Homogeneous Coordinate Ring
|
|
- Christina Wheeler
- 5 years ago
- Views:
Transcription
1 Students: Kaiserslautern University Algebraic Group June 14, 2013
2 Outline Quotients in Algebraic Geometry 1 Quotients in Algebraic Geometry 2 3 4
3 Outline Quotients in Algebraic Geometry 1 Quotients in Algebraic Geometry 2 3 4
4 Definition Let G be a group acting on a variety X = Spec(R), R is a K-algebra. Then the following map G R R (g, f ) g.f defined by (g.f )(x) = f (g 1.x) for all x X is an action of G on R.
5 Remark: a) The group acting as above is induced by the group acting of G on X. b) The above acting gives two objects, namely the set G-orbits X /G = {G.x x X } and the ring of invariants R G = {f R g.f = f, for all g G}.
6 Definition Let G act on X and let π : X Y be morphism that is constant on G-orbits. Then π is called a good categorical quotient if: a) If U Y is open, then the natural map O Y (U) O X (π 1 (U)) induces an isomorphism O Y (U) O X (π 1 (U)) G. b) If W X is closed and G-invariant, then π(w ) Y is closed. c) If W 1, W 2 are closed, disjoint, and G-invariant in X, then π(w 1 ) and π(w 2 ) are disjoint in Y. We often write a good categorical quotient as π : X X //G.
7 Theorem: Let π : X X //G be a good categorical quotient. Then: a) Given any diagram where φ is a morphism of varieties such that φ(g.x) = φ(x) for g G and x X, there is unique morphism φ making the diagram commute, i.e., φ π = φ. b) π is surjective. c) A subset U X //G is open iff π 1 (U) X is open. d) x, y X, we have π(x) = π(y) G.x G.y.
8 Definition a) A subgroup G of GL n (C) is called an affine algebraic group if G is a subvariety of GL n (C). b) Let G be an affine algebraic group acting on a variety X. The G-action is called algebraic action if the action defines a morphism. G X X (g, x) g.x
9 Proposition Let an affine algebraic group G act algebraically on a variety X, and assume that a good categorical quotient π : X X //G. Then: a) If p X //G, then π 1 (p) contains a unique closed G-orbit. b) π induces a bijection {closed G-orbits in X } X //G.
10 Proposition Let π : X X //G be a good categorical quotient. Then the following are equivalent: a) All G-orbits are closed in X. b) Given x, y X, we have π(x) = π(y) x and y lie in the same G-orbit. c) π induces a bijection {G-orbits in X } X //G.
11 Definition A good categorical quotient is called a good geometric quotient if it satisfies the condition of the above proposition. We write a good geometric quotient as π : X X /G.
12 Definition An affine algebraic group G is called reductive if its maximal connected solvable subgroup is a torus. Proposition Let G be a reductive group acting algebraically on an affine variety X = Spec(R). Then a) R G is a finely generated C-algebra. b) The morphism π : X Spec(R G ) induced by R G R is a good categorical quotient.
13 Proposition Let G act on X and let π : X Y be a morphism of varieties that is constant on G-orbits. If Y has an open cover Y = α V α such that π π 1 (V α) : π 1 (V α ) V α is a good categorical quotient for every α, then π : X Y is a good categorical quotient.
14 Example: Let C act on C 2 \{0} by scalar multiplication, where C 2 = Spec(C[x 0, x 1 ]). Then C 2 \{0} = U 0 U 1, where U 0 = C 2 \V (x 0 ) = Spec(C[x ±1 0, x 1]) U 1 = C 2 \V (x 1 ) = Spec(C[x 0, x ±1 1 ]) U 0 U 1 = C 2 \V (x 0 x 1 ) = Spec(C[x ±1 0, x ±1 1 ]) The rings of invariants are C[x ±1 0, x 1] C = C[x 1 /x 0 ] C[x 0, x ±1 1 ]C = C[x 0 /x 1 ] C[x ±1 0, x ±1 1 ]C = C[(x 1 /x 0 ) ±1 ]
15 It follows that the V i = U i //C glue together in the usual way to create P 1. Since C -orbits are closed in C 2 \{0}, it follows that is a good geometric quotient. P 1 = (C 2 \{0})/C
16 Outline Quotients in Algebraic Geometry 1 Quotients in Algebraic Geometry 2 3 4
17 Let X Σ be the toric variety of a fan Σ in N R. The goal is to construct X Σ as a good categorical quotient X Σ (C r \Z)//G for an appropriate of affine space C r, exceptional set Z C r, and reductive group G.
18 Definition Let X Σ be the toric variety of fan Σ in N (R). Assume that X Σ has no torus factor. We define G = Hom Z (Cl(X Σ ), C ) where Cl (X Σ ) = Div (X Σ) /Div0 (X Σ ). Remark: By the above definition, we have the following short exact sequence of affine algebraic group 1 G (C ) Σ(1) T N 1.
19 Lemma Let G be as in the above definition. Then: a) Cl(X Σ ) is the character group of G. b) G is isomorphic to a product of a torus and a finite Abelian group. In particular, G is reductive. c) Give a basis e 1,..., e n of M. We have G = {(t ρ ) (C ) Σ(1) ρ = {(t ρ ) (C ) Σ(1) ρ t m,uρ ρ = 1 for all m M} t e i,u ρ ρ = 1 for 1 i n}.
