Toroidal Embeddings and Desingularization

Size: px
Start display at page:

Download "Toroidal Embeddings and Desingularization"

Transcription

1 California State University, San Bernardino CSUSB ScholarWorks Electronic Theses, Projects, and Dissertations Office of Graduate Studies Toroidal Embeddings and Desingularization LEON NGUYEN Follow this and additional works at: Part of the Geometry and Topology Commons Recommended Citation NGUYEN, LEON, "Toroidal Embeddings and Desingularization" (2018). Electronic Theses, Projects, and Dissertations This Thesis is brought to you for free and open access by the Office of Graduate Studies at CSUSB ScholarWorks. It has been accepted for inclusion in Electronic Theses, Projects, and Dissertations by an authorized administrator of CSUSB ScholarWorks. For more information, please contact

2 Toroidal Embeddings and Desingularization A Thesis Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degree Master of Arts in Mathematics by Leon Nguyen June 2018

3 Toroidal Embeddings and Desingularization A Thesis Presented to the Faculty of California State University, San Bernardino by Leon Nguyen June 2018 Approved by: Dr.Wenxiang Wang, Committee Chair Dr. Rolland Trapp, Committee Member Dr. Corey Dunn, Committee Member Dr. Charles Stanton, Chair, Department of Mathematics Dr. Corey Dunn, Graduate Coordinator

4 iii Abstract Algebraic geometry is the study of solutions in polynomial equations using objects and shapes. Differential geometry is based on surfaces, curves, and dimensions of shapes and applying calculus and algebra. Desingularizing the singularities of a variety plays an important role in research in algebraic and differential geometry. Toroidal Embedding is one of the tools used in desingularization. Therefore, Toroidal Embedding and desingularization will be the main focus of my project. In this paper, we first provide a brief introduction on Toroidal Embedding, then show an explicit construction on how to smooth a variety with singularity through Toroidal Embeddings.

5 iv Acknowledgements I would like to thank Dr. Wang for all his help and support in the development and information to contributing to this project and taking the time out of his day to meet with me and explaining each steps specifically to help me understand better. I also would like to thank Dr. Trapp and Dr. Dunn on their commentaries to make my project and better and improve. Your assistance was very much appreciated. Finally, without my family s encouragement and support this would have not been done.

6 v Table of Contents Abstract Acknowledgements List of Figures iii iv vi 1 Introduction Introduction Definitions Theorems Toroidal Embeddings 9 3 Desingularization 12 Bibliography 30

7 vi List of Figures 3.1 σ ˆσ The sub-cones and their faces of σ σ ˆσ σ ˆσ σ ˆσ τ ˆτ τ ˆτ τ ˆτ τ ˆτ

8 1 Chapter 1 Introduction 1.1 Introduction An affine variety is the zero set of a family of polynomials in C n. It often has singularities. To be able to desingularize the singularities of a variety is the key to many research in algebraic and differential geometry. There are different techniques in desingularization. Toroidal Embedding is one of them. In this paper, we will first introduce the Toroidal Embedding, then apply it to smooth a variety with singularity. First, we introduce some definitions in algebraic and differential geometry, and recall some theorems we need. For the further details on these definitions and theorems, see [GH] and [H].

9 2 1.2 Definitions Definition 1.1: Diffeomorphism Let f(x 1, x 2,..., x n ) = ( f 1 (x 1, x 2,..., x n ),..., f n (x 1, x 2,..., x n ) ) be a mapping between two open sets of R n. f is said to be a C r mapping if each f i can be differentiated up to r times, i = 1, 2,..., n. f is said to be C -differentiable if it is C mapping. f is called a diffeomorphism if f is a bijection, and f and f 1 are both C - differentiable. Definition 1.2: Holomorphism Let f(z) : U C be a function of one complex variable, where U is an open set of C, f(z) = u(x, y) + iv(x, y) f(z) is said to be holomorphic, if f (z) exists for every z 0 = (x 0, y 0 ) U, Where f (z 0 ) = lim z z 0 f(z) f(z 0 ) z z 0 = u x (x 0, y 0 ) + i v x (x 0, y 0 ) = i u y (x 0, y 0 ) + v y (x 0, y 0 ) A function f(z 1, z 2,..., z n ) of n complex variables is said to be holomorphic if f(z 1, z 2,..., z n ) is holomorphic in each of variables.

