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

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 $n\geq 3$.

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