The homogeneous ideals of higher secant varieties

✍ Scribed by Michael L. Catalano-Johnson

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 83 KB
• Volume: 158
• Category: Article
• ISSN: 0022-4049
• DOI: 10.1016/s0022-4049(00)00041-4

✦ Synopsis

Let P N be a projective space over an algebraically closed ÿeld of characteristic zero. Let X ⊂ P N be a closed, irreducible subvariety, not lying on a hyperplane. The kth higher secant variety of X , denoted X k , is the closure of the union of all linear spaces spanned by k points of X . We prove that I (X k ), the homogeneous ideal of X k , is contained in the kth symbolic power of I (X ). As a consequence, X k lies on no hypersurface of degree less than k + 1. Furthermore, if X is a curve, and deg X k = k + 1, we prove that X is a rational normal curve.