×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Audin, M.,The Topology of Torus Actions on Symplectic Manifolds. Birkhäuser, Basel, 1991. · Zbl 0726.57029
[2] Cox, D. A., The homogeneous coordinate ring of a toric variety. To appear inJ. Algebraic Geom. · Zbl 0846.14032
[3] Cox, D. A., The functor of a smooth toric variety. To appear inTôhoku Math. J. · Zbl 0828.14035
[4] Dabrowski, A., On normality of the closure of a generic torus orbit inG/P. Preprint. · Zbl 0969.14034
[5] Dold, A. &Thom, R., Quasifaserungen und unendliche symmetrische Produkte.Ann. of Math., 67 (1958), 239–281. · Zbl 0091.37102 · doi:10.2307/1970005
[6] Flaschka, H. &Haine, L., Torus orbits inG/P.Pacific J. Math., 149 (1991), 251–292. · Zbl 0788.22017
[7] Fulton, W.,Introduction to Toric Varieties. Ann. of Math. Stud., 131. Princeton Univ. Press, Princeton, N.J., 1993. · Zbl 0813.14039
[8] Guest, M. A., Kozlowski, A. & Yamaguchi, K., The topology of spaces of coprime polynomials. To appear inMath. Z. · Zbl 0861.55015
[9] Gelfand, I. M. &MacPherson, R. D., Geometry in Grassmannians and a generalization of the dilogarithm.Adv. in Math., 44 (1982), 279–312. · Zbl 0504.57021 · doi:10.1016/0001-8708(82)90040-8
[10] Guest, M. A., On the space of holomorphic maps from the Riemann sphere to the quadric cone.Quart. J. Math. Oxford Ser. (2), 45 (1994), 57–75. · Zbl 0802.58012 · doi:10.1093/qmath/45.1.57
[11] Guest, M. A., Instantons, rational maps, and harmonic maps, inProceedings of Workshop on the Geometry and Topology of Gauge Fields, Campinas, 1991.Matematica Contemporanea, 2 (1992), 113–155.
[12] Harris, J.,Algebraic Geometry. Graduate Texts in Math., 133. Springer-Verlag, New York-Berlin, 1992.
[13] Hausmann, J.-C., &Husemoller, D., Acyclic maps.Enseign. Math. (2), 25 (1979), 53–75. · Zbl 0412.55008
[14] Lawson, H. B., Algebraic cycles and homotopy theory.Ann. of Math. (2), 129 (1989), 253–291. · Zbl 0688.14006 · doi:10.2307/1971448
[15] McDuff, D., Configuration spaces of positive and negative particles.Topology, 14 (1975), 91–107. · Zbl 0296.57001 · doi:10.1016/0040-9383(75)90038-5
[16] Oda, T.,Convex Bodies and Algebraic Geometry: An Introduction to the Theory of Toric Varieties. Springer-Verlag, Berlin-Heidelberg, 1988. · Zbl 0628.52002
[17] –, Geometry of toric varieties, inProceedings of the Hyderabad Conference on Algebraic Groups (S. Ramanan, ed.), pp. 407–440. Manoj Prakashan, Madras, 1991.
[18] Segal, G. B., The topology of spaces of rational functions.Acta Math., 143 (1979), 39–72. · Zbl 0427.55006 · doi:10.1007/BF02392088
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.