Secant Varieties of Grasmann Varieties

In this paper we discuss the dimensions of the (higher) secant varieties to the Grassmann varieties, embedded via the Plucker embeddings. We use Terracini's Lemma and the duality in the exterior algebra of a finite dimensional vector space to translate the problem into that of finding the dimension of a graded piece of a " fat" ideal in the exterior algebra. Among other things, our methods prove that (apart from the chordal varieties to the Grassmannians of lines) all the chordal varieties of Grassmannians have the expected dimension. We also discuss some Grassmannians with deficient secant varieties, giving new proofs that G(3,7) and G(3,9) are of this type. We also find a new Grassmannian, namely G(4,8) with deficient secant varieties.