Bilinear secants and birational geometry of blowups of $\mathbb P^n \times \mathbb P^{n+1}$

TL;DR

This paper introduces bilinear secant varieties and joins to study the birational geometry of blowups of products of projective spaces, revealing their log Fano property and computing their cones for specific cases.

Contribution

It generalizes classical secant varieties to bilinear secant varieties, applying this to analyze the birational geometry of certain blowups of product spaces.

Findings

X^{n,n+1}_s is log Fano for certain s and n.

Effective and movable cones are computed for specific blowups.

Bilinear secant varieties are central to understanding these geometric properties.

Abstract

We introduce bilinear secant varieties and joins of subvarieties of products of projective spaces, as a generalisation of the classical secant varieties and joins of projective varieties. We show that the bilinear secant varieties of certain rational normal curves of $\mathbb P^n \times \mathbb P^{n+1}$ play a central role in the study of the birational geometry of $X^{n,n+1}_s$, its blowup in $s$ points in general position. We show that $X^{n,n+1}_s$ is log Fano, and we compute its effective and movable cones, for $s\le n+2$ and $n\ge 1$ and for $s\le n+3$ and $n\le 2$, and we compute the effective and movable cones of $X^{3,4}_6$.