The projectivity of the moduli space of stable curves. II: The stacks \(M_{g,n}\). (English) Zbl 0544.14020

Knudsen, Finn F.

Math. Scand. 52, 161-199 (1983).

[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

Keywords:

families of stable curves with n base points; contraction morphism

Citations:

Zbl 0544.14021; Zbl 0343.14008