20 Example: The ray generators of the fan for P n are u 0 = n e i, u 1 = e 1,..., u n = e n. i=1 By the above lemma, (t 0,..., t n ) (C ) n+1 lies in G if and only if t m, e 1... e n 0 t m,e t m,en n = 1 for all m M = Z n. Taking m equal to e 1,..., e n, we see that G is defined by t0 1 t 1 =... = t0 1 t n. Thus G = {(λ,..., λ) λ C } C, which is the action of C on C n+1 given by scalar multiplication.
21 Example: The fan for P 1 P 1 has ray generators u 1 = e 1, u 2 = e 1, u 3 = e 2, u 4 = e 2 in N = Z 2. By this lemma, (t 1, t 2, t 3, t 4 ) (C ) 4 lies in G if and only if t m,e 1 1 t m, e 1 2 t m,e 2 3 t m, e 2 4 = 1 for all m M = Z 2. Taking m equal to e 1, e 2, we obtain t 1 t 1 2 = t 3 t 1 4 = 1. Thus G = {(µ, µ, λ, λ) µ, λ C } (C ) 2.
22 Definition Let X Σ be the toric variety of fan Σ in N (R). S := C[x ρ ρ Σ(1)] is called the homogeneous coordinate ring of X Σ.
23 Definition Let X Σ be the toric variety of fan Σ in N (R). a) For each cone σ Σ, define the monomial x ˆσ = ρ/ σ(1) x ρ. b) B(Σ) := x ˆσ σ Σ S is called irrelevant ideal. Remark: a) Spec(S) = C Σ(1). b) x ˆτ is the multiple of x ˆσ whenever τ is a face of σ.
24 c) B(Σ) = x ˆσ σ Σ max. where Σ max is the set of maximal cones of Σ. Now define Z(Σ) = V (B(Σ)) C Σ(1). Example: The fan for P n consists of cones generated by proper subsets of {u 0,..., u n }, where u 0 = n i=1 e i, u 1 = e 1,..., u n = e n. Let u i generate ρ i for 0 i n and x i be the corresponding variable in the total coordinate ring. The maximal cones of the fan are σ i = Cone(u 0,..., û i,..., u n ). Then x ˆσ i = x i, so that B(Σ) = x 0,..., x n. Hence Z(Σ) = {0}.
25 Definition A subset P Σ(1) is a primitive collection if: a) P σ(1) for all σ Σ. b) For every proper subset Q P, there is a σ Σ with Q σ(1).
26 Proposition The Z(Σ) as a union of irreducible components is given by Z(Σ) = P V (x ρ ρ P), where the union is over all primitive collections P Σ(1). Example: The fan for P n consists of cones generated by proper subsets of {u 0,..., u n }, where u 0 = n i=1 e i, u 1 = e 1,..., u n = e n. The only primitive collection is {ρ 0,..., ρ n }, so Z(Σ) = V (x 0,..., x n ) = {0}.
27 Example: The fan for P 1 P 1 has ray generators u = e 1, u 2 = e 1, u 3 = e 2, u 4 = e 2. each u i gives a ray ρ i and a variable x i. We compute Z(Σ) in two ways: * The maximal cone Cone(u 1, u 3 ) gives the monomial x 2 x 4 and the others give x 1 x 4, x 1 x 3, x 2 x 3. Thus B(Σ) = x 2 x 4, x 1 x 4, x 1 x 3, x 2 x 3. We can check that Z(Σ) = {0} C 2 C 2 {0}. * The only primitive collections are {ρ 1, ρ 2 } and {ρ 3, ρ 4 }, so that Z(Σ) = V (x 1, x 2 ) V (x 3, x 4 ) = {0} C 2 C 2 {0} by the proposition, where B(Σ) = x 1, x 2 x 3, x 4.
28 Let {e ρ ρ Σ(1)} be the standard basis of the lattice Z Σ(1). For each σ Σ, define the cone σ = Cone(e ρ ρ σ(1)) R Σ(1). These cones and their faces form a fan Σ = { σ σ Σ} in (Z Σ(1) ) R = R Σ(1). This fan has the following properties.
29 Proposition Let Σ be the fan as above. a) C Σ(1) \Z(Σ) is the toric variety of the fan Σ. b) The map e ρ u ρ defines a map of lattices Z Σ(1) N that is compatible with the fans Σ and Σ in N R. c) The resulting toric morphism is constant on G-orbits. π : C Σ(1) \Z(Σ) X Σ
30 Theorem Let X Σ be the toric variety without torus factors and consider the toric morphism π : C Σ(1) \Z(Σ) X Σ from the above proposition. Then: a) π is a good categorical quotient for the action of G on C Σ(1) \Z(Σ), so that X Σ (C Σ(1) \Z(Σ))//G. b) π is a good geometric quotient if and only if Σ is simplicial.
31 We have a commutative diagram: X Σ (C Σ(1) \Z(Σ))//G T N (C ) Σ(1) /G
32 Example: From some examples above, the quotient representation of P n is P n = (C n+1 \{0})/C, where C acts by scalar multiplication. This is a good geometric quotient since Σ is a smooth and hence simplicial.
33 Example: Also from some example before, the quotient representation P 1 P 1 is C 1 C 1 = (C 4 \({0} C 2 C 2 {0}))/(C ) 2, where (C ) 2 acts via (µ, λ).(a, b, c, d) = (µa, µb, λa, λd). This is again a good geometric quotient.
34 Outline Quotients in Algebraic Geometry 1 Quotients in Algebraic Geometry 2 3 4
35 In this section we will explore how this ring relates to the algebra and geometry of X Σ.
36 Definition Let X Σ be a toric variety without torus factor. Its total coordinate ring is S = C[x ρ ρ Σ(1)]. We have the sequence: 0 M Z Σ(1) Cl(X Σ ) 0 where α = (a ρ ) Z Σ(1) maps to the class [Σ ρ a ρ D ρ ] Cl (X Σ ). Given x α = ρ x ρ aρ S, we define its degree: deg (x α ) = [Σ ρ a ρ D ρ ] Cl (X Σ ). For β Cl (X Σ ), S β denotes the corresponding grade piece of S.
