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Compactifying the space of stable maps. (English) Zbl 0991.14007
Let $${\mathcal M}\to S$$ be a proper Deligne-Mumford stack admitting a coarse moduli space $${\mathbf M}\to S$$ with a fixed polarization. If $$C$$ is a nodal projective curve, a morphism $$C\to{\mathcal M}$$ is said to be a stable map of degree $$d$$ if the associated morphism $$C\to{\mathbf M}$$ is a stable map of degree $$d$$. The results proved in this paper imply that the category of stable maps into $$\mathcal M$$ is a Deligne-Mumford stack. However, this stack is not complete. The main aim of the paper under review is to correct this deficiency. In order to do that the authors enlarge the category of stable maps into $$\mathcal M$$. The source curve $$\mathcal C$$ of a new stable map $${\mathcal C}\to {\mathcal M}$$ will acquire an orbispace structure at its nodes, and the authors endow it with a structure of a Deligne-Mumford stack. Specifically, the authors define the category $${\mathcal K}_{g,n}({\mathcal M},d)$$, fibered over $$\text{Sch}/S$$, of twisted stable $$n$$-pointed maps $${\mathcal C}\to {\mathcal M}$$ of genus $$g$$ and degree $$d$$ in two different (but equivalent) ways:
(a) as a category of stable twisted $${\mathcal M}$$-valued objects over nodal pointed curves endowed with atlases of orbispace charts, and
(b) as a category of representable maps from pointed nodal Deligne-Mumford stacks into $$\mathcal M$$, such that the map on coarse moduli spaces is stable.
Then the main result proved in this paper (whose proof makes use of both ways of defining $${\mathcal K}_{g,n}({\mathcal M},d)$$) is the following theorem:
(i) The category $${\mathcal K}_{g,n}({\mathcal M},d)$$ is a proper algebraic stack.
(ii) The coarse moduli space $${\mathbf K}_{g,n}({\mathcal M},d)$$ of $${\mathcal K}_{g,n}({\mathcal M},d)$$ is projective.
(iii) There are canonical maps $$f:{\mathcal K}_{g,n}({\mathcal M},d)\to {\mathcal K}_{g,n}({\mathbf M},d)$$, $$g:{\mathcal K}_{g,n}({\mathbf M},d)\to {\mathbf K}_{g,n}({\mathbf M},d)$$, $$h:{\mathcal K}_{g,n}({\mathcal M},d)\to {\mathbf K}_{g,n}({\mathcal M},d)$$, and $$k:{\mathbf K}_{g,n}({\mathcal M},d)\to {\mathbf K}_{g,n}({\mathbf M},d)$$ such that $$g\circ f=k\circ h$$, $$f$$ is proper, quasifinite, relatively of Deligne-Mumford type and tame, and $$k$$ is finite.
In particular, if $${\mathcal K}_{g,n}({\mathbf M},d)$$ is a Deligne-Mumford stack, so is $${\mathcal K}_{g,n}({\mathcal M},d)$$. The authors’ approach was suggested by Kontsevich’s moduli of stable maps.

##### MSC:
 14D20 Algebraic moduli problems, moduli of vector bundles 14H10 Families, moduli of curves (algebraic)
##### Keywords:
coarse moduli space; stable curve; Deligne-Mumford stack
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##### References:
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