Projective Varieties Swept Out by Rational Normal Curves

Abstract

Let X ⊂ ℙ ⁿ be a complex nondegenerate projective variety of dimension m ≥ 2. For t ≤ n − m and a general q ∈ ℙ ⁿ , the linear space L _q spanned by q and t general points of X meets X in a finite set of points. We classify those X ⊂ ℙ ⁿ for which there exists a point q ∈ ℙ ⁿ such that L _q meets X in a positive dimensional variety. If this occurs, there exists d ≤ n − m such that a degree d rational normal curve through d general points of X is contained in X. Examples of this situation are provided. An infinitesimal generalization of part of the main result is also stated.

Key Words:

• Rational normal curves
• Veronese varieties

2000 Mathematics Subject Classification:

• 14N05
• 14M20

ACKNOWLEDGMENTS

The first author was partially supported by MIUR and GNSAGA of INdAM (Italy). The second author was partially supported by the project MTM2006-04785 of the Spanish Government.

Notes

Communicated by R. Piene.