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
type: Item
instance of: scholarly article
title: 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
zbMATH Open document ID: 0826.14035
DOI: 10.1007/BF02392803
author: Martin A. Guest
published in: Acta Mathematica
publication date: 1995-05-11T00:00:00Z
review text: 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 \textit{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.
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14F35
zbMATH DE Number: 734731
zbMATH Keywords: toric variety
zbMATH Keywords: homotopy of the space of rational curves
zbMATH Keywords: fun
reviewed by: Konrad Drechsler
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 734731