Equivariant Embeddings of Algebraic Groups

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Table of Contents Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Toric varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Luna–Vust theory and spherical varieties . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 The “wonderful compactification” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Affine group embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Chapter 2 Toric varieties and group embeddings . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1 Toric varieties as group embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.1 Affine toric varieties and one-parameter subgroups . . . . . . . . . . . . . . . 12 2.1.2 Toric varieties from fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1.3 The torus action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.4 Equivariant divisors in toric varieties . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Toric varieties in group embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.1 Regular compactifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.2 A family of left-equivariant SL n -embeddings . . . . . . . . . . . . . . . . . . 19 2.3 Group embeddings from toric varieties . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.1 Mumford’s group embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.2 Group embeddings constructed using flag varieties . . . . . . . . . . . . . . . 23 Chapter 3 Affine group embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.1 Group actions on affine varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2 Classification of affine group embeddings . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2.1 Limits of one-parameter subgroups . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2.2 States and strongly convex lattice cones . . . . . . . . . . . . . . . . . . . . . 48 3.2.3 Classification theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.3 Functoriality of our classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3.1 Equivariant morphisms between affine G-embeddings . . . . . . . . . . . . . . 58 3.3.2 Biequivariant resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 vi
Chapter 1 Introduction A basic problem in algebraic geometry is the study of algebraic actions. Even actions with a dense orbit are not well understood. A quasihomogeneous variety is a normal G-variety X possessing an open orbit isomorphic to G/H, for some closed subgroup H of G. Toric varieties are an important family of quasihomogeneous varieties, where G is an algebraic torus and H is its trivial subgroup. A partial classification of quasihomogeneous varieties was obtained in the important paper of Luna and Vust [30], written twenty years ago. Their classification seeks to generalize that of toric varieties, but it is only feasible in special cases. The goal of this dissertation is to solve the equivariant classification problem for one case not covered in [30], namely the classification of quasihomogeneous varieties in which H is trivial. In this introduction, we review the history of and some of the results that are essential to our problem. As mentioned above, toric varieties are the natural place to begin, so we discuss them in the first section. After that, we present a short overview of Luna and Vust’s paper [30] and some of the work that has been done in an attempt to exploit the Luna–Vust classification in cases that initially appear to be outside the scope of their original work. This chapter concludes with an overview of our results. First, we establish a few conventions that will be used throughout this thesis and notation that we use freely hereafter. We will always work over a ground field k, which we assume to be algebraically closed and of characteristic zero. The term variety will refer to a separated, integral scheme of finite type over k. All algebraic groups are assumed to be linear and defined over k, and will be denoted by letters such as G and H. In particular, G will refer to a connected reductive linear algebraic group defined over k, unless otherwise stipulated. 1
Page 1 and 2: c○ Copyright by David Charles Mur
Page 3 and 4: Abstract We classify embeddings of
Page 5: Acknowledgements This thesis would
Page 9 and 10: x i ≠ 0} for i = 0, 1, 2, the one
Page 11 and 12: complement of U in G/H. Remark 1 ([
Page 13 and 14: G. The wonderful compactification o
Page 15 and 16: X is to use one-parameter subgroups
Page 17 and 18: Chapter 2 Toric varieties and group
Page 19 and 20: Proposition 3 ([28], Proposition I.
Page 21 and 22: and every toric variety for T arise
Page 23 and 24: To see this, suppose D is a T -stab
Page 25 and 26: Conversely, every subdivision of C
Page 27 and 28: 0 : 0 : 1 : 1], [0 : 0 : 0 : 1 : 1]
Page 29 and 30: Mumford’s embedding is constructe
Page 31 and 32: Moreover, Ind(X) is a G-embedding,
Page 33 and 34: the T -action with respect to the c
Page 35 and 36: from X with a line bundle from the
Page 37 and 38: G/B × G/B − corresponding to the
Page 39 and 40: since G (and hence G × G) is semi-
Page 41 and 42: 1. There is a finite-dimensional G-
Page 43 and 44: the following (obviously) commutati
Page 45 and 46: Now, to prove the claim, by Lemma 7
Page 47 and 48: ational polyhedral cone in X ∗ (T
Page 49 and 50: ⋃ P ⊃T P • (σ ∩ ∆ P (T )
Page 51 and 52: Section 3.2.3, is a uniqueness theo
Page 53 and 54: Therefore, the classification of af
Page 55 and 56: For each k-point g of G, we have ma
Page 57 and 58:
is over a finite set of integers J.
Page 59 and 60:
i ≥ 0, 〈χ ij , γ (0,1) 〉 =
Page 61 and 62:
Lemma 12. If γ 1 , γ 2 ∈ X ∗
Page 63 and 64:
an integrally closed left-invariant
Page 65 and 66:
of the subalgebras A Γ(X,x0 ) ⊂
Page 67 and 68:
on varieties discussed in this sect
Page 69 and 70:
Proof. Let Γ be a strongly convex
Page 71 and 72:
that Γ(X, [I 2 , (1, 1)]) GL 2 = {
Page 73 and 74:
References [1] A. Bialynicki-Birula
Page 75 and 76:
[24] B. Iversen, Cohomology of Shea
Page 77:
Vita David Charles Murphy was born
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Equivariant Embeddings of Algebraic Groups

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