37 Remark: The grading on S is closely related to G = Hom Z (Cl (X Σ ), C ). Cl (X Σ ) is the character group of G, where as usual β Cl (X Σ ) gives the character χ β : G C. The action of G on C Σ(1) induces an action on S with the following property: For given f S f S β g.f = χ β ( g 1) f for all g G f (g.x) = χ β (g) f (x) for all g G, x C Σ(1). We say that f S β is homogeneous of degree β.
38 Example: The total coordinate ring of P n is C[x 0,..., x n ]. The map Z n+1 Z is (a 0,..., a n ) a a n. This gives the grading on C[x 0,.., x n ] where each variable x i has degree 1, so that homogeneous polynomial has the usual meaning.
39 Example: The fan for P n P m is the product of the fans of P n and P m. The class group is Cl(P n P m ) Cl(P n ) Cl(P m ) Z 2. The total coordinate ring is C[x 0,..., x n, y 0,..., y m ], where deg(x i ) = (1, 0) deg(y i ) = (0, 1). For this ring, homogeneous polynomial means bihomogeneous polynomial.
40 Proposition Let S be the total coordinate ring of the simplicial toric variety X Σ. Then: a) If I S is a homogeneous ideal, then V (I ) = {π(x) X Σ f (x) = 0 for all f I } is a closed subvarieties of X Σ. b) All closed subvarieties of X Σ arise this way.
41 Proposition (The Toric Nullstellensazt) Let X Σ be the simplicial toric variety with total coordinate ring S and irrelevant ideal B(Σ) S. If I S is a homogeneous ideal, then V (I ) = in X Σ B(Σ) k I for some k 0.
42 Proposition (The Toric Ideal-Variety Correspondence) Let X Σ be a simplicial toric variety. Then there is a bijective correspondence { } radical homogeneous ideals {closed subvarieties of X Σ } I B(Σ) S
43 When X Σ is not simplicial, there is still a relation between ideals in the total coordinate ring and closed subvarieties of X Σ. Proposition Let S be the total coordinate ring of the toric variety X Σ. Then: a) If I S is a homogeneous ideal, then V (I ) = { p X Σ there is a x π 1 (p), f (x) = 0 f I } is a closed subvariety of X Σ. b) All closed subvarieties of X Σ arise this way.
44 Outline Quotients in Algebraic Geometry 1 Quotients in Algebraic Geometry 2 3 4
45 Definition a) A polytope M R is the convex hull of affine set of points. b) The dimension of is the dimension of the subspace spanned by the difference {m 1 m 2 m 1, m 2 }. c) is called integral if the vertice of lie in M. d) Let 1,..., k be polytopes. We define k = {m m k m i i ; i = 1,..., k}. We also denote k := for k N. We have k = {km m }.
46 Definition Let be a polytope. We define a) t k x m is called monomial, where m k. b) The monomials multiply by c) The degree of t k x m is d) The polytope ring of is t k x m.t l x n := t k+l x m+n. deg(t k x m ) := k. S := C[t k x m k N, m k ].
47 Remark: a) The definition of the monomials multiply is well-defined. Indeed, because m k, m l we get m + m (k + l). b) S = C[t k x m m k ] is a grade ring. Hence S = (S ) k. k=0
48 Definition Let S be a polytope ring of. a) S + := k 1 (S ) k is called irrelevant ideal. b) T := {P P is a homogeneous prime ideal of S and P S + }. P T is called relevant prime ideal of S. c) For any homogeneous ideal I of S, we define Z(I ) := {P P is a relevant prime of S and P I }.
49 Proposition a) If {I j } is a family of homogeneous ideals in S then j Z(I j ) = Z( j I j ). b) If I 1, I 2 are homogeneous ideals then Z(I 1 ) Z(I 2 ) = Z(I 1 I 2 ).
50 T is a topological space whose closed sets are Z(I ), I is a homogeneous ideal of S. Let f be any homogeneous element of S of degree 1. We set U f := T \Z( f ). We may identity U f with the topological space Spec(S [f 1 ] 0 ) and give it the corresponding structure of an affine scheme, where S [f 1 ] 0 = { g f s f, g homogeneous in S and deg(g) = deg(f s )}. We will write (Proj(S )) f for this open affine subscheme of Proj(S ). Let P = Proj(S ).
51 Definition Let be a polytope and F be a nonempty face of. We define a) σ v F := {λ(m m ) m, m F, λ 0} M R is a cone and its dual is a cone σ F N R. b) NF ( ) := {σ F F is nonempty face of } is normal face of.
52 Theorem P = X (NF ( )).
Toric Varieties. Madeline Brandt. April 26, 2017
Toric Varieties Madeline Brandt April 26, 2017 Last week we saw that we can define normal toric varieties from the data of a fan in a lattice. Today I will review this idea, and also explain how they can
More informationMath 418 Algebraic Geometry Notes
Math 418 Algebraic Geometry Notes 1 Affine Schemes Let R be a commutative ring with 1. Definition 1.1. The prime spectrum of R, denoted Spec(R), is the set of prime ideals of the ring R. Spec(R) = {P R
More informationLecture 1. Toric Varieties: Basics
Lecture 1. Toric Varieties: Basics Taras Panov Lomonosov Moscow State University Summer School Current Developments in Geometry Novosibirsk, 27 August1 September 2018 Taras Panov (Moscow University) Lecture
More informationRepresentations and Linear Actions
Representations and Linear Actions Definition 0.1. Let G be an S-group. A representation of G is a morphism of S-groups φ G GL(n, S) for some n. We say φ is faithful if it is a monomorphism (in the category
More informationInstitutionen för matematik, KTH.
