On morphisms from $\protect \mathbb{P}^3$ to $\protect \mathbb{G}(1,3)$

Abstract

Every morphism from $\mathbb{P}^n$ to $\mathbb{G}(k,m)$ is constant if $m, and nonconstant morphisms from $\mathbb{P}^n$ to $\mathbb{G}(k,n)$ rarely appear when $0. In this setting, Tango proved that a morphism from $\mathbb{P}^n$ to $\mathbb{G}(1,n)$ is constant if $n\notin \lbrace 3,5\rbrace $. Here we focus on the case $n=3$ and show that if $\phi :\mathcal{O}_{\mathbb{P}^3}^{\oplus 4}\rightarrow E$ is the surjection onto a rank $2$ vector bundle $E$ inducing a morphism $\varphi :\mathbb{P}^3\rightarrow \mathbb{G}(1,3)$, then $h^1(E^*)\le 1$. Furthermore, a complete classification is given if equality holds.