Illinois Journal of Mathematics

Polarized varieties whose points are joined by rational curves of small degrees

Abstract

Let $X$ be a projective variety with $\mathbb{Q}$-factorial singularities, over an algebraically closed field $k$ of characteristic $0$, $L$ an ample Cartier divisor on $X$, and $x$ a non-singular point of $X$. We prove that if for two general points $y,z \in X$ there exists a rational curve $C$ passing through $x, y, z$ such that $(L.C) = 2$, then $(X.L) \simeq (\mathbb{P}^{n}.\mathcal{O}(1))$ or $(Q^{n}.\mathcal{O}(1))$, a hyperquadric.

Article information

Source
Illinois J. Math., Volume 43, Issue 2 (1999), 350-390.

Dates
First available in Project Euclid: 19 October 2009

https://projecteuclid.org/euclid.ijm/1255985220

Digital Object Identifier
doi:10.1215/ijm/1255985220

Mathematical Reviews number (MathSciNet)
MR1703193

Zentralblatt MATH identifier
0939.14008

Citation

Kachi, Yasuyuki; Sato, Eiichi. Polarized varieties whose points are joined by rational curves of small degrees. Illinois J. Math. 43 (1999), no. 2, 350--390. doi:10.1215/ijm/1255985220. https://projecteuclid.org/euclid.ijm/1255985220