Institutionen för matematik, KTH. Contents 7 Affine Varieties 1 7.1 The polynomial ring....................... 1 7.2 Hypersurfaces........................... 1 7.3 Ideals...............................
More informationToric Geometry. An introduction to toric varieties with an outlook towards toric singularity theory
Toric Geometry An introduction to toric varieties with an outlook towards toric singularity theory Thesis project for the research track of the Master Mathematics Academic year 2014/2015 Author: Geert
More informationIntroduction to toric geometry
Introduction to toric geometry Ugo Bruzzo Scuola Internazionale Superiore di Studi Avanzati and Istituto Nazionale di Fisica Nucleare Trieste ii Instructions for the reader These are work-in-progress notes
More informationwhere m is the maximal ideal of O X,p. Note that m/m 2 is a vector space. Suppose that we are given a morphism
8. Smoothness and the Zariski tangent space We want to give an algebraic notion of the tangent space. In differential geometry, tangent vectors are equivalence classes of maps of intervals in R into the
More information(1) is an invertible sheaf on X, which is generated by the global sections
7. Linear systems First a word about the base scheme. We would lie to wor in enough generality to cover the general case. On the other hand, it taes some wor to state properly the general results if one
More informationAlgebraic group actions and quotients
23rd Autumn School in Algebraic Geometry Algebraic group actions and quotients Wykno (Poland), September 3-10, 2000 LUNA S SLICE THEOREM AND APPLICATIONS JEAN MARC DRÉZET Contents 1. Introduction 1 2.
More informationToric Varieties and the Secondary Fan
Toric Varieties and the Secondary Fan Emily Clader Fall 2011 1 Motivation The Batyrev mirror symmetry construction for Calabi-Yau hypersurfaces goes roughly as follows: Start with an n-dimensional reflexive
More informationarxiv: v1 [math.ag] 4 Sep 2018
QUOTIENTS OF SMOOTH PROJECTIVE TORIC VARIETIES BY µ p IN POSITIVE CHARACTERISTICS p arxiv:1809.00867v1 [math.ag] 4 Sep 2018 TADAKAZU SAWADA Abstract. In this paper we show that quotients of smooth projective
More informationarxiv: v1 [quant-ph] 19 Jan 2010
A toric varieties approach to geometrical structure of multipartite states Hoshang Heydari Physics Department, Stockholm university 10691 Stockholm Sweden arxiv:1001.3245v1 [quant-ph] 19 Jan 2010 Email:
More informationAlgebraic Varieties. Notes by Mateusz Micha lek for the lecture on April 17, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra
Algebraic Varieties Notes by Mateusz Micha lek for the lecture on April 17, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra Algebraic varieties represent solutions of a system of polynomial
More informationV (f) :={[x] 2 P n ( ) f(x) =0}. If (x) ( x) thenf( x) =
20 KIYOSHI IGUSA BRANDEIS UNIVERSITY 2. Projective varieties For any field F, the standard definition of projective space P n (F ) is that it is the set of one dimensional F -vector subspaces of F n+.
More informationSpherical varieties and arc spaces
Spherical varieties and arc spaces Victor Batyrev, ESI, Vienna 19, 20 January 2017 1 Lecture 1 This is a joint work with Anne Moreau. Let us begin with a few notations. We consider G a reductive connected
More informationBinomial Exercises A = 1 1 and 1
Lecture I. Toric ideals. Exhibit a point configuration A whose affine semigroup NA does not consist of the intersection of the lattice ZA spanned by the columns of A with the real cone generated by A.
More informationTropical Varieties. Jan Verschelde
Tropical Varieties Jan Verschelde University of Illinois at Chicago Department of Mathematics, Statistics, and Computer Science http://www.math.uic.edu/ jan jan@math.uic.edu Graduate Computational Algebraic
More informationProjective Varieties. Chapter Projective Space and Algebraic Sets
Chapter 1 Projective Varieties 1.1 Projective Space and Algebraic Sets 1.1.1 Definition. Consider A n+1 = A n+1 (k). The set of all lines in A n+1 passing through the origin 0 = (0,..., 0) is called the
More information12. Hilbert Polynomials and Bézout s Theorem
12. Hilbert Polynomials and Bézout s Theorem 95 12. Hilbert Polynomials and Bézout s Theorem After our study of smooth cubic surfaces in the last chapter, let us now come back to the general theory of
More informationCombinatorics and geometry of E 7
Combinatorics and geometry of E 7 Steven Sam University of California, Berkeley September 19, 2012 1/24 Outline Macdonald representations Vinberg representations Root system Weyl group 7 points in P 2
More informationToric Varieties. David Cox John Little Hal Schenck
Toric Varieties David Cox John Little Hal Schenck DEPARTMENT OF MATHEMATICS, AMHERST COLLEGE, AMHERST, MA 01002 E-mail address: dac@cs.amherst.edu DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE, COLLEGE
More informationPROBLEMS, MATH 214A. Affine and quasi-affine varieties
PROBLEMS, MATH 214A k is an algebraically closed field Basic notions Affine and quasi-affine varieties 1. Let X A 2 be defined by x 2 + y 2 = 1 and x = 1. Find the ideal I(X). 2. Prove that the subset
More informationk k would be reducible. But the zero locus of f in A n+1
Math 145. Bezout s Theorem Let be an algebraically closed field. The purpose of this handout is to prove Bezout s Theorem and some related facts of general interest in projective geometry that arise along
More informationPROJECTIVIZED RANK TWO TORIC VECTOR BUNDLES ARE MORI DREAM SPACES
PROJECTIVIZED RANK TWO TORIC VECTOR BUNDLES ARE MORI DREAM SPACES JOSÉ LUIS GONZÁLEZ Abstract. We prove that the Cox ring of the projectivization PE of a rank two toric vector bundle E, over a toric variety
More information1. Group Theory Permutations.
