Cover Page
The handle
http://hdl.handle.net/1887/82073
holds various files of this Leiden University
dissertation.
Author: Akeyr, G.
Dual complexes of semistable varieties
Proefschrift
ter verkrijging van
de graad van Doctor aan de Universiteit Leiden, op gezag van Rector Magnificus prof.mr. C.J.J.M. Stolker,
volgens besluit van het College voor Promoties te verdedigen op dinsdag 17 december 2019
klokke 12.30 uur
door
Garnet Jonathan Akeyr
promotor: prof.dr. Bas Edixhoven Co-promotor: dr. David Holmes Samenstelling van de promotiecommissie:
Contents
Introduction 1
1 Separated quotients of Picard schemes 5
1.1 Introduction . . . 5
1.1.1 Idea of proof . . . 6
1.1.2 Overview of chapter . . . 7
1.2 Semistable morphisms and alignment . . . 7
1.2.1 Definition of semistable morphism . . . 7
1.2.2 Graphs and alignment . . . 10
1.2.3 Alignment of a semistable morphism . . . 11
1.3 Behaviour of strata under specialisation and generisation . . 12
1.3.1 Specialisation map on vertices . . . 12
1.3.2 The specialisation map on edges . . . 14
1.3.3 The specialisation map . . . 18
1.4 Cartier divisors and alignment . . . 20
1.4.1 Constructing Cartier divisors on X . . . 20
1.4.2 Cartier labellings on graphs . . . 22
1.4.3 Alignment and Cartier divisors . . . 24
1.5 The Picard scheme and alignment . . . 27
2 Families of dual complexes of normal crossings varieties 37 2.1 Introduction . . . 37
2.1.1 Overview of chapter . . . 38
2.2 Semistable morphisms . . . 39
2.2.1 Basic definitions . . . 39
2.2.2 Strata of fibres . . . 40
2.3 Dual complexes of semistable varieties . . . 46
2.3.1 Generalised ∆-complexes . . . 47
2.3.2 Dual delta complexes associated to a semistable mor-phism . . . 50
2.3.3 Generalised cone complexes . . . 52
CONTENTS
3 Artin fans and dual graphs 63
3.1 Introduction . . . 63
3.1.1 Overview of chapter . . . 65
3.2 Basics of logarithmic geometry . . . 65
3.2.1 Facts on monoids . . . 66
3.2.2 Logarithmic schemes . . . 69
3.2.3 Log smoothness . . . 72
3.2.4 Artin fans . . . 74
3.3 Setup . . . 76
3.4 Artin fans and the universal deformation . . . 80
3.5 Artin fans of log schemes . . . 88
3.6 Dual graph from the Artin fan . . . 91
Bibliography 97
Summary 101
Nederlandse samenvatting 105
Acknowledgements 109
