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Stable $$n$$-pointed trees of projective lines. (English) Zbl 0698.14019
The object of this paper is the classification of stable $$n$$-pointed trees. Stable n-pointed trees arise in a natural way if one tries to find moduli for totally degenerate curves. The authors prove in particular the existence of a fine moduli space $$B_ n$$ of stable n-pointed trees.
A connected projective variety C over a field k together with an n-tuple $$\phi =(\phi_ 1,...,\phi_ n)$$ of distinct k-rational points of C is called a stable n-pointed tree $$(C,\phi)$$ of projective lines over k if
(1) every component of C is isomorphic to the projective line over k;
(2) every singular point of C is k-rational and an ordinary double point;
(3) the intersection graph of the components of C is a tree;
(4) the set $$\{\phi_ 1,...,\phi_ n\}\cup \{\text{singular points of C}\}$$ has at least 3 points on every component of C, and
(5) $$\phi_ 1,...,\phi_ n$$ are regular points on $$C.$$
$$B_ n$$ is a closed subscheme of a product of projective lines over $${\mathbb{Z}}$$ given by a multihomogeneous ideal which is written down in the paper. The k-valued points of $$B_ n$$ are in 1-1-correspondence with the isomorphy classes of stable n-pointed trees of projective lines over k. $$B_ n$$ is smooth and of relative dimension 2n-3 over $${\mathbb{Z}}$$. The canonical projection $$B_{n+1}\to B_ n$$ is the universal family of stable n-pointed trees. The Picard group of $$B_ n$$ is free of rank $$2^{n-1}-(n+1)-n(n-3)/2.$$ The authors give a method to compute the Betti numbers of B($${\mathbb{C}})$$ (see the following review). Further, it turns out that $$B_ n$$ is a blow-up of the quotient of semi-stable points in $${\mathbb{P}}^ n_ 1$$ for the action of fractional linear transformations in every component. The authors describe the blow-up in several steps where at each stage the obtained space is interpreted as a solution to a certain moduli problem.
Reviewer: K.Drechsler

##### MSC:
 14H10 Families, moduli of curves (algebraic) 14F45 Topological properties in algebraic geometry 14C22 Picard groups 14D22 Fine and coarse moduli spaces