Tangent bundle of $\PP^2$ and morphism from $\PP^2$ to $\text{Gr}(2, \CC^{4})$

Abstract

In this note we study the image of $\PP^2$ in $\text{Gr}(2, \CC^{4})$ given by tangent bundle of $\PP^2. $ We show that there is component $\mathcal{H}$ of the Hibert scheme of surfaces in $\text{Gr}(2, \CC^{4})$ with no point of it corresponds to a smooth surface.