Twisted cubic curves in the Segre variety

Abstract

Let X=P1×P1×P1 be the Segre variety. Let S be the space of twisted cubic curves in X with tri-degree (1, 1, 1). In this note, we prove that S is a rational, smooth variety of dimension 6. Also, we compute the Poincaré polynomial of S by stratifying the space into projective space fibration over some base spaces.