# zbMATH — the first resource for mathematics

The projectivity of the moduli space of stable curves. II: The stacks $$M_{g,n}$$. (English) Zbl 0544.14020
[For part I see ibid. 39, 19-55 (1976; Zbl 0343.14008).]
The object of study in this article are families of stable curves with n base points. A family of curves with arithmetic genus g and n base points is an algebraic stack $$M_{g,n}$$. If in a family with $$n+1$$ base points we forget the last base point we in general no longer have a family of stable n-pointed curves. The main result of this paper is that there is a unique way of contracting extra stuff so as to get n-pointed stable curves from $$(n+1)$$-pointed curves. The contraction morphism $$M_{g,n+1}\to M_{g,n}$$ is in fact the universal n-pointed stable curve of genus g, i.e. it is representable. - On the stack $$M_{g,n}$$ the singular curves form a divisor. This divisor splits into a certain number of irreducible components. Each component is the image of a clutching map $$M_{g_ 1,g_ 1+1}\times M_{g_ 2,n_ 2+1}\to M_{g_ 1+g_ 2,n_ 1+n_ 2}$$ or $$M_{g-1,n+2}\to M_{g,n}$$ which is obtained by gluing together two base points. These clutching maps are also representable morphisms. In fact they are almost always closed immersions.

##### MSC:
 14H10 Families, moduli of curves (algebraic) 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14D20 Algebraic moduli problems, moduli of vector bundles
Full Text: