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Moduli space of stable maps to projective space via GIT. (English) Zbl 1191.14015

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.

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
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