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

Abstract

We classify all the embeddings of ${\mathbb{P}}_n$ in a Grassmannian $Gr(1,N)$ such that the composition with Plücker 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 $n\ge 3$.

arXiv comment12 pages, many changes

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• Vector spaces of skew-symmetric matrices of constant rank