The Linear Span of Four Points in the Pl\"ucker's Quadric in P5

Abstract

Given four (distinct) lines ${\ell }_1$, ${\ell }_2$, ${\ell }_3$, ${\ell }_4$ in ${\text{p}}^3$. Let $P_i$ ($i=1,..,4$)  be the image of ${\ell }_i$ in thePl\"ucker's quadric $\mathcal{Q}\subset {\text{p}}^5$ under the Pl\"ucker embedding $\mathcal{P}$ (in ($\text{???}$)). Set $\mathrm{\Lambda }=⟨P_1,...,P_4⟩$ be the linear span of those four points in ${\text{p}}^5$. The purpose of this article is to write specifically what kind of quadric $\mathrm{\Lambda }\cap \mathcal{Q}$ can be, takingunder considerations all possible configurations of these four lines in ${\text{p}}^3$. 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 ${\text{p}}^3$ are in general position if and only if $\mathrm{\Lambda }$ is a 3-plane and $\mathrm{\Lambda }\cap \mathcal{Q}$ is an irreducible quadric surface. In fact, we prove that there are exactly two solutions if and only if $\mathrm{\Lambda }$ is a 3-plane and $\mathrm{\Lambda }\cap \mathcal{Q}$ is a nonsingular quadric.