CONFIGURATION-SPACES AND THE SPACE OF RATIONAL CURVES ON A TORIC VARIETY

The space of holomorphic maps from S-2 to a complex algebraic variety X, i.e. the space of parametrized rational curves on X, arises in seve ral areas of geometry. It is a well known problem to determine an inte ger n(D) such that the inclusion of this space in the corresponding sp ace of continuous maps induces isomorphisms of homotopy groups up to d imension n(D), where D denotes the homotopy class of the maps. The sol ution to this problem is known for an important but special class of v arieties, the generalized flag manifolds: such an integer may be compu ted, and n(D) --> infinity as D --> infinity. We consider the problem for another class of varieties, namely, toric varieties. For smooth to ric varieties and certain singular ones, n(D) may be computed, and n(D ) --> infinity as D --> infinity. For other singular toric varieties, however, it turns out that n(D) cannot always be made arbitrarily larg e by a suitable choice of D.