Stable n-pointed trees of projective lines

Gerritzen, Lothar; Herrlich, Frank; Vanderput, M

doi:10.1016/s1385-7258(88)80024-6

Cited by 28 publications

(59 citation statements)

References 6 publications

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“…We call the points of ∂M 0,n (Q p ) stable trees of dendrograms or, by abuse of language, simply stable. In fact, these are the so-called stable n-pointed trees of projective lines [11]. Such are algebraic curves C which are unions of projective lines L together with n points X = {x 1 , .…”

Section: Allowing Collisionsmentioning

confidence: 99%

Degenerating Families of Dendrograms

Bradley

2008

J Classif

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Abstract. Dendrograms used in data analysis are ultrametric spaces, hence objects of nonarchimedean geometry. It is known that there exist p-adic representation of dendrograms. Completed by a point at infinity, they can be viewed as subtrees of the Bruhat-Tits tree associated to the p-adic projective line. The implications are that certain moduli spaces known in algebraic geometry are p-adic parameter spaces of (families of) dendrograms, and stochastic classification can also be handled within this framework. At the end, we calculate the topology of the hidden part of a dendrogram.

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Section: Allowing Collisionsmentioning

confidence: 99%

Degenerating Families of Dendrograms

Bradley

2008

J Classif

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“…It is known that M 0; r is represented by a smooth projective scheme over Z, see [11]. M 0; r be the natural forgetful morphism.…”

Section: Completementioning

confidence: 99%

Reduction of covers and Hurwitz spaces

Bouw¹,

Wewers²

2004

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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In this paper we study the reduction of Galois covers of curves, from characteristic zero to positive characteristic. The starting point is a recent result of Raynaud, which gives a criterion for good reduction for covers of the projective line branched at three points. We use the ideas of Raynaud to study the case of covers of the projective line branched at four points. Under some condition on the Galois group, we generalize the criterion for good reduction of Raynaud. As a new ingredient, we use the Hurwitz space of such covers. Combining our results on reduction of covers with the Hurwitz space approach, we are able to describe the reduction of the Hurwitz space modulo p and compute the number of covers with good reduction.In this paper, we follow the approach of Raynaud. We consider the reduction of Gcovers f K : Y K ! P 1 K branched at four points. We suppose that pk jGj, but that p does not divide the ramification indices of f K . This is the next case to study after the result of Raynaud. However, we put a much stronger condition on the group G. In particular, we assume that n ¼ 2. The criterion for good reduction given by Raynaud extends to our situa-Brought to you by | University of Connecticut Authenticated Download Date | 5/30/15 11:48 PM Brought to you by | University of Connecticut Authenticated Download Date | 5/30/15 11:48 PM Definition 1.1.2. Let f R : Y R ! X R and f 0; R : Y R ! X 0; R be as above. We call Bouw and Wewers, Reduction of covers and Hurwitz spaces 5 Brought to you by | University of Connecticut Authenticated Download Date | 5/30/15 11:48 PM 1.2.1. The ambient stack. Let ðX =S; CÞ and ðY =S; DÞ be stably marked curves, defined over the same scheme S. A morphism of stably marked curves is an S-morphism Bouw and Wewers, Reduction of covers and Hurwitz spaces 7 Brought to you by | University of Connecticut Authenticated Download Date | 5/30/15 11:48 PM

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“…The morphism π admits a well-known projective completion that is still denoted by π. This is the canonical morphism from the (projective) moduli space of curves of genus zero with 5 marked points M 0,5 to the one with 4 marked points M 0,4 (we refer to [GHP88] for a highly comprehensive study of the spaces M 0,n from an algebraic view point. A lot of ideas contained in this paper are used here).…”

Section: Explicit Computationmentioning

confidence: 99%

“…To begin with, let's recall that there exist four sections s 1 , s 2 , s 3 and s 4 of π, corresponding to the four marked points (see [GHP88], Section 3). We define x and y to be the coordinates on M 0,5 such that x = y = ∞ on s 1 , x = 1 on s 2 , x = y = 0 on s 4 and y = 1 on s 3 .…”

Section: Explicit Computationmentioning

confidence: 99%

“…Since the key point of this section is the explicit algebraic structure of the compactification of moduli spaces of curves of genus zero, we will assume that the reader has some familiarities with these notions. We just recall (see [GHP88], Section 3) that the fiber C b = π −1 (b) is a stable 4-pointed tree of projective lines made of two irreducible components. Let us denote by C b,1 the irreducible component of C b containing the two closed points z 1 and z 2 , and C b,2 the other one.…”

Section: Degenerate Covers and Their Computationmentioning

confidence: 99%

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