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

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Abstract

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.

Original languageEnglish
Pages (from-to)191-196
Number of pages6
JournalBulletin of the American Mathematical Society
Volume31
Issue number2
DOIs
Publication statusPublished - 1994
Externally publishedYes

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ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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