Zak’s Theorem on Superadditivity

This paper is concerned with an important theorem due to Fyodor L. Zak, which appeared in [Za1]: Let X ⊂ P be a (reduced and irreducible) subvariety. A k – secant space to X is a k – dimensional linear subspace of P which is spanned by k+1 points from X, the k – secant variety of X in P is the (closure of the) union of all the k – secant spaces of X. Zak denotes this space by S(X), we shall also use a different terminology: Whenever X and Y are subvarieties of P , we define their join XY in P as the closure of the union of all lines in P spanned by a point from X and a point from Y . This defines a commutative and associative operation on the set of subvarieties of P , making it into a commutative monoid, see [Å] for details. We have S(X) = X. Zak considers a relative secant defect, defined as