Stable n-pointed trees of projective lines

Stable n-pointed trees arise in a natural way if one tries to find moduli for totally degenerate curves: Let C be a totally degenerate stable curve of genus g ≥ 2 over a field k. This means that C is a connected projective curve of arithmetic genus g satisfyingo

1. (a) every irreducible component of C is a rational curve over κ.

2. (b) every singular point of C is a κ-rational ordinary double point.

3. (c) every nonsingular component L of C meets C−L in at least three points. It is always possible to find g singular points P₁,..., P_g on C such that the blow up C of C at P₁,..., P_g is a connected projective curve with the following properties:o

   1. (i) every irreducible component of C is isomorphic to P_k¹

   2. (ii) the components of C intersect in ordinary κ-rational double points

   3. (iii) the intersection graph of C is a tree.

The morphism φ : C → C is an isomorphism outside 2g regular points Q₁, Q₁′, Q_g, Q_g′ and identifies Q_i with Q_j′. is uniquely determined by the g pairs of regular κ-rational points (Q_i, Q_i′). A curve C satisfying (i)-(iii) together with n κ-rational regular points on it is called a n-pointed tree of projective lines. C is stable if on every component there are at least three points which are either singular or marked. The object of this paper is the classification of stable n-pointed trees. We prove in particular the existence of a fine moduli space B_n of stable n-pointed trees. The discussion above shows that there is a surjective map πB_2g → D_g of B_2g onto the closed subscheme D_g of the coarse moduli scheme M_g of stable curves of genus g corresponding to the totally degenerate curves. By the universal property of M_g, π is a (finite) morphism. π factors through B_2g = B_2g mod the action of the group of pair preserving permutations of 2g elements (a group of order 2^gg, isomorphic to a wreath product of S_g and ℤ/2ℤ

The induced morphism π: B_2g → D_g is an isomorphism on the open subscheme of irreducible curves in D_g, but in general there may be nonequivalent choices of g singular points on a totally degenerated curve for the above construction, so π has nontrivial fibres. In particular, π is not the quotient map for a group action on B_2g. This leads to the idea of constructing a Teichmüller space for totally degenerate curves whose irreducible components are isomorphic to B_2g and on which a discontinuous group acts such that the quotient is precisely D_g; π will then be the restriction of this quotient map to a single irreducible component. This theory will be developped in a subsequent paper.

In this paper we only consider stable n-pointed trees and their moduli theory. In § 1 we introduce the abstract cross ratio of four points (not necessarily on the same projective line) and show that for a field κ the κ-valued points in the projective variety B_n of cross ratios are in 1 − 1 correspondence with the isomorphy classes of stable n-pointed trees of projective lines over κ. We also describe the structure of the subvarieties B(T, ψ) of stable n-pointed trees with fixed combinatorial type.

We generalize our notion in § 2 to stable n-pointed trees of projective lines over an arbitrary noetherian base scheme S and show how the cross ratios for the fibres fit together to morphisms on S. This section is closely related to [Kn], but it is more elementary since we deal with a special case.

§ 3 contains the main result of the paper: the canonical projection B_{n + 1} → B_n is the universal family of stable n-pointed trees. As a by-product of the proof we find that B_n is a smooth projective scheme of relative dimension 2n - 3 over ℤ. We also compare B_n to the fibre product B_n−1 × _Bn-2 B_{n − 1} and investigate the singularities of the latter.

In § 4 we prove that the Picard group of B_n is free of rank 2ⁿ⁻1−(n+1)−n(n−3)/2.

We also give a method to compute the Betti numbers of the complex manifold B_n(ℂ).

In § 5 we compare B_n to the quotient Q_n: = ℙ_ssⁿ/PGL₂ of semi-stable points in ℙ₁ⁿ for the action of fractional linear transformations in every component. This orbit space has been studied in greater detail by several authors, see [GIT], [MS], [G]. It turns out that B_n is a blow-up of Q_n, and we describe the blow-up in several steps where at each stage the obtained space is interpreted as a solution to a certain moduli problem.