10 3 Definition 1.3: Manifold Let X be a topological space satisfying the Hansdorff separation axiom. A differentiable structure on X of dimension n is a collection of open charts {(U i, φ i )}, i is ranging in some index I, satisfying the following conditions (i) X = i I U i (ii) Each of φ i is a bijection of U i onto an open set of R n. (iii) φ j φ 1 i : φ i (U i U j ) φ j (U i U j ) is a diffeomorphism. X is called a R-manifold of dimension n if X admits a differentiable structure of dim n, {(U i, φ i ), i I}. Definition 1.4: Complex Manifold from A complex manifold M is a differentiable manifold admitting an open covering {U α } and coordinate maps φ α : U α C n such that φ α φ 1 β a biholomorphism φ β (U α U β ) to φ α (U α U β ). Example 1.1 Real Manifold Let X = {(x, y) : x 2 + y 2 = 1}, a subset of R 2 U 1 = {(x, y) : 1 < x < 1, y = 1 x 2 }, U 2 = {(x, y) : 1 < x < 1, y = 1 x 2 }, U 3 = {(x, y) : 1 < y < 1, x = 1 y 2 }, U 4 = {(x, y) : 1 < y < 1, x = 1 y 2 } defined by φ 1 : U 1 R, φ 1 (x, y) = x

11 4 φ 3 : U 3 R, defined by φ 3 (x, y) = y φ 2 and φ 4 can be defined similarly. Then φ 1 φ 1 1 : φ 1 (U 1 ), φ 1 (U 1 ), (φ 1 φ 1 1 )(t) = t φ 3 φ 1 1 : φ 1 (U 1 U 3 ) φ 3 (U 1 U 3 ) (φ 3 φ 1 1 )(t) = φ 3(φ 1 (t)) = φ 3 (t, 1 t 2 ) = 1 t 2, where 0 < t < 1 {(U i, φ i ), i = 1, 2, 3, 4} is a differential structure on X. Therefore, X is a R-manifold of dimension 1. Example 1.2 Complex Manifold The unit two-sphere S 2, which is the subset of R 3, defined by x 2 + y 2 + z 2 = 1 is a complex manifold. One can use stereographic projection from the North Pole to the real plane R 2 with coordinates X, Y given by (X, Y ) = ( x 1 z, y ). 1 z This can be done for any point except the North Pole itself (corresponding to z = 1). To include the North Pole, we introduce a second chart, in which we stereographically project from the South Pole: (U, V ) = ( x 1 + z, y ), 1 + z

12 5 which holds for any point on S 2 except for the South Pole(atz = 1). In both patches, we can now define complex coordinates Z = X + iy, Z = X iy, W = U iv, W = U + iv, and show that on the overlap of the two patches, the transition function is holomorphic. Indeed, on the overlap we compute that W = 1 Z. This expression relates the coordinates W to Z in a holomorphic way. Hence the two-sphere is a complex manifold which can be identified with C. Definition 1.5: Jacobian (i). Holomorphic Jacobian Let U C n be an open set of C n and let f : U C n be a holomorphic mapping, that is, f = (f 1, f 2,..., f n ) with each f j holomorphic. Let w j = f j (z), where z = (z 1,..., z n ). The Holomorphic Jacobian of f is the matrix J C f = (w 1,..., w n ) (z 1,..., z n ) w 1 w z z n = w n z 1... w n z n z = x + iy Recall: For a complex valued function f(z) = u + iv of a complex variable f z = 1 ( f ) 2 x i f y and = 1 2 [ ( u x + v v ) + i( y x u ) ] y f z = 1 ( f ) 2 x + i f y

13 6 (ii). Real Jacobian Let z j = x j + iy j, w k = u k + iv k, j = 1,..., n, k = 1,..., n. The Real Jacobian of f is the matrix One can prove that J R f = (u 1, v 1,..., u n, v n ) (x 1, y 1,..., x n, y n ) = u 1 x 1, u 1 u 1 x n, u 1 y 1... y n v n x 1, vn y 1... v n x n, vn y n detj R f = detj C f detj C f. Definition 1.6: Variety over C 1. X is called an analytic variety if X is the common zero locus of a collection of holomorphic functions of n variables, i.e X = {(z 1, z 2,..., z n ); f 1 (z 1,..., z n ) = 0,..., f k (z 1,..., z n ) = 0} C n, k n, where each f i is a holomorphic function. z 0 = (z 1,..., z n ) X is called a non-singular point if rank [J C (f)( z 0 )] = k, (i.e. there is a k k sub-matrix A of J C (f)(z 0 ), such that deta 0.)

