Stable \(n\)-pointed trees of projective lines.

*(English)*Zbl 0698.14019The 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.

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 |