The Linear Span of Four Points in the Pl\"ucker's Quadric in ${\mathbb P}^5$

Abstract

Given four (distinct) lines $\ell_1$, $\ell_2$, $\ell_3$, $\ell_4$ in $\p^3$. Let $P_i$ ($i=1,..,4$)  be the image of $\ell_i$ in thePl\"ucker's quadric ${\cal Q}\subset\p^5$ under the Pl\"ucker embedding ${\cal P}$ (in (\ref{pluckerG24})). Set $\Lambda =\left\langle P_1,..., P_4 \right\rangle$ be the linear span of those four points in $\p^5$. The purpose of this article is to write specifically what kind of quadric $\Lambda\cap{\cal Q}$ can be, takingunder considerations all possible configurations of these four lines in $\p^3$. In particular, having in mind the classical problem in Schubert Calculus: {\it How many lines in  3-space meet four given lines in general position}? whose answer is 2 (see p. 272 in \cite{Fulton} or p. 746 in \cite{GriffithsHarris}). We verified that four lines in $\p^3$ are in general position if and only if $\Lambda$ is a 3-plane and $\Lambda\cap {\cal Q}$ is an irreducible quadric surface. In fact, we prove that there are exactly two solutions if and only if $\Lambda$ is a 3-plane and $\Lambda\cap {\cal Q}$ is a nonsingular quadric.