On the varieties of the second row of the split Freudenthal-Tits Magic Square

TL;DR

This paper provides a unified geometric characterization of analogues of complex Severi varieties over arbitrary fields, extending classical results to a broader algebraic setting.

Contribution

It offers a generalization of Mazzocca and Melone's approach, characterizing Severi varieties through projective properties as axioms over arbitrary fields.

Findings

Unified geometric framework for Severi varieties over arbitrary fields

Extension of classical complex results to algebraic varieties

Generalization of Mazzocca and Melone's axiomatic approach

Abstract

Our main aim is to provide a uniform geometric characterization of the analogues over arbitrary fields of the four complex Severi varieties, i.e.~the quadric Veronese varieties in 5-dimensional projective spaces, the Segre varieties in 8-di\-men\-sional projective spaces, the line Grassmannians in 14-dimensional projective spaces, and the exceptional varieties of type $\mathsf{E}_{6}$ in 26-dimensional projective space. Our theorem can be regarded as a far-reaching generalization of Mazzocca and Melone's approach to finite quadric Veronesean varieties. This approach takes projective properties of complex Severi varieties as smooth varieties as axioms.