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

Sierra, José Carlos

Comptes Rendus. Mathématique, Tome 359 (2021) no. 7, pp. 853-860

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

Classification : 14J60, 14M15, 14N05

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