Secant Varieties of Grassmann Varieties

A classical and still unsolved functional Waring problem asks for the minimal number r sufficient to represent any form of degree d of n+1 variables as a sum of powers of linear forms. For general forms, this problem reduces to the geometric problem of computing the dimensions of higher secant varieties of the Veronese variety vd(Pn)?P(n+dn)-1; this last problem was solved by J. E. Alexander and A. Hirschowitz [J. Algebraic Geom. 4 (1995), no. 2, 201--222; MR1311347 (96f:14065)]. In the paper under review the authors consider a similar problem for skew-symmetric tensors. More precisely, let G=G(k,n)?P(n+1k+1)-1 be the Grassmann variety of k-dimensional linear subspaces in the projective space Pn. The expected (or, equivalently, maximal possible) dimension of the s-th self-join Gs is equal to min{(n+1k+1)-1,s(k+1)(n-k)+s-1}; G is called s-defective iff dimGs is smaller than expected. For k=1, n=5 all Grassmannians G(k,n) are defective, but for k>1 the situation is different and largely unexplored. In their main Theorem 2.1 the authors show that, for 1=k=n-12, G(k,n) has expected dimension provided that s=n+1k+1. In particular, for 2=k=n-3, the chordal variety G(k,n)2 has expected dimension. In the final section the authors give a list of three defective Grassmannians G(k,n), 2=k=n-12, viz. G(2,6), G(3,7), and G(2,8); the first three constructed by analogy with the exceptional Veronese varieties in the Alexander-Hirschowitz list, and the last one guessed by M. Catalano-Johnson. Of course, it is intriguing to consider whether there are further examples of this kind. The proofs are based on the Terracini lemma and exterior algebra computations.