A note on the codimension of the linear section of the Lagrangian-Grassmannian L(6,12)

2017/01/28 by Jesús Carrillo-Pacheco, Fausto Jarquín-Zárate, Carrillo-Pacheco, Jesús +5
Mathematics · #Finite Group Theory Research #Advanced Topics in Algebra #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.1701.08242

Abstract

Consider a 2n-dimensional symplectic vector space E over an arbitrary field \mathbbF. Given a contraction map f: \wedgeⁿ E → \wedge^n-2 E such that the Lagrangian--Grassmannian L(n,2n)=G(n,2n)∩\mathbb P(ker f), where \wedge^r E denotes the r-th exterior power of E and \mathbb P(ker f) is the projectivization of ker f. In this paper, for a symplectic vector space E of dimension n=6, we prove that the surjectivity of the contraction map f:\wedge⁶ E → \wedge⁴ E depends on the characteristic of the base field and we calculate the codimension of the linear section \mathbb P(ker f)⊆ \mathbb P(\wedge⁶E) for any characteristic.

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