Lines on Projective Varieties

I prove two theorems: Let X ⊂ P be a hypersurface and let x ∈ X be a general point. If the set of lines having contact to order k with X at x is of dimension greater than expected, then the lines having contact to order k are actually contained in X. A variety X is said to be covered by lines if there exist a finite number of lines in X passing through a general point. Let X ⊂ P be a variety covered by lines. Then there are at most n! lines passing through a general point of X. Definition. A variety (resp. manifold) X ⊂ P (or A ) is covered by lines if through a general point (resp. every point) x ∈ X there passes a finite number of lines contained in X. It was known classically that a surface covered by lines can contain at most two lines through a general point. In [3] it was shown that a 3-fold covered by lines contains at most 6 lines through a general point. I prove: Theorem 1. Let X ⊂ P be covered by lines. Then there are at most n! lines passing through a general point of X. The theorem holds over any field of characteristic zero. The conclusion is valid for analytic varieties in affine or projective space if one asks that the lines be contained in the closure of X. The conclusion holds in the C∞ category if one replaces “general point” by “every point” in the hypotheses. Theorem 1 will be a consequence of theorem 2: Theorem 2. Let X ⊂ P be a hypersurface and let x ∈ X be a general point. Let Σ ⊂ PTxX denote the tangent directions to lines having contact to order λ with X at x. (The notation is such that Σ = PTxX.) If there is an irreducible component Σ k 0 ⊂ Σ k such that dimΣ0 > n− k then Σ k 0 ⊂ Σ ∞, i.e., all lines corresponding to points of Σ0 are contained in X. Note that the expected dimension of Σ is n− k. Corollary. Let X ⊂ P be a variety such that through a general point x ∈ X there is a p-plane having contact to order n− p+ 2. Then X is uniruled by P’s. The corollary is not expected to be optimal for most values of k, see [2], theorem 4. 1991 Mathematics Subject Classification. primary 53, secondary 14.