Abstract. We determine which tangential varieties of homogeneously embedded rational homogeneous varieties are spherical. We determine the homogeneous coordinate rings and rings of covariants of the tangential varieties of homogenously embedded compact Hermitian symmetric spaces (CHSS). We give bounds on the degrees of generators of the ideals of tangential varieties of CHSS and obtain more explicit infomation about the ideals in certain cases. 1.

Abstract. Let Gr(k, n) be the Plücker embedding of the Grassmann variety of projective k-planes in P n. For a projective variety X, let σs(X) denote the variety of its s − 1 secant planes. More precisely, σs(X) denotes the Zariski closure of the union of linear spans of s-tuples of points lying on X. We exhibit two functions s0(n) ≤ s1(n) such that σs(Gr(2, n)) has the expected dimension whenever n ≥ 9 and either s ≤ s0(n) or s1(n) ≤ s. Both s0(n) and s1(n) are asymptotic to n2. This yields, asymptotically, the typical rank of 18 an element of ∧3 Cn+1. Finally, we classify all defective σs(Gr(k, n)) for s ≤ 6 and provide geometric arguments underlying each defective case. 1.

We study the secant variety of the spinor variety, focusing on its equations of degree three and four. We show that in type Dn, cubic equations exist if and only if n ≥ 9. In general the ideal has generators in degrees at least three and four. Finally we observe that the other Freudenthal varieties exhibit strikingly similar behaviors.