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The topology of the space of rational curves on a toric variety. - Appendix 1: $$\pi_ 0 Q^ X_ D$$ and $$\pi_ 1Q^ X_ D$$. -Appendix 2: Representation of holomorphic maps by polynomials. (English) Zbl 0826.14035
Toric varieties are good examples to check properties of algebraic varieties $$X$$ which do not hold in general but which are true in special cases (e.g. for $$X = \mathbb{P}^n_\mathbb{C})$$. The present paper is concerned with a theorem of G. Segal [Acta Math. 143, 39-72 (1979; Zbl 0427.55006)] about the homotopy of the space of rational curves in $$\mathbb{P}^n_\mathbb{C}$$. The generalization to toric varieties $$X$$ was possible by a configuration space description of the space of all holomorphic maps from the Riemann sphere to $$X$$. This is based on the special fan construction of the toric varieties. The author presents the complete theorem for nonsingular projective toric varieties. For arbitrary (compact) toric varieties he has to make several assumptions. Some instructive examples and two appendices demonstrate the technique and the problems.

##### MSC:
 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 14H10 Families, moduli of curves (algebraic) 14F35 Homotopy theory and fundamental groups in algebraic geometry
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