14 7 where f = (f 1,..., f k ) Otherwise, z 0 is called a singular point. If X sig = {all singular point of X},then X sig as a submanifold of C n. polynomial in 2. X is called affine variety if X is the common zero locus of a collection of C[z 1,..., z n ]. 3. X is called an algebraic variety (projective variety) if X is the common zero locus of a collection of homogenous polynomials.

15 8 1.3 Theorems Theorem 1.1: Implicit Function Theorem Let (f 1, f 2,..., f k ) be a holomorphic function of n complex variable (z 1,..., z n ), k n. If det(j C (f) k (z 0 )) 0, where and J C (f) k = (f 1,..., f k ) (z 1,..., z k ) z 0 = (z 0 1, z 0 2,..., z 0 n) then there exist holomorphic function φ 1,..., φ k of n k variables such that in a neighborhood of z 0, f 1 (z 1,..., z n ) = f 2 (z 1,..., z n ) = = f k (z 1,..., z n ) = 0 if and only if z j = φ j (z k+1,..., z n ), j = 1,..., k. Theorem 1.2: Inverse Function Theorem Let U C n be an open set of C n and f : U C n be a holomorphic mapping with detj C f(z 0 ) 0, z 0 U. Then f is one-to-one in a neighborhood of z 0, and f 1 is holomorphic at f(z 0 ).

16 9 Chapter 2 Toroidal Embeddings Toroidal Embeddings We first give a brief introduction on toroidal embedding. For the further details on the subject, see [N]. Definition. Let T be an n-dimensional complex torus, i.e., T = (C ) n. (1) A torus embedding of T is an algebraic variety X such that (a) X contains T as a Zariski open dense subset; (b) T acts on X extending the natural action on itself defined by translation. (2) A morphism between torus embedding X and X is a map f : X X such that the following diagram commutes, T X f X X We can describe the torus embedding combinatorially. Let T = (C ) n = Spec(C[T 1, T1 1,..., T n, Tn 1 ]) as a scheme. For a commutative ring R, Spec(R) is the set of all prime ideals of R. Let M = Hom(T, C ). M is called the character Group of T. Then M Z n with the following mapping, for r = (r 1, r 2,...r n ) Z n, χ r M, where χ r (t 1, t 2,..., t n ) = t r 1 1 tr 2 2 trn n. Let N = Hom(C, T ). N is called the group of one-parameter subgroup in T. Then N Z n with the following mapping,

17 10 for a = (a 1, a 2,..., a n ) Z n, λ a N, where λ a (t) = (t a 1, t a 2,..., t an ). M and N are dual to each other under the pairing, : M N Z, r, a = Σ n i=1r i a i. Notice that χ r (λ a (t)) = t r,a for r M, a N, and t C. If we identify χ r with the monomial Π n i=1 T r i i, for a subsemigroup S of M containing 0, then C[χ r ] r S is a subring of C[M] = C[T 1, T1 1,..., T n, Tn 1 ]. Let N c = N C, then N C C n and T = Spec(C[M]) = N C /N. a line. Then Let σ be a convex rational polyhedral cone in N R = N R = R n not containing The dual of σ, σ = {a N R ; r i, a 0, i = 1,..., k, r i M}. ˆσ = {r M R ; r, a 0, for all a σ}, is the cone in M R. X σ is defined to be Spec(C[ˆσ M]). Then X σ is an affine normal torus embedding of T through Spec(C[M]) Spec(C[ˆσ M]). Let {r 1,..., r m } be a subset of M which generates ˆσ M, i.e., ˆσ M = Z + r Z + r m. (m n, since σ does not contain a line) Then X σ = Spec(C[χ r 1,..., χ rm ]) C m. The embedding of T into X σ is defined by i : T C m, i(t) = (χ r 1 (t),..., χ rm (t)), where t = (t 1,..., t n ) T.

