal to the dimensions of certain representations of the automorphism groups of the corresponding Severi varieties. We construct a complex involving these representations, which should be considered as a geometric version of the (putative) triangulations.