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

Abstract

Consider a $2n$-dimensional symplectic vector space $E$ over an arbitrary field $\mathbb{F}$. Given a contraction map $f:{\wedge }^{n}E\to {\wedge }^{n-2}E$ such that the Lagrangian--Grassmannian $L(n,2n)=G(n,2n)\cap \mathbb{P}(\ker f)$, 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 $f:{\wedge }^{6}E\to {\wedge }^{4}E$ depends on the characteristic of the base field and we calculate the codimension of the linear section $\mathbb{P}(\ker f)\subseteq \mathbb{P}({\wedge }^{6}E)$ for any characteristic.

6 pages