18 11 X σ is the scheme-theoretic closure of i(t ) in C m. T acts on X σ as t x = (χ r 1 (t)x 1,..., χ rm (t)x m ) for t T, x = (x 1,..., x m ) X σ. X σ can be decomposed as the disjoint union of T -orbits under this action, and {T orbits in X σ } {all faces of σ}. If τ is a face of σ ( we will write it as τ < σ), let N(τ) be the subset {r i ; r i, τ = 0} of {r 1,..., r m }, and O τ be the T -orbit in X σ corresponding to τ. Then O τ = {(x 1,..., x m ) X σ ; x i 0 iff r i N(τ)}, and dimτ + dimo τ = dimt = n O 0 = T. A finite rational partial polyhedral decomposition of N R is a finite collection Σ = {σ i } such that (i) the face of σ is in Σ if σ Σ; (ii) σ i σ j is a face of both σ i and σ j for σ i, σ j Σ. For a finite rational partial polyhedral decomposition Σ, we can patch all X σi, σ i Σ together to form a normal torus embedding of T, X Σ, by the T -orbits. In fact, if τ < σ, then X τ X σ and the inclusion X τ X σ is an open immersion in the following diagram, T T X τ X σ. X Σ is the disjoint union of all T -orbits in X Σ. X σ is smooth if and only if σ is regular, i.e.,σ is generated by a part of a Z-basis of N. X Σ is smooth if and only if each member σ of Σ is regular. For a non-regular σ, one can find a finite rational polyhedral decomposition Σ of σ such that each member of Σ is regular. Then X Σ will be a smooth variety which is a blowing-up of X σ at its singularity.

19 12 Chapter 3 Desingularization Consider S = {(x, y, z) C 3 ; F (x, y, z) = z 3 xy = 0}, then S is an analytic hypersurface in C 3. Since J C F = [ y, x, 3z 2 ]. It is clear that 0 = (0, 0, 0) is the only singular point of S, i.e. S 0 = S \{0} is a complex manifold. S is the natural completion of S 0 in C 3, yet singular. In algebraic geometry, it is known that one can obtain a canonical smooth completion S 0 of S 0 by blowing up the singular point 0. In this project, we are going to explicitly construct S 0, by using toroidal embeddings. Let σ = {(x, y) R 2 ; x y 0, x + 4y 0}. σ is called a rational convex polyhedral cone. Figure 3.1: σ

20 13 Notice that σ = R + {(4, 1), (1, 1)}, where R + is the set of all non-negative real numbers. Let <, > denote the usual inner product in R 2. The dual of σ is defined as the following, ˆσ = {r R 2 ; < r, a > 0, a σ}. ˆσ = { r = (r 1, r 2 ); r 1 a 1 + r 2 a 2 0, a = (a 1, a 2 ) σ}. Let r 1 = (1, 1) and r 2 = ( 1, 4) Claim: ˆσ = R + { r 1, r 2 } Proof: We will prove it by showing two inclusions, R + { r 1, r 2 } ˆσ and ˆσ R + { r 1, r 2 }. (i) First, we prove R + { r 1, r 2 } ˆσ By the definition of σ, it is clear that r 1 ˆσ and Therefore, r 2 ˆσ. R + { r 1, r 2 } ˆσ (ii) Second, we prove ˆσ R + { r 1, r 2 }. For any r = (r 1, r 2 ) ˆσ, then < r, a > 0, a σ.

21 14 Since { r 1 }, { r 2 } forms a basis for R 2, then r = α r 1 + β r 2, for some α R, and β R. For a 1 = (1, 1) σ, < r 1, a 1 >= 0, < r 2, a 1 >= 3, therefore, It implies that Similarly, < r, a 1 >= 3β 0. β 0. a 2 = (4, 1) σ, < r 1, a 1 >= 3, < r 2, a 2 >= 0, then It implies that Then Therefore, We have proved that < r, a 2 >= 3α 0. α 0 r R + { r 1, r 2 } ˆσ R + { r 1, r 2 } ˆσ = R + { r 1, r 2 }

22 15 Figure 3.2: ˆσ Let ˆσ z denote the set the all integer points of ˆσ, i.e., ˆσ z = ˆσ Z 2, where Z = the set of integers. Then ˆσ Z = Z + {r 1 = (1, 1), r 2 = ( 1, 4), r 3 = (0, 1)}, i.e. {r 1, r 2, r 3 } forms a Z + -basis for ˆσ Z Each r i corresponds to a monomial m ri in C[T 1, T 1 1, T 2, T 1 2 ], m r1 = T 1 T 1 2, m r2 = T 1 1 T 4 2, m r3 = T 2. They induce a map i σ : T C 3, where, T = (C ) 2, the complex torus of dim 2. i σ (t 1, t 2 ) = (t 1 t 1 2, t 1 1 t4 2, t 2 ) =: (x, y, z). It is clear that i σ (T ) S. In fact, S = Spec(C[m r1, m r2, m r3 ]),

