Bilinear secants and birational geometry of blowups of \mathbb Pⁿ × \mathbb Pⁿ⁺¹

2024/12/26 by Postinghel, Elisa, Prendergast-Smith, Artie
#14C20 #14J45 #14J70 #14N07 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2412.19364

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 Pⁿ × \mathbb Pⁿ⁺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≤ n+2 and n≥ 1 and for s≤ n+3 and n≤ 2, and we compute the effective and movable cones of X^3,4₆.

Related

• Secant Varieties and Birational Geometry
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• Birational geometry of the twofold symmetric product of a Hirzebruch surface via secant maps
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• Birational geometry of algebraic varieties