A four-dimensional cousin of the Segre cubic

TL;DR

This paper introduces a special Fano fourfold linked to the Segre cubic, highlighting its unique properties, construction methods, and explicit Chow ring description, enriching the understanding of complex algebraic varieties.

Contribution

It presents a new Fano fourfold related to the Segre cubic, detailing its construction, properties, and explicit Chow ring, and explores its symmetry and rigidity.

Findings

The fourfold is infinitesimally rigid.

It has Picard number six.

Explicit Chow ring description provided.

Abstract

This note is devoted to a special Fano fourfold defined by a four-dimensional space of skew-symmetric forms in five variables. This fourfold appears to be closely related with the classical Segre cubic and its Cremona-Richmond configuration of planes. Among other exceptional properties, it is infinitesimally rigid and has Picard number six. We show how to construct it by blow-up and contraction, starting from a configuration of five planes in a four-dimensional quadric, compatibly with the symmetry group $S_5$. From this construction we are able to describe the Chow ring explicitly.