23 16 the Scheme-theoretic closure of i σ (T ) in C 3. We will denote Spec(C[m r1, m r2, m r3 ]) by X σ. It is called the Torus embedding associated with a rational convex polyhedral cone σ. It is known that, from the general theory of Toroidal Embeddings, X φ is smooth if and only if φ is regular, for any rational convex polyhedral cone φ of R 2. φ is called regular if φ Z 2 can be generated by a R + -basis of φ. The cone σ above is not regular because the R + -basis {(4, 1), (1, 1)} can not generate σ Z 2. Hence, X σ = S has a singularity at 0 = (0, 0, 0) Now we consider the sub-cones and their faces of σ. Figure 3.3: The sub-cones and their faces of σ σ 1 = R + {(2, 1), (1, 1)} σ 2 = R + {(2, 1), (3, 1)} σ 3 = R + {(3, 1), (4, 1)} τ 1 = R + {(1, 1)} τ 2 = R + {(2, 1)} τ 3 = R + {(3, 1)} τ 4 = R + {(4, 1)}

24 17 They are all regular. Σ = {0, τ 1, τ 2, τ 3, τ 4, σ 1, σ 2, σ 3 } is called a regular rational convex polyhedral decomposition of σ. For each cone in Σ, there is a Torus embedding of T associated to it. We are going to construct some of them here in detail. Considering the sub-cone σ 1 Figure 3.4: σ 1 It can be verified that the dual ˆσ 1 =: {r R 2, < r, a > 0, a σ 1 } = R + {(1, 1), ( 1, 2)}. (So, σ 1 is regular.) ˆσ 1 Z 2 = Z + {(1, 1), ( 1, 2)}. Therefore, the embedding i σ1 : T C 2 is given by i σ1 (t 1, t 2 ) = (t 1 t 1 2, t 1 1 t2 2).

25 18 Figure 3.5: ˆσ 1 Let X σ1 denote the closure of i σ1 (T ) in C 3, then X σ1 = C 2. Considering the sub-cone σ 2 Figure 3.6: σ 2

26 19 It can be verified that the dual ˆσ 2 =: {r R 2, < r, a > 0, a σ 2 } = R + {(1, 2), ( 1, 3)}. Figure 3.7: ˆσ 2 (So, σ 2 is regular.) ˆσ 2 Z 2 = Z + {(1, 2), ( 1, 3)}. Therefore, the embedding i σ2 : T C 2 is given by i σ2 (t 1, t 2 ) = (t 1 t 2 2, t 1 1 t3 2). Let X σ2 denote the closure of i σ2 (T ) in C 3, then X σ2 = C 2.

27 20 Considering the sub-cone σ 3 Figure 3.8: σ 3 It can be verified that the dual ˆσ 3 =: {r R 2, < r, a > 0, a σ 3 } = R + {(1, 3), ( 1, 4)}. (So, σ 3 is regular.) ˆσ 3 Z 2 = Z + {(1, 3), ( 1, 4)}. Therefore, the embedding i σ3 : T C 2 is given by i σ3 (t 1, t 2 ) = (t 1 t 3 2, t 1 1 t4 2). For the face τ 1 = R + {(1, 1)} of σ, it is found that ˆτ 1 = R + {( 1, 1), (1, 1), (0, 1)},

28 21 Figure 3.9: ˆσ 3 Figure 3.10: τ 1 ˆτ 1 Z 2 = Z + {( 1, 1), (1, 1), (0, 1)}, and i τ1 defines an embedding of T into C 3, where i τ1 (t 1, t 2 ) = (t 1 t 1 2, t 1 1 t 2, t 2 ). Hence, X τ1 = {(x, y, z) C 3, xy = 1} is a smooth surface in C 3.

29 22 Figure 3.11: ˆτ 1 For the face τ 2 = R + {(2, 1)} of σ, Figure 3.12: τ 2 it is found that ˆτ 2 = R + {( 1, 2), (1, 2), (1, 0)}, ˆτ 2 Z 2 = Z + {( 1, 2), (1, 2), (1, 0)},

30 23 Figure 3.13: ˆτ 2 and i τ2 defines an embedding of T into C 3, where i τ2 (t 1, t 2 ) = (t 1 1 t2 2, t 1 t 2 2, t 1). Hence, X τ2 = {(x, y, z) C 3, xy = 1} is a smooth surface in C 3. For the face τ 3 = R + {(3, 1)} of σ, Figure 3.14: τ 3

