Stable n-pointed trees of projective lines
Stable n-pointed trees of projective lines
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稳定的 n 点投影线树
DOI：
10.1016/s1385-7258(88)80024-6
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发表时间：
1988
期刊：
影响因子：
--
通讯作者：
M. Put
中科院分区：
文献类型：
--
作者：
L. Gerritzen;F. Herrlich;M. Put
如果我们试图寻找完全退化曲线的模，那么稳定点树以一种自然的方式出现：设C是域k上亏格g ≥ 2的完全退化稳定曲线。这意味着C是算术亏格的连通射影曲线，满足：（a）C的每个不可约分支是κ上的有理曲线。(b)C的每一个奇点都是κ-有理常偶点。(c)每个非奇异分支LofC至少满足C-Lin三点。总有可能找到g个奇异点P1，...，PgonC使得爆炸超过CatP 1，...，Pk是一条连通的射影曲线，它具有以下性质：o（i）C的每一不可约分支同构于Pk 1，（ii）C的分支相交于普通的κ-有理二重点，（iii）C的交图是一棵树，态射φ：C → C是2个顶点Q1，Q1′，Qg，Qg′外的同构，Qi与Qj ′相同.是由g对正则κ-有理点（Qi，Qi′）唯一确定的。一条满足（i）-（iii）的曲线C上有n个κ-有理正则点，称为射影线的点树，C是稳定的，如果在它的每个分支上至少有三个点是奇异点或标记点。本文的研究对象是稳定点树的分类。我们特别证明了稳定点树的细模空间Bn的存在性。上述讨论表明，存在一个满射π B2 g → DgofB 2g到亏格g的稳定曲线对应的粗模概型Mg的闭子概型Dg上。根据Mg的泛性质，π是一个（有限）态射。π因子通过B2 g = B2 gmod 2g元的保对置换群的作用（一个2gg阶群，同构于Sg与π/2的圈积）诱导态射π：B2 g → D是D g中不可约曲线的开子概型上的同构，但一般地，对于上述构造，在完全退化曲线上可能存在g个奇点的非等价选择，因此π具有非平凡纤维。特别地，π不是B2 g上群作用的商映射。这就引出了为完全退化的曲线构造一个Teichmüller空间的想法，这些曲线的不可约分支同构于B2 g，并且一个不连续群作用在其上使得商精确地为Dg; π将是这个商映射到单个不可约分支的限制。本文只考虑稳定点树及其模理论。在§ 1中，我们引入了四个点（不一定在同一条射影线上）的抽象交比，并证明了对于域κ，交比的射影簇Bn中的κ值点与κ上射影线的稳定点树的同构类是1 − 1对应的。我们还描述了具有固定组合类型的稳定点树的子变量B（T，T）的结构，并将§ 2中的概念推广到任意Noether基概型上的投射线的稳定点树，并说明了纤维的交比如何与S上的态射相吻合。这一节与[Kn]密切相关，但由于我们处理的是一种特殊情况，所以它更基本。§ 3包含了本文的主要结果：标准投影Bn + 1→ Bn是稳定点树的泛族。作为证明的一个副产品，我们发现Bn是一个相对维数为2n- 3的光滑投影概型。在§ 4中，我们证明了Bn的Picard群不含秩2n −1 −（n +1）− n（n − 3）/2，并给出了一种方法，证明了Bn的Picard群不含秩2n −1−（n+1）−n（n−3）/2。
Stablen-pointed trees arise in a natural way if one tries to find moduli for totally degenerate curves: LetCbe a totally degenerate stable curve of genusg ≥ 2over a field k. This means thatCis a connected projective curve of arithmetic genusgsatisfyingo(a) every irreducible component ofCis a rational curve over κ.(b) every singular point ofCis a κ-rational ordinary double point.(c) every nonsingular componentLofCmeetsC−Lin at least three points. It is always possible to find g singular pointsP1,..., PgonCsuch that the blow upCofCatP1,..., Pgis a connected projective curve with the following properties:o(i) every irreducible component ofCis isomorphic to Pk1(ii) the components ofCintersect in ordinary κ-rational double points(iii) the intersection graph ofCis a tree.The morphism φ : C → C is an isomorphism outside 2gregular pointsQ1, Q1′, Qg, Qg′and identifiesQiwithQj′. is uniquely determined by the g pairs of regular κ-rational points (Qi, Qi′). A curveCsatisfying (i)-(iii) together withnκ-rational regular points on it is called an-pointed tree of projective lines.Cis 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 stablen-pointed trees. We prove in particular the existence of a fine moduli spaceBnof stablen-pointed trees. The discussion above shows that there is a surjective mapπB2g→ DgofB2gonto the closed subschemeDgof the coarse moduli schemeMgof stable curves of genusgcorresponding to the totally degenerate curves. By the universal property ofMg, π is a (finite) morphism. π factors throughB2g= B2gmod the action of the group of pair preserving permutations of 2gelements (a group of order 2gg, isomorphic to a wreath product ofSgand ℤ/2ℤThe induced morphismπ: B2g→ Dgis an isomorphism on the open subscheme of irreducible curves inDg, but in general there may be nonequivalent choices ofgsingular 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 onB2g. This leads to the idea of constructing a Teichmüller space for totally degenerate curves whose irreducible components are isomorphic toB2gand on which a discontinuous group acts such that the quotient is preciselyDg; 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 stablen-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 varietyBnof cross ratios are in 1 − 1 correspondence with the isomorphy classes of stablen-pointed trees of projective lines over κ. We also describe the structure of the subvarietiesB(T, ψ) of stablen-pointed trees with fixed combinatorial type.We generalize our notion in § 2 to stablen-pointed trees of projective lines over an arbitrary noetherian base schemeSand show how the cross ratios for the fibres fit together to morphisms onS. 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 projectionBn + 1→ Bnis the universal family of stablen-pointed trees. As a by-product of the proof we find thatBnis a smooth projective scheme of relative dimension 2n- 3 over ℤ. We also compareBnto the fibre productBn−1×Bn-2Bn − 1and investigate the singularities of the latter.In § 4 we prove that the Picard group ofBnis free of rank2n−1−(n+1)−n(n−3)/2.We also give a method …