Rational curves on homogeneous cones

Abstract

A homogeneous cone $X$ is the cone over a homogeneous variety $G/P$ embedded thanks to an ample line bundle $L$. In this article, we describe the irreducible components of the scheme of morphisms of class $\alpha \in A_1(X)$ from a rational curve to X. The situation depends on the line bundle L : if the projectivised tangent space to the vertex contains lines then the irreducible components are described by the difference between Cartier and Weil divisors. On the contrary if there is no line in the projectivised tangent space to the vertex then there are new irreducible components corresponding to the multiplicity of the curve through the vertex.