31 24 it is found that ˆτ 3 = R + {( 1, 3), (1, 3), (1, 0)}, ˆτ 3 Z 2 = Z + {( 1, 3), (1, 3), (1, 0)}, Figure 3.15: ˆτ 3 and i τ3 defines an embedding of T into C 3, where i τ3 (t 1, t 2 ) = (t 1 1 t3 2, t 1 t 3 2, t 1). Hence, X τ3 = {(x, y, z) C 3, xy = 1} is a smooth surface in C 3. For the face τ 4 = R + {(4, 1)} of σ, it is found that ˆτ 4 = R + {( 1, 4), (1, 4), (1, 0)},

32 25 Figure 3.16: τ 4 ˆτ 4 Z 2 = Z + {( 1, 4), (1, 4), (1, 0)}, and i τ4 defines an embedding of T into C 3, where i τ4 (t 1, t 2 ) = (t 1 1 t4 2, t 1 t 4 2, t 1). Hence, X τ4 = {(x, y, z) C 3, xy = 1} is a smooth surface in C 3. X τ2, X τ3 and X τ4 will be the same type of surface as X τ1. We are now going to show how can these X τi and X σj be patched together through the orbit decompositions. Consider the sub-cone σ 1 of σ first. It is clear that ˆσ ˆσ 1 and Recall that ˆσ Z 2 ˆσ 1 Z 2. ˆσ 1 Z 2 = Z + {s 1, s 2 } and

33 26 Figure 3.17: ˆτ 4 ˆσ Z 2 = Z + {r 1, r 2, r 3 }, where s 1 = (1, 1) and s 2 = ( 1, 2). Therefore, the set {s 1, s 2 } should generates the set {r 1, r 2, r 3 } over Z +. In fact, r 1 = s 1, r 2 = s 1 + 2s 2, r 3 = s 1 + s 2. Their monomials are related as in the following, m r1 = m s1, m r2 = m s1 m 2 s 2, m r3 = m s1 m s2. They induce a homomorphism α 1 from X σ1 to X σ. α 1 (u, v) = (u, uv 2, uv) =: (x, y, z). It can be verified that α 1 is 1-1 on X σ {z 0}, and the following diagram

34 27 commutes. α T X σ1 α 1 X σ X σ On the other hand, each X has a canonical decomposition in terms of all possible faces of the cone. For instance, X σ = O 0 O τ1 O τ4 O σ, where O 0 = {(x, y, z); z 3 = xy, xyz 0} = (C ) 2 = T O τ1 = {(x, 0, 0); x 0} = C O τ4 = {(0, y, 0); y 0} = C O σ = {(0, 0, 0)} This is called the orbits decompositions of X σ. We explain in the following how the orbit associated to a face of σ, for instance O τ1, is determined. Recall that the embedding is given by the monomials {m r1, m r2, m r3 }. i σ : T X σ C 3 i σ (t 1, t 2 ) = (x, y, z) where and x = m r1 (t 1, t 2 ), y = m r2 (t 1, t 2 ), z = m r3 (t 1, t 2 ), {r 1 = (1, 1), r 2 = ( 1, 3), r 3 = (0, 1)} ˆσ. Notice that r 1 = 0 on τ 1, r 2 and r 3 are non-zero on τ 1. Hence, the orbit O τ1 is determined by the conditions x 0, y = z = 0. Similarly, X σ1 = C 2 = O0 1 O1 τ 1 Oτ 1 2 O σ1, where O0 1 = {(u, v), uv 0} = (C ) 2 = T Oτ 1 1 = {(u, 0); u 0} = C Oτ 1 2 = {(0, v); v 0} = C O σ1 = {(0, 0)}

35 28 σ and σ 1 have common faces τ 1 and 0 ( the origin ). The induced homomorphism α 1 becomes an isomorphism when it is restricted to O 1 0 or O1 τ 1. In the mean time, α 1 will collapse O 1 τ 2 O σ1 into a single point O σ. This is illustrated in the following diagrams. X σ1 = O 1 0 O 1 τ 1 O 1 τ 2 O σ1 X σ = O 0 O τ1 O σ O τ4 The orbit decompositions for X σ2 are stated in the following, X σ2 = O 2 0 O 2 τ 2 O 2 τ 3 O σ2. The induced homomorphism α 2 from X σ2 to X σ will collapse O 2 τ 2 O 2 τ 3 O σ2 into a single point O σ,while {α 2 : O 2 0 O 0} is isomorphism. X σ2 = O 2 0 O 2 τ 2 O 2 τ 3 O σ2 X σ = O 0 O τ1 O σ O τ4 Each X τi has an orbit decompositions with two orbits only. For instance, X τ2 = O 0 O τ 2. Since τ 2 is a face of σ 1, there is an induced immersion from X τ2 into X σ1 which will isomorphically map O 0 onto O1 0, and O τ 2 onto O 1 τ 2. Therefore, under the isomorphisms, we may write X τ2 = O 0 O τ 2 and X σ1 = O 0 O τ 1 O τ 2 O σ1. Similarly, X σ2 = O 0 O τ 2 O τ 3 O σ2. X σ3 = O 0 O τ 3 O τ 4 O σ3. X τ1 = O 0 O τ 1,

