Equivariant Embeddings of Algebraic Groups

More documents

Recommendations

Info

$Math 214, Calculus III Final Exam Solutions 1. (25 points) Suppose ...$

$Math 320, Real Analysis I Quiz 2 — Solutions 1. Give an example of ...$

$Math 113, Section 2 Test 2 Solutions 1. (25 points) Consider the ...$

$Math 320, Real Analysis I Quiz 1 — Solutions 1. (a) Define what it ...$

$Math 440, Linear Algebra II Solutions to Homework 9 Problems 11-5 ...$

[10] D. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4:1 (1995), 17–50. [11] D. Cox and S. Katz, Mirror Symmetry and Algebraic Geometry, Mathematical Surveys and Monographs 68, American Mathematical Society, 1999. [12] V.I. Danilov, The geometry of toric varieties, Uspekhi Mat. Nauk. 33:2 (1978), 85–134. [13] M. Demazure, Sous-groupes algébriques de rang maximum du groupe de Cremona, Ann. Sci. École Norm. Sup. (4) 3 (1970), 507–588. [14] G. Ewald, Combinatorial Convexity and Algebraic Geometry, Graduate Texts in Mathematics 168, Springer-Verlag, 1996. [15] R. Fossum, The Divisor Class Group of a Krull Domain, Ergebnisse d. Math. 74, Springer- Verlag, 1973. [16] R. Fossum and B. Iversen, On Picard groups of algebraic fibre spaces, J. Pure Appl. Algebra 3 (1973), 269–280. [17] W. Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies 131, Princeton University Press, 1993. [18] R. Godement, Topologie Algébrique et Théorie des Faisceaux, Actualit’es Sci. Ind., no. 1252, Publ. Math. Univ. Strasbourg, no. 13, Hermann, 1958. [19] F. Grosshans, Algebraic homogeneous spaces and invariant theory, Lecture Notes in Math. 1673, Springer-Verlag, 1997. [20] A. Grothendieck, Éléments de Géométrie Algébrique, I-IV, Inst. Hautes Études Sci. Publ. Math. 4,8,11,17,20,24,28,32 (1960–1967). [21] W. Haboush, Reductive groups are geometrically reductive, Ann. of Math. 102 (1975), 67–83. [22] R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer-Verlag, 1977. [23] B. Iversen, A fixed point formula for action of tori on algebraic varieties, Inv. Math. 16 (1972), 229–236. 68
[24] B. Iversen, Cohomology of Sheaves, Lecture Notes Series 55, Aarhus Universitet, Matematisk Institut, 1984. [25] N. Iwahori and H. Matsumoto, On some Bruhat decomposition and the structure of Hecke ring of p-adic Chevalley groups, Publ. Math. IHES 25 (1965), 5–48. [26] J. Jantzen, Representations of Algebraic Groups, Pure and Applied Mathematics 131, Academic Press, 1987. [27] G. Kempf, Instability in invariant theory, Ann. of Math. (2) 108 (1978), no. 2, 299–316. [28] G. Kempf, F. Knudsen, D. Mumford, and B. Saint-Donat, Toroidal Embeddings I, Lecture Notes in Mathematics 339, Springer-Verlag, 1973. [29] F. Knop, The Luna-Vust theory of spherical embeddings, Proc. of the Hyderabad Conf. on Algebraic Groups (Hyderabad, 1989), 225–249, Manoj Prakashan, Madras, 1991. [30] D. Luna and Th. Vust, Plongements d’espaces homogènes, Comment. Math. Helv. 58 (1983), 186–245. [31] H. Matsumura, Commutative Ring Theory, Cambridge Studies in Adv. Math. 8, Cambridge, 1986. [32] D. Mumford, J. Fogarty and F. Kirwan, Geometric Invariant Theory, 3rd Enlarged Edition, Ergebnisse d. Math. Ser. 3 34, Springer-Verlag, 1994. [33] T. Oda, Convex Bodies and Algebraic Geometry: An introduction to the theory of toric varieties, Ergebnisse d. Math. Ser. 3 15, Springer-Verlag, 1988. [34] D.I. Panyushev, Complexity and rank of homogeneous spaces, Geometriae Dedicata 34 (1990), 249–269. [35] V.L. Popov, Quasihomogeneous affine varieties of the group SL(2), Izv. Akad. Nauk SSSR Ser. Mat 37 (1973), 792–832. [36] V.L. Popov, Groups, Generators, Syzygies, and Orbits in Invariant Theory, Translations of Mathematical Monographs 100, American Mathematical Society, 1992. 69
Page 1 and 2:
c○ Copyright by David Charles Mur
Page 3 and 4:
Abstract We classify embeddings of
Page 5 and 6:
Acknowledgements This thesis would
Page 7 and 8:
Chapter 1 Introduction A basic prob
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: References [1] A. Bialynicki-Birula
Page 77: Vita David Charles Murphy was born
show all

Equivariant Embeddings of Algebraic Groups

Create successful ePaper yourself

Delete template?

Save as template?