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A geometric invariant theory construction of moduli spaces of stable maps. (English) Zbl 1155.14009

Let \(\overline{M}_{g, n}({\mathbb{P}}^r, d)\) be the Kontsevich–Manin coarse moduli space of stable maps of degree \(d\) from \(n\)-pointed genus \(g\) curves into the \(r\)-dimensional projective space \({\mathbb{P}}^r\). The authors construct \(\overline{M}_{g, n}({\mathbb{P}}^r, d)\) via geometric invariant theory. The method follows the one used in the case \(n=0\) in [D. Gieseker, Lectures on moduli of curves. Lectures on Mathematics and Physics. Mathematics, 69. Tata Institute of Fundamental Research, Bombay. Berlin-Heidelberg-New York: Springer (1982; Zbl 0534.14012)], but the proof that the semistable set is nonempty is entirely different. The construction is only valid over \(\text{Spec}\,{\mathbb{C}}\), but a special case, a GIT presentation of the moduli space of stable curves of genus \(g\) with \(n\) marked points, is valid over \(\text{Spec}\,{\mathbb{Z}}\).

MSC:

14D20 Algebraic moduli problems, moduli of vector bundles
14H10 Families, moduli of curves (algebraic)

Citations:

Zbl 0534.14012
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