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Let V 5 be the del Pezzo 3-fold defined by the 6-dimensional linear section of the Grassmannian variety Gr (2 , 5) under the Plücker embedding. In this paper, we present an explicit birational relation of compactifications of degree-two rational curves (i.e. conics) in V 5. By a product, we obtain the virtual Poincaré polynomial of compactified moduli spaces. [ABSTRACT FROM AUTHOR]

Let X = P 1 × P 1 × P 1 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. [ABSTRACT FROM AUTHOR]