TY - JOUR

T1 - Configuration spaces and the space of rational curves on a toric variety

AU - Guest, M. A.

PY - 1994/10

Y1 - 1994/10

N2 - The space of holomorphic maps from S2 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) → ∞ as D → ∞. 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) → ∞ as D → ∞. For other singular toric varieties, however, it turns out that n(D) cannot always be made arbitrarily large by a suitable choice of D.

AB - The space of holomorphic maps from S2 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) → ∞ as D → ∞. 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) → ∞ as D → ∞. 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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U2 - 10.1090/S0273-0979-1994-00515-4

DO - 10.1090/S0273-0979-1994-00515-4

M3 - Comment/debate

AN - SCOPUS:84967710293

SN - 0273-0979

VL - 31

SP - 191

EP - 196

JO - Bulletin of the American Mathematical Society

JF - Bulletin of the American Mathematical Society

IS - 2

ER -