Kiem, Young-Hoon; Moon, Han-Bom Moduli space of stable maps to projective space via GIT. (English) Zbl 1191.14015 Int. J. Math. 21, No. 5, 639-664 (2010). Summary: We compare the Kontsevich moduli space \(\overline {M}_{0,0}(\mathbb P^{n-1},d)\) of stable maps to projective space with the quasi-map space \(\mathbb P(\text{Sym}^{d}(\mathbb C^{2}) \otimes \mathbb C^{n})//SL(2)\). Consider the birational map \[ \bar {\psi}: \mathbb P(\text{Sym}^d(\mathbb C^2) \otimes \mathbb C^n) // \mathrm{SL}(2) \to \overline {M}_{0,0}(\mathbb P^{n-1},d) \] which assigns to an \(n\) tuple of degree \(d\) homogeneous polynomials \(f_{1}, \cdots , f_{n}\) in two variables, the map \(f = (f_{1} : \cdots : f_{n}) : \mathbb P^{1} \rightarrow \mathbb P^{n-1}\). In this paper, for \(d = 3\), we prove that \(\bar {\psi }\) is the composition of three blow-ups followed by two blow-downs. Furthermore, we identify the blow-up/down centers explicitly in terms of the moduli spaces \(\overline {M}_{0,0}(\mathbb P^{n-1},d)\) with \(d = 1, 2\). In particular, \(\overline {M}_{0,0}(\mathbb P^{n-1},3)\) is the \(\mathrm{SL}(2)\)-quotient of a smooth rational projective variety. The degree two case \(\overline {M}_{0,0}(\mathbb P^{n-1},2)\), which is the blow-up of \(\mathbb P(\mathrm{Sym}^{2}\mathbb C^{2} \otimes \mathbb C^{n})//\mathrm{SL}(2)\) along \(\mathbb P^{n-1}\), is worked out as a preliminary example. Cited in 1 ReviewCited in 6 Documents MSC: 14D20 Algebraic moduli problems, moduli of vector bundles 14L24 Geometric invariant theory 14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) 55N91 Equivariant homology and cohomology in algebraic topology Keywords:moduli space; stable map; geometric invariant theory; equivariant cohomology PDFBibTeX XMLCite \textit{Y.-H. Kiem} and \textit{H.-B. Moon}, Int. J. Math. 21, No. 5, 639--664 (2010; Zbl 1191.14015) Full Text: DOI arXiv References: [1] DOI: 10.1007/s00222-003-0308-5 · Zbl 1092.14019 [2] DOI: 10.1016/j.top.2004.12.002 · Zbl 1081.14045 [3] Chung K., Amer. J. Math. [4] DOI: 10.1007/BF02698859 · Zbl 1001.14018 [5] DOI: 10.1007/BF01850655 · Zbl 0689.14012 [6] DOI: 10.1090/S1056-3911-06-00425-5 · Zbl 1114.14032 [7] DOI: 10.1002/9781118032527 [8] DOI: 10.1007/978-1-4757-3849-0 [9] DOI: 10.1007/s00031-003-0510-y · Zbl 1080.14537 [10] DOI: 10.1215/S0012-7094-07-13636-6 · Zbl 1119.14033 [11] Kirwan F. C., Math. Notes 31, in: Cohomology of Quotients in Symplectic and Algebraic Geometry (1985) [12] DOI: 10.2307/1971369 · Zbl 0592.14011 [13] DOI: 10.1090/S0894-0347-1992-1145826-8 [14] DOI: 10.1007/978-3-642-57916-5 · Zbl 0797.14004 [15] DOI: 10.1090/S0002-9947-99-01909-1 · Zbl 0911.14028 [16] DOI: 10.1090/S0894-0347-96-00204-4 · Zbl 0874.14042 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.