Cremona linearizations of some classical varieties

TL;DR

This paper introduces a method using Cremona transformations to linearize rational varieties, simplifying their secant and tangential varieties, with applications to classical varieties like Veronese, Segre, and Grassmann, and connections to algebraic statistics.

Contribution

It presents a new effective approach for linearizing rational varieties via Cremona transformations, unifying classical and statistical methods.

Findings

Successfully linearized classical varieties such as Veronese, Segre, and Grassmann.

Simplified equations of secant and tangential varieties using Cremona transformations.

Connected algebraic statistics concepts like cumulant Cremonas to geometric transformations.

Abstract

In this paper we present an effective method for linearizing rational varieties of codimension at least two under Cremona transformations, starting from a given parametrization. Using these linearizing Cremonas, we simplify the equations of secant and tangential varieties of some classical examples, including Veronese, Segre and Grassmann varieties. We end the paper by treating the special case of the Segre embedding of the n-fold product of projective spaces, where cumulant Cremonas, arising from algebraic statistics, appear as specific cases of our general construction.