M ar 1 99 9 A LOCAL VERSION OF THE FULTON - HANSEN THEOREM RELATING SECANT AND TANGENTIAL VARIETIES

An important theorem in the study of subvarieties of projective space is the following theorem of Fulton and Hansen [FH]: If X n ⊂ PV is a variety, τ (X) its tangential variety and σ(X) its secant variety, then either dim τ (X) = 2n and dim σ(X) = 2n + 1 or τ (X) = σ(X) (see below for definitions). τ (X) is the union of tangent stars of X, but from the perspective of differential geometry , one would like to work with τ c (X), the closure of the union of embedded tangent spaces at smooth points of X, as its dimension can be calculated from infinitesimal data at a general point of X. However, the theorem fails to hold for τ c (X) [L1]. In this paper we determine a geometric criterion that implies the Fulton-Hansen theorem holds for τ c (X), namely that the Gauss image of τ c (X) be " as small as possible " (corollary 2). We also prove this criterion holds in certain special cases (theorems 4 and 5). showed that if M n ⊂ PV is a manifold with degenerate tangential manifold τ (M) ⊂ PV , then the Gauss map of τ (M) has at least two dimensional fibers. They then remark that they suspect this result is related to the Fulton-Hansen theorem. Later in the same paper, they state a way to calculate dim σ(M) from differential invariants at a general point of M. In [L1], it was pointed out since the result on the fibers of γ(τ (X)) was purely local, it could not be sufficient to recover the Fulton-Hansen theorem. In fact, it was proven that the Fulton-Hansen theorem implied that when X is a smooth variety and τ (X) is a hypersurface, the Gauss map of τ (X) must have at least three dimensional fibers [L1, 13.10]. The announcement was incorrectly stated for arbitrary codimension of τ (X), as the proof is given only in the case τ (X) is a hypersurface. Moreover, a proof in the general case would not have been possible as the formula for calculating dim σ(X) used was incorrect (see proposition 6 and the remark below it). Despite all this, we show that the intuition of Griffiths and Harris was indeed correct, and the degeneracy of γ(τ (X)) is directly related to the Fulton-Hansen theorem. Definitions. Let V = C …