1.1. Permutations. 1. Group Theory Problem 1.1. Let G be a subgroup of S n of index 2. Show that G = A n. Problem 1.2. Find two elements of S 7 that have the same order but are not conjugate. Let π S 7
More informationMath 797W Homework 4
Math 797W Homework 4 Paul Hacking December 5, 2016 We work over an algebraically closed field k. (1) Let F be a sheaf of abelian groups on a topological space X, and p X a point. Recall the definition
More informationCohomology groups of toric varieties
Cohomology groups of toric varieties Masanori Ishida Mathematical Institute, Tohoku University 1 Fans and Complexes Although we treat real fans later, we begin with fans consisting of rational cones which
More informationAN EXPOSITION OF THE RIEMANN ROCH THEOREM FOR CURVES
AN EXPOSITION OF THE RIEMANN ROCH THEOREM FOR CURVES DOMINIC L. WYNTER Abstract. We introduce the concepts of divisors on nonsingular irreducible projective algebraic curves, the genus of such a curve,
More informationDemushkin s Theorem in Codimension One
Universität Konstanz Demushkin s Theorem in Codimension One Florian Berchtold Jürgen Hausen Konstanzer Schriften in Mathematik und Informatik Nr. 176, Juni 22 ISSN 143 3558 c Fachbereich Mathematik und
More informationNeural Codes and Neural Rings: Topology and Algebraic Geometry
Neural Codes and Neural Rings: Topology and Algebraic Geometry Ma191b Winter 2017 Geometry of Neuroscience References for this lecture: Curto, Carina; Itskov, Vladimir; Veliz-Cuba, Alan; Youngs, Nora,
More informationINTERSECTION THEORY CLASS 12
INTERSECTION THEORY CLASS 12 RAVI VAKIL CONTENTS 1. Rational equivalence on bundles 1 1.1. Intersecting with the zero-section of a vector bundle 2 2. Cones and Segre classes of subvarieties 3 2.1. Introduction
More informationBetti numbers of abelian covers
Betti numbers of abelian covers Alex Suciu Northeastern University Geometry and Topology Seminar University of Wisconsin May 6, 2011 Alex Suciu (Northeastern University) Betti numbers of abelian covers
More informationSummer Algebraic Geometry Seminar
Summer Algebraic Geometry Seminar Lectures by Bart Snapp About This Document These lectures are based on Chapters 1 and 2 of An Invitation to Algebraic Geometry by Karen Smith et al. 1 Affine Varieties
More informationMath 210C. The representation ring
Math 210C. The representation ring 1. Introduction Let G be a nontrivial connected compact Lie group that is semisimple and simply connected (e.g., SU(n) for n 2, Sp(n) for n 1, or Spin(n) for n 3). Let
More informationTORIC WEAK FANO VARIETIES ASSOCIATED TO BUILDING SETS
TORIC WEAK FANO VARIETIES ASSOCIATED TO BUILDING SETS YUSUKE SUYAMA Abstract. We give a necessary and sufficient condition for the nonsingular projective toric variety associated to a building set to be
More informationOn Mordell-Lang in Algebraic Groups of Unipotent Rank 1
On Mordell-Lang in Algebraic Groups of Unipotent Rank 1 Paul Vojta University of California, Berkeley and ICERM (work in progress) Abstract. In the previous ICERM workshop, Tom Scanlon raised the question
More information3. The Sheaf of Regular Functions
24 Andreas Gathmann 3. The Sheaf of Regular Functions After having defined affine varieties, our next goal must be to say what kind of maps between them we want to consider as morphisms, i. e. as nice
More informationRiemann surfaces with extra automorphisms and endomorphism rings of their Jacobians
Riemann surfaces with extra automorphisms and endomorphism rings of their Jacobians T. Shaska Oakland University Rochester, MI, 48309 April 14, 2018 Problem Let X be an algebraic curve defined over a field
More informationOn the singular elements of a semisimple Lie algebra and the generalized Amitsur-Levitski Theorem
On the singular elements of a semisimple Lie algebra and the generalized Amitsur-Levitski Theorem Bertram Kostant, MIT Conference on Representations of Reductive Groups Salt Lake City, Utah July 10, 2013
More informationMATH 8253 ALGEBRAIC GEOMETRY WEEK 12
MATH 8253 ALGEBRAIC GEOMETRY WEEK 2 CİHAN BAHRAN 3.2.. Let Y be a Noetherian scheme. Show that any Y -scheme X of finite type is Noetherian. Moreover, if Y is of finite dimension, then so is X. Write f
More informationFILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS.
FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS. Let A be a ring, for simplicity assumed commutative. A filtering, or filtration, of an A module M means a descending sequence of submodules M = M 0
More informationWe can choose generators of this k-algebra: s i H 0 (X, L r i. H 0 (X, L mr )
MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 43 5.3. Linearisations. An abstract projective scheme X does not come with a pre-specified embedding in a projective space. However, an ample line bundle
More informationPOLYNOMIAL IDENTITY RINGS AS RINGS OF FUNCTIONS
POLYNOMIAL IDENTITY RINGS AS RINGS OF FUNCTIONS Z. REICHSTEIN AND N. VONESSEN Abstract. We generalize the usual relationship between irreducible Zariski closed subsets of the affine space, their defining
More informationIN POSITIVE CHARACTERISTICS: 3. Modular varieties with Hecke symmetries. 7. Foliation and a conjecture of Oort
FINE STRUCTURES OF MODULI SPACES IN POSITIVE CHARACTERISTICS: HECKE SYMMETRIES AND OORT FOLIATION 1. Elliptic curves and their moduli 2. Moduli of abelian varieties 3. Modular varieties with Hecke symmetries
More informationAlgebraic Geometry. Question: What regular polygons can be inscribed in an ellipse?
Algebraic Geometry Question: What regular polygons can be inscribed in an ellipse? 1. Varieties, Ideals, Nullstellensatz Let K be a field. We shall work over K, meaning, our coefficients of polynomials
More informationArithmetic Algebraic Geometry
Arithmetic Algebraic Geometry 2 Arithmetic Algebraic Geometry Travis Dirle December 4, 2016 2 Contents 1 Preliminaries 1 1.1 Affine Varieties.......................... 1 1.2 Projective Varieties........................
More informationSection Projective Morphisms
Section 2.7 - Projective Morphisms Daniel Murfet October 5, 2006 In this section we gather together several topics concerned with morphisms of a given scheme to projective space. We will show how a morphism
More information10. Smooth Varieties. 82 Andreas Gathmann
82 Andreas Gathmann 10. Smooth Varieties Let a be a point on a variety X. In the last chapter we have introduced the tangent cone C a X as a way to study X locally around a (see Construction 9.20). It
More informationGeometry and combinatorics of spherical varieties.
Geometry and combinatorics of spherical varieties. Notes of a course taught by Guido Pezzini. Abstract This is the lecture notes from a mini course at the Winter School Geometry and Representation Theory
More informationNOTES ON ALGEBRAIC GEOMETRY MATH 202A. Contents Introduction Affine varieties 22
NOTES ON ALGEBRAIC GEOMETRY MATH 202A KIYOSHI IGUSA BRANDEIS UNIVERSITY Contents Introduction 1 1. Affine varieties 2 1.1. Weak Nullstellensatz 2 1.2. Noether s normalization theorem 2 1.3. Nullstellensatz
More informationGeometric Structure and the Local Langlands Conjecture
Geometric Structure and the Local Langlands Conjecture Paul Baum Penn State Representations of Reductive Groups University of Utah, Salt Lake City July 9, 2013 Paul Baum (Penn State) Geometric Structure
More informationdiv(f ) = D and deg(d) = deg(f ) = d i deg(f i ) (compare this with the definitions for smooth curves). Let:
Algebraic Curves/Fall 015 Aaron Bertram 4. Projective Plane Curves are hypersurfaces in the plane CP. When nonsingular, they are Riemann surfaces, but we will also consider plane curves with singularities.
More informationMath 216A. A gluing construction of Proj(S)
Math 216A. A gluing construction o Proj(S) 1. Some basic deinitions Let S = n 0 S n be an N-graded ring (we ollows French terminology here, even though outside o France it is commonly accepted that N does
More informationELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS
ELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS J. WARNER SUMMARY OF A PAPER BY J. CARLSON, E. FRIEDLANDER, AND J. PEVTSOVA, AND FURTHER OBSERVATIONS 1. The Nullcone and Restricted Nullcone We will need
More informationπ X : X Y X and π Y : X Y Y
Math 6130 Notes. Fall 2002. 6. Hausdorffness and Compactness. We would like to be able to say that all quasi-projective varieties are Hausdorff and that projective varieties are the only compact varieties.
More informationSection Divisors
Section 2.6 - Divisors Daniel Murfet October 5, 2006 Contents 1 Weil Divisors 1 2 Divisors on Curves 9 3 Cartier Divisors 13 4 Invertible Sheaves 17 5 Examples 23 1 Weil Divisors Definition 1. We say a
More informationH(G(Q p )//G(Z p )) = C c (SL n (Z p )\ SL n (Q p )/ SL n (Z p )).
92 19. Perverse sheaves on the affine Grassmannian 19.1. Spherical Hecke algebra. The Hecke algebra H(G(Q p )//G(Z p )) resp. H(G(F q ((T ))//G(F q [[T ]])) etc. of locally constant compactly supported
More informationAlgebraic Geometry (Math 6130)
Algebraic Geometry (Math 6130) Utah/Fall 2016. 2. Projective Varieties. Classically, projective space was obtained by adding points at infinity to n. Here we start with projective space and remove a hyperplane,
More informationToroidal Embeddings and Desingularization
California State University, San Bernardino CSUSB ScholarWorks Electronic Theses, Projects, and Dissertations Office of Graduate Studies 6-2018 Toroidal Embeddings and Desingularization LEON NGUYEN 003663425@coyote.csusb.edu
More informationMathematics 7800 Quantum Kitchen Sink Spring 2002
Mathematics 7800 Quantum Kitchen Sink Spring 2002 4. Quotients via GIT. Most interesting moduli spaces arise as quotients of schemes by group actions. We will first analyze such quotients with geometric
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationMath 440 Problem Set 2
Math 440 Problem Set 2 Problem 4, p. 52. Let X R 3 be the union of n lines through the origin. Compute π 1 (R 3 X). Solution: R 3 X deformation retracts to S 2 with 2n points removed. Choose one of them.
