On triple Veronese embeddings of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {P}^n$\end{document} in the Grassmannians

Abstract

We classify all the embeddings of Pn in a Grassmannian Gr(1, N ) such that the composition with the Plücker embedding is given by a linear system of cubics on Pn . As a direct corollary, we prove that every vector bundle giving such an embedding, splits if n ≥ 3.