36 29 X τ3 = O 0 O τ 3 and X τ4 = O 0 O τ 4 Let X Σ = 3 i=1 X σ i 4 j=1 X τ j be the union of X σi and X τj, patching through the orbits. Hence, X Σ has an orbit decomposition as the following, X Σ = O 0 O τ 1 O τ 3 O τ 2 O τ 4 O σ1 O σ2 O σ3. X Σ is a smooth variety, since Σ = {0, τ 1, τ 2, τ 3, τ 4, σ 1, σ 2, σ 3 } is a regular rational convex polyhedral decomposition of σ. There is a homomorphism β from X Σ to X σ, which will be an isomorphism on O 0, O τ 1 and O τ 4 respectively, O 0 = O 0 O τ 1 = Oτ1 O τ 3 = Oτ4. β will map O τ 2 O τ 3 O σ1 O σ2 O σ3 have the following diagrams. into the single point O σ. Therefore, we X Σ = O 0 O τ 1 O τ 4 D X σ = O 0 O τ1 O τ4 O σ where D = O τ 2 O τ 3 O σ1 O σ2 O σ3 Notice that O σ = {0}, the singular point of S. We have constructed a smooth completion X Σ of the surface S with a mapping β : X Σ X σ = S. β is the blowing-up of S at its singular point 0. is the exceptional divisor. β 1 (0) = O τ 2 O τ 3 O σ1 O σ2 O σ3,

37 30 Bibliography [AMRT75] Ash, A., Mumford, D., Rapoport, M., Tai, Y. : Smoothy compactification of locally symmetric varieties. Brookline, Math. Sci. Press [BB66] Baily, W., Borel, A. : Compactifications of arithmetic quotients of bounded symmetric domains. Ann. Math., Vol 84, (1966). [GH78] Griffiths, P., Harris, J. : Principles of Algebraic Geometry. John Wiley & Sons, [Hel78] Helgason, S. : Differential Geometry Lie Groups, and Symmetric Spaces. Academic Press, [Nam80] Namikawa, Y. : Toroidal compactifications of Siegel spaces, Lecture Notes in Math., Vol 812, Springer, Berlin, [Wan93] Wang, W.-X. : On the smooth compactification of Siegel spaces. J. of Differential Geometry, Vol 38 (1993) [Wan99] Wang, W.-X. : A note on the logarithmic canonical line bundle of locally symmetric Hermitian spaces. Commentarii Mathematici (Japan), Vol 48 (1999) 1-5.

Lecture 1. Toric Varieties: Basics

Lecture 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 information

Riemann Surfaces and Algebraic Curves

Riemann Surfaces and Algebraic Curves Riemann Surfaces and Algebraic Curves JWR Tuesday December 11, 2001, 9:03 AM We describe the relation between algebraic curves and Riemann surfaces. An elementary reference for this material is [1]. 1

More information

where 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

where 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

INTRODUCTION TO TORIC VARIETIES

INTRODUCTION TO TORIC VARIETIES INTRODUCTION TO TORIC VARIETIES Jean-Paul Brasselet IML Luminy Case 97 F-13288 Marseille Cedex 9 France jpb@iml.univ-mrs.fr The course given during the School and Workshop The Geometry and Topology of

More information

Math 797W Homework 4

Math 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 information

PROBLEMS, MATH 214A. Affine and quasi-affine varieties

PROBLEMS, 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 information

Subgroups of Lie groups. Definition 0.7. A Lie subgroup of a Lie group G is a subgroup which is also a submanifold.