More informationC n.,..., z i 1., z i+1., w i+1,..., wn. =,..., w i 1. : : w i+1. :... : w j 1 1.,..., w j 1. z 0 0} = {[1 : w] w C} S 1 { },
Complex projective space The complex projective space CP n is the most important compact complex manifold. By definition, CP n is the set of lines in C n+1 or, equivalently, CP n := (C n+1 \{0})/C, where
More informationQuotients by algebraic groups Properties of actions.
72 JENIA TEVELEV 6. Quotients by algebraic groups Let G be an algebraic group acting algebraically on an algebraic variety X, i.e. the action map G X! X is a morphism of algebraic varieties. 6.0.1. EXAMPLE.
More informationAN INTRODUCTION TO AFFINE SCHEMES
AN INTRODUCTION TO AFFINE SCHEMES BROOKE ULLERY Abstract. This paper gives a basic introduction to modern algebraic geometry. The goal of this paper is to present the basic concepts of algebraic geometry,
More informationAPPENDIX 3: AN OVERVIEW OF CHOW GROUPS
APPENDIX 3: AN OVERVIEW OF CHOW GROUPS We review in this appendix some basic definitions and results that we need about Chow groups. For details and proofs we refer to [Ful98]. In particular, we discuss
More informationPIECEWISE POLYNOMIAL FUNCTIONS, CONVEX POLYTOPES AND ENUMERATIVE GEOMETRY
PARAMETER SPACES BANACH CENTER PUBLICATIONS, VOLUME 36 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1996 PIECEWISE POLYNOMIAL FUNCTIONS, CONVEX POLYTOPES AND ENUMERATIVE GEOMETRY MICHEL
More informationSynopsis of material from EGA Chapter II, 4. Proposition (4.1.6). The canonical homomorphism ( ) is surjective [(3.2.4)].
Synopsis of material from EGA Chapter II, 4 4.1. Definition of projective bundles. 4. Projective bundles. Ample sheaves Definition (4.1.1). Let S(E) be the symmetric algebra of a quasi-coherent O Y -module.
More informationGEOMETRIC INVARIANT THEORY AND SYMPLECTIC QUOTIENTS
GEOMETRIC INVARIANT THEORY AND SYMPLECTIC QUOTIENTS VICTORIA HOSKINS 1. Introduction In this course we study methods for constructing quotients of group actions in algebraic and symplectic geometry and
More informationThe rational cohomology of real quasi-toric manifolds
The rational cohomology of real quasi-toric manifolds Alex Suciu Northeastern University Joint work with Alvise Trevisan (VU Amsterdam) Toric Methods in Homotopy Theory Queen s University Belfast July
More informationSystems of linear equations. We start with some linear algebra. Let K be a field. We consider a system of linear homogeneous equations over K,
Systems of linear equations We start with some linear algebra. Let K be a field. We consider a system of linear homogeneous equations over K, f 11 t 1 +... + f 1n t n = 0, f 21 t 1 +... + f 2n t n = 0,.
More informationINTRODUCTION TO GEOMETRIC INVARIANT THEORY
INTRODUCTION TO GEOMETRIC INVARIANT THEORY JOSÉ SIMENTAL Abstract. These are the expanded notes for a talk at the MIT/NEU Graduate Student Seminar on Moduli of sheaves on K3 surfaces. We give a brief introduction
More informationarxiv:math.ag/ v1 7 Jan 2005
arxiv:math.ag/0501104 v1 7 Jan 2005 Asymptotic cohomological functions of toric divisors Milena Hering, Alex Küronya, Sam Payne January 7, 2005 Abstract We study functions on the class group of a toric
More informationChern classes à la Grothendieck
Chern classes à la Grothendieck Theo Raedschelders October 16, 2014 Abstract In this note we introduce Chern classes based on Grothendieck s 1958 paper [4]. His approach is completely formal and he deduces
More informationADVANCED TOPICS IN ALGEBRAIC GEOMETRY
ADVANCED TOPICS IN ALGEBRAIC GEOMETRY DAVID WHITE Outline of talk: My goal is to introduce a few more advanced topics in algebraic geometry but not to go into too much detail. This will be a survey of
More informationGEOMETRIC STRUCTURES OF SEMISIMPLE LIE ALGEBRAS
GEOMETRIC STRUCTURES OF SEMISIMPLE LIE ALGEBRAS ANA BALIBANU DISCUSSED WITH PROFESSOR VICTOR GINZBURG 1. Introduction The aim of this paper is to explore the geometry of a Lie algebra g through the action
More informationTopology of Toric Varieties, Part II
Topology of Toric Varieties, Part II Daniel Chupin April 2, 2018 Abstract Notes for a talk leading up to a discussion of the Hirzebruch-Riemann-Roch (HRR) theorem for toric varieties, and some consequences
More informationarxiv: v2 [math.ag] 29 Aug 2009
LOGARITHMIC GEOMETRY, MINIMAL FREE RESOLUTIONS AND TORIC ALGEBRAIC STACKS arxiv:0707.2568v2 [math.ag] 29 Aug 2009 ISAMU IWANARI Abstract. In this paper we will introduce a certain type of morphisms of
More informationSingular Toric Varieties
Singular Toric Varieties by Bernt Ivar Utstøl Nødland THESIS for the degree of MASTER OF SCIENCE (Master i Matematikk) Det matematisk- naturvitenskapelige fakultet Universitetet i Oslo March 2015 Faculty
More information12. Linear systems Theorem Let X be a scheme over a ring A. (1) If φ: X P n A is an A-morphism then L = φ O P n
12. Linear systems Theorem 12.1. Let X be a scheme over a ring A. (1) If φ: X P n A is an A-morphism then L = φ O P n A (1) is an invertible sheaf on X, which is generated by the global sections s 0, s
