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

TL;DR

This paper investigates the embedding of the projective plane into the Grassmannian via its tangent bundle, revealing a component of the Hilbert scheme with no smooth surface points.

Contribution

It identifies a specific component of the Hilbert scheme related to the tangent bundle embedding with unique properties.

Findings

Existence of a Hilbert scheme component with no smooth surface points

Characterization of the tangent bundle image of $ ext{PP}^2$ in the Grassmannian

Insights into the geometry of tangent bundle embeddings

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.