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Configuration spaces and the space of rational curves on a toric variety

Author: M. A. Guest
Journal: Bull. Amer. Math. Soc. 31 (1994), 191-196
MSC: Primary 55P99; Secondary 14M25, 55Q99
MathSciNet review: 1260521
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Abstract: 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 several areas of geometry. It is a well known problem to determine an integer $ n(D)$ such that the inclusion of this space in the corresponding space of continuous maps induces isomorphisms of homotopy groups up to dimension $ n(D)$, where D denotes the homotopy class of the maps. The solution to this problem is known for an important but special class of varieties, the generalized flag manifolds: such an integer may be computed, and $ n(D) \to \infty $ as $ D \to \infty $. We consider the problem for another class of varieties, namely, toric varieties. For smooth toric varieties and certain singular ones, $ n(D)$ may be computed, and $ n(D) \to \infty $ as $ D \to \infty $. For other singular toric varieties, however, it turns out that $ n(D)$ cannot always be made arbitrarily large by a suitable choice of D.

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