Stable n-pointed trees of projective lines

Abstract

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 satisfying o (a) every irreducible component of C is a rational curve over κ. (b) every singular point of C is a κ-rational ordinary double point. (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,..., Pg on C such that the blow up C of C at P1,..., Pg is a connected projective curve with the following properties: o (i) every irreducible component of C is isomorphic to Pk1 (ii) the components of C intersect in ordinary κ-rational double points (iii) the intersection graph of C is a tree. The morphism φ : C → C is an isomorphism outside 2g regular points Q1, Q1′, Qg, Qg′ and identifies Qi with Qj′. is uniquely determined by the g pairs of regular κ-rational points (Qi, Qi′). 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 Bn of stable n-pointed trees. The discussion above shows that there is a surjective map πB2g → Dg of B2g onto the closed subscheme Dg of the coarse moduli scheme Mg of stable curves of genus g corresponding to the totally degenerate curves. By the universal property of Mg, π is a (finite) morphism. π factors through B2g = B2g mod the action of the group of pair preserving permutations of 2g elements (a group of order 2gg, isomorphic to a wreath product of Sg and ℤ/2ℤ The induced morphism π: B2g → Dg is an isomorphism on the open subscheme of irreducible curves in Dg, 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 B2g. This leads to the idea of constructing a Teichmuller space for totally degenerate curves whose irreducible components are isomorphic to B2g and on which a discontinuous group acts such that the quotient is precisely Dg; 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 Bn 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 Bn + 1 → Bn is the universal family of stable n-pointed trees. As a by-product of the proof we find that Bn is a smooth projective scheme of relative dimension 2n - 3 over ℤ. We also compare Bn to the fibre product Bn−1 × Bn-2 Bn − 1 and investigate the singularities of the latter. In § 4 we prove that the Picard group of Bn 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 Bn(ℂ). In § 5 we compare Bn to the quotient Qn: = ℙssn/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 Bn is a blow-up of Qn, 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.