Lines on algebraic varieties

2001/11/04 by J. M. Landsberg, Landsberg, J. M.
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG

paper · pdf · doi:10.48550/arxiv.math/0111039

3 pages

arxiv created 2001/11/04 · arxiv updated 2009/11/30

Abstract

A variety X is covered by lines if there exist a finite number of lines contained in X passing through each general point. I prove two theorems. Theorem 1:Let Xⁿ⊂ P^M be a variety covered by lines. Then there are at most n! lines passing through a general point of X. Theorem 2:Let Xⁿ\subsetPⁿ⁺¹ 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.

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