On Triple Veronese Embeddings of $\PP_n$ in the Grassmannians

TL;DR

This paper classifies specific cubic embeddings of projective spaces into Grassmannians and proves a splitting criterion for vector bundles in this context, contributing to the understanding of algebraic embeddings and vector bundle structures.

Contribution

It provides a complete classification of triple Veronese embeddings of projective spaces into Grassmannians and establishes a splitting result for associated vector bundles when the dimension is at least three.

Findings

Classified all triple Veronese embeddings into Grassmannians.

Proved vector bundles giving such embeddings split for n ≥ 3.

Connected results to the Hartshorne conjecture.

Abstract

We classify all the embeddings of $\mathbb{P}_n$ in a Grassmannian $Gr(1,N)$ such that the composition with Pl\"{u}cker embedding is given by a linear system of cubics on $\mathbb{P}_n$. As a corollary in the direction of the Hartshorne conjecture, we prove that every vector bundle giving such an embedding, splits if $n\geq 3$.