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**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.

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.