More informationAlgebraic Geometry I Lectures 14 and 15
Algebraic Geometry I Lectures 14 and 15 October 22, 2008 Recall from the last lecture the following correspondences {points on an affine variety Y } {maximal ideals of A(Y )} SpecA A P Z(a) maximal ideal
More informationBirational geometry and deformations of nilpotent orbits
arxiv:math/0611129v1 [math.ag] 6 Nov 2006 Birational geometry and deformations of nilpotent orbits Yoshinori Namikawa In order to explain what we want to do in this paper, let us begin with an explicit
More informationROST S DEGREE FORMULA
ROST S DEGREE FORMULA ALEXANDER MERKURJEV Some parts of algebraic quadratic form theory and theory of simple algebras with involutions) can be translated into the language of algebraic geometry. Example
More informationOn the Hodge Structure of Projective Hypersurfaces in Toric Varieties
On the Hodge Structure of Projective Hypersurfaces in Toric Varieties arxiv:alg-geom/9306011v1 25 Jun 1993 Victor V. Batyrev Universität-GH-Essen, Fachbereich 6, Mathematik Universitätsstr. 3, Postfach
More informationResolution of Singularities in Algebraic Varieties
Resolution of Singularities in Algebraic Varieties Emma Whitten Summer 28 Introduction Recall that algebraic geometry is the study of objects which are or locally resemble solution sets of polynomial equations.
More informationh M (T ). The natural isomorphism η : M h M determines an element U = η 1
MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 7 2.3. Fine moduli spaces. The ideal situation is when there is a scheme that represents our given moduli functor. Definition 2.15. Let M : Sch Set be a moduli
More informationRings and groups. Ya. Sysak
Rings and groups. Ya. Sysak 1 Noetherian rings Let R be a ring. A (right) R -module M is called noetherian if it satisfies the maximum condition for its submodules. In other words, if M 1... M i M i+1...
More informationAN INTRODUCTION TO TORIC SURFACES
AN INTRODUCTION TO TORIC SURFACES JESSICA SIDMAN 1. An introduction to affine varieties To motivate what is to come we revisit a familiar example from high school algebra from a point of view that allows
More informationA course in. Algebraic Geometry. Taught by Prof. Xinwen Zhu. Fall 2011
A course in Algebraic Geometry Taught by Prof. Xinwen Zhu Fall 2011 1 Contents 1. September 1 3 2. September 6 6 3. September 8 11 4. September 20 16 5. September 22 21 6. September 27 25 7. September
More information7.3 Singular Homology Groups
184 CHAPTER 7. HOMOLOGY THEORY 7.3 Singular Homology Groups 7.3.1 Cycles, Boundaries and Homology Groups We can define the singular p-chains with coefficients in a field K. Furthermore, we can define the
More informationSmooth projective horospherical varieties with Picard number 1
arxiv:math/0703576v1 [math.ag] 20 Mar 2007 Smooth projective horospherical varieties with Picard number 1 Boris Pasquier February 2, 2008 Abstract We describe smooth projective horospherical varieties
More informationAlgebraic varieties. Chapter A ne varieties
Chapter 4 Algebraic varieties 4.1 A ne varieties Let k be a field. A ne n-space A n = A n k = kn. It s coordinate ring is simply the ring R = k[x 1,...,x n ]. Any polynomial can be evaluated at a point
More informationLECTURE 3: REPRESENTATION THEORY OF SL 2 (C) AND sl 2 (C)
LECTURE 3: REPRESENTATION THEORY OF SL 2 (C) AND sl 2 (C) IVAN LOSEV Introduction We proceed to studying the representation theory of algebraic groups and Lie algebras. Algebraic groups are the groups
More informationExploring the Exotic Setting for Algebraic Geometry
Exploring the Exotic Setting for Algebraic Geometry Victor I. Piercey University of Arizona Integration Workshop Project August 6-10, 2010 1 Introduction In this project, we will describe the basic topology
More informationCourse 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra
Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................
More informationUrsula Whitcher May 2011
K3 Surfaces with S 4 Symmetry Ursula Whitcher ursula@math.hmc.edu Harvey Mudd College May 2011 Dagan Karp (HMC) Jacob Lewis (Universität Wien) Daniel Moore (HMC 11) Dmitri Skjorshammer (HMC 11) Ursula
More informationALGEBRA EXERCISES, PhD EXAMINATION LEVEL
ALGEBRA EXERCISES, PhD EXAMINATION LEVEL 1. Suppose that G is a finite group. (a) Prove that if G is nilpotent, and H is any proper subgroup, then H is a proper subgroup of its normalizer. (b) Use (a)
More informationNOTES FOR DRAGOS: MATH 210 CLASS 12, THURS. FEB. 22
NOTES FOR DRAGOS: MATH 210 CLASS 12, THURS. FEB. 22 RAVI VAKIL Hi Dragos The class is in 381-T, 1:15 2:30. This is the very end of Galois theory; you ll also start commutative ring theory. Tell them: midterm
More informationSubgroups of Linear Algebraic Groups
Subgroups of Linear Algebraic Groups Subgroups of Linear Algebraic Groups Contents Introduction 1 Acknowledgements 4 1. Basic definitions and examples 5 1.1. Introduction to Linear Algebraic Groups 5 1.2.
More information