Totally degenerated formal schemes [Elektronische Ressource] / vorgelegt von Rolf Stefan Wilke

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• Mathematics

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• Publié par: universitat_ulm
• Publié le: 01 janvier 2009
• Nombre de lectures: 25
• Langue: English

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Totally Degenerated Formal Schemes
Dissertation zur Erlangung des Doktorgrades Dr. rer. nat.
der Fakultät für Mathematik und Wirtschaftswissenschaften
der Universität Ulm
Vorgelegt von Rolf Stefan Wilke aus Arolsen
Ulm, 2009Dekan: Prof. Dr. Werner Kratz
Erster Gutachter: Prof. Dr. Werner Lütkebohmert
Zweiter Prof. Dr. Irene I. Bouw
Tag der Promotion: 4. Dezember 2009“It is a mistake to think you can solve any major problems just with potatoes.”
Douglas Adams: Life, The Universe and EverythingContents
Introduction iii
1 Formal and Rigid Geometry 1
1.1 Rigid Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Admissible Formal Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Formal Cartier and Weil Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Convex Geometry and Toric Varieties 9
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Toric Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Simplicial Homology and Cohomology . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Polytopal Complexes with Integral Structure . . . . . . . . . . . . . . . . . . . 15
n3 Polytopal Domains in G 17m
3.1 Definitions and First Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Subdivisions and Admissible Formal Blowing Ups . . . . . . . . . . . . . . . . 25
3.3 Cartier Divisors, Line Bundles and Polyhedral Functions . . . . . . . . . . . . 31
3.4 Strictly Semi-Stable Formal Models . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.5 Ampleness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.6 on the Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4 Totally Degenerated Formal Schemes 49
4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 The Universal Covering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3 The Picard Variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4 Automorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.5 General Polytopal Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5 Examples 81
5.1 Mumford Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
iii Contents
5.2 Analytic Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.3 The Hopf Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.4 A Rigid Analytic Klein Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.5 The Sheared Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.6 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6 Affinoid Polytopal Domains are Factorial 107
6.1 Van der Put’s Base Change Theorem . . . . . . . . . . . . . . . . . . . . . . . . 108
6.2 The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Bibliography 123
Zusammenfassung (deutsch) 127Introduction
In this thesis, we introduce a new class of rigid analytic varieties over a complete non-
archimedean field K ; namely those which have a totally degenerated formal model. These are
natural generalizations of the well known Mumford curves to arbitrary dimension. Similar
to the one-dimensional case, we will show that the Picard variety of these varieties is given
g g
by a quotient G =M , where M is a lattice in G , not necessarily of full rank.m;K m;K
To any smooth projective curve X over C (or, equivalently: a compact Riemann sur-
face) of genus g , one can associate its Jacobian variety Jac(X) ; an abelian variety which
parametrizes the equivalence classes of divisors on X of degree 0 . The well-known Torelli
theorem states that X is uniquely determined by its (principal polarized) Jacobian. This
makes the Jacobian a very important object for the study of Riemann surfaces. If X is a
Riemann surface of genus 1 , i.e. an analytic torus C= , the Jacobian Jac(X) is canon-
ically isomorphic to X itself. In general, the Jacobian is analytically isomorphic to a g -
g gdimensional analytic torus C =M , where M is a lattice in C of rank g , the so-called
period lattice.
Over a complete non-archimedean valued field K , such as the p -adic numbers Q , thep
above situation does not extend without modification. In general, it is not true that the
g
Jacobian of a curve is given by an analytic torus G =M . This is related to the fact thatm;K
only a certain class of p -adic curves has a complex analog; namely, the so-called Mumford
curves. These curves X have a formal model X over the valuation ring R such thatK
1every irreducible component of the special fibre X is isomorphic to P and X has only0 0
ordinary double points as singularities. Mumford proved in [28] that these are precisely the
curves which have a Schottky uniformization
= , where PGL(2;K) is a SchottkyK
1group, and
P is the set of points where acts discontinuously; this is a directK K
analog of the classical Schottky uniformization over the complex numbers. In [26], Manin
and Drinfeld proved that, as in the complex case, the Jacobian variety of a Mumford curve
g
of genus g is again isomorphic to an analytic torus G =M , where M is a multiplicativem;K
glattice in G of rank g .m;K
iiiiv Introduction
0If dimX > 1 , the analog of the Jacobian variety Jac(X) is the Picard variety Pic (X) ,
which represents certain isomorphy classes of line bundles. The existence of the Picard
variety of proper algebraic schemes over a field has been proven in the 1960s. An analogous
result in the category of proper rigid analytic varieties over a complete discretely-valued
field K was established much later, in 2000, by Hartl and Lütkebohmert [21].
0In this thesis, we will deal with the question when the Picard variety Pic (X ) of a properK
grigid-analytic variety X over K is again an analytic torus G =M . The example ofK m;K
Mumford curves already shows that one can expect this to be true only in very special
cases. In the work of Hartl and Lütkebohmert [21], it becomes apparent that the
fibre of a suitable formal model plays a key role in determining the structure of the Picard
variety. This motivates the following generalization of Mumford curves:
We say a proper rigid-analytic variety X over K has a totally degenerated model X over RK
if the special fibre X of X consists of smooth rational components with normal crossings;0
i.e. locally, X looks like the intersection of some coordinate hyperplanes in the affine space0
r
A (see Definition 4.1.1 for the precise conditions).
Theorem 4.3.5. Let X be the generic fibre of a totally degenerated formal scheme which is proper.K
0On the category of smooth and connected rigid spaces, the Picard functor Pic is represented byX =KK
a quotient T =M , where T is a split torus, and M is a lattice in T such that M\T =f1g .K K K K
0If X is algebraizable, it is well-known that Pic (X ) is always proper; i.e. M has fullK K
rank g . If X is not algebraizable, however, this need not be true. A standard example isK
the Hopf surface, introduced in the rigid analytic framework by Mustafin [29], which also
has a totally degenerate model.
Generalizing the techniques for Mumford curves, we construct a suitable uniformization
X =
= . As in the case of analytic tori, we show that any line bundle on X whichK K K
0corresponds to a point of Pic (X ) pulls back to the trivial line bundle on
. Hence,K K
line bundles on X can be described by -linearizations of constant type of the trivial lineK
0bundle on
. This allows us to describe Pic (X ) in terms of automorphic functions:K K
gb e eTheorem 4.4.12. Let J := Hom( ;G ) G , where is the free part of the abelianization=m;K m;K
=[ ; ] of , and let
bM :=fc2J ; c is the factor of automorphy for an invertible functionf on
gK
0b bThen M is a lattice in J , and the quotient J :=J=M represents Pic (X ) .K