Subgroups of Lie groups. Definition 0.7. A Lie subgroup of a Lie group G is a subgroup which is also a submanifold. Recollections from finite group theory. The notion of a group acting on a set is extremely useful. Indeed, the whole of group theory arose through this route. As an example of the abstract power of this

More information

Homogeneous Coordinate Ring

Homogeneous Coordinate Ring Students: Kaiserslautern University Algebraic Group June 14, 2013 Outline Quotients in Algebraic Geometry 1 Quotients in Algebraic Geometry 2 3 4 Outline Quotients in Algebraic Geometry 1 Quotients in

More information

1. Algebraic vector bundles. Affine Varieties

1. Algebraic vector bundles. Affine Varieties 0. Brief overview Cycles and bundles are intrinsic invariants of algebraic varieties Close connections going back to Grothendieck Work with quasi-projective varieties over a field k Affine Varieties 1.

More information

Combinatorics and geometry of E 7

Combinatorics 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 information

Toric 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 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 information

Introduction to toric geometry

Introduction 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 information

Resolution of Singularities in Algebraic Varieties

Resolution 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 information

REALIZING TOURNAMENTS AS MODELS FOR K-MAJORITY VOTING

REALIZING TOURNAMENTS AS MODELS FOR K-MAJORITY VOTING California State University, San Bernardino CSUSB ScholarWorks Electronic Theses, Projects, and Dissertations Office of Graduate Studies 6-016 REALIZING TOURNAMENTS AS MODELS FOR K-MAJORITY VOTING Gina

More information

Delzant s Garden. A one-hour tour to symplectic toric geometry

Delzant s Garden. A one-hour tour to symplectic toric geometry Delzant s Garden A one-hour tour to symplectic toric geometry Tour Guide: Zuoqin Wang Travel Plan: The earth America MIT Main building Math. dept. The moon Toric world Symplectic toric Delzant s theorem

More information

QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday 10 February 2004 (Day 1)

QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday 10 February 2004 (Day 1) Tuesday 10 February 2004 (Day 1) 1a. Prove the following theorem of Banach and Saks: Theorem. Given in L 2 a sequence {f n } which weakly converges to 0, we can select a subsequence {f nk } such that the

More information

Spherical varieties and arc spaces

Spherical 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 information

Algebraic v.s. Analytic Point of View

Algebraic v.s. Analytic Point of View Algebraic v.s. Analytic Point of View Ziwen Zhu September 19, 2015 In this talk, we will compare 3 different yet similar objects of interest in algebraic and complex geometry, namely algebraic variety,

More information

Monomial equivariant embeddings of quasitoric manifolds and the problem of existence of invariant almost complex structures.

Monomial equivariant embeddings of quasitoric manifolds and the problem of existence of invariant almost complex structures. Monomial equivariant embeddings of quasitoric manifolds and the problem of existence of invariant almost complex structures. Andrey Kustarev joint work with V. M. Buchstaber, Steklov Mathematical Institute

More information

TitleOn manifolds with trivial logarithm. Citation Osaka Journal of Mathematics. 41(2)

TitleOn manifolds with trivial logarithm. Citation Osaka Journal of Mathematics. 41(2) TitleOn manifolds with trivial logarithm Author(s) Winkelmann, Jorg Citation Osaka Journal of Mathematics. 41(2) Issue 2004-06 Date Text Version publisher URL http://hdl.handle.net/11094/7844 DOI Rights

More information

4. The concept of a Lie group

4. The concept of a Lie group 4. The concept of a Lie group 4.1. The category of manifolds and the definition of Lie groups. We have discussed the concept of a manifold-c r, k-analytic, algebraic. To define the category corresponding

More information

Introduction to Arithmetic Geometry Fall 2013 Lecture #23 11/26/2013

Introduction to Arithmetic Geometry Fall 2013 Lecture #23 11/26/2013 18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #23 11/26/2013 As usual, a curve is a smooth projective (geometrically irreducible) variety of dimension one and k is a perfect field. 23.1

More information

10. Smooth Varieties. 82 Andreas Gathmann

10. 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 information

Geometry of moduli spaces

Geometry of moduli spaces Geometry of moduli spaces 20. November 2009 1 / 45 (1) Examples: C: compact Riemann surface C = P 1 (C) = C { } (Riemann sphere) E = C / Z + Zτ (torus, elliptic curve) 2 / 45 (2) Theorem (Riemann existence

More information

EKT of Some Wonderful Compactifications

EKT of Some Wonderful Compactifications EKT of Some Wonderful Compactifications and recent results on Complete Quadrics. (Based on joint works with Soumya Banerjee and Michael Joyce) Mahir Bilen Can April 16, 2016 Mahir Bilen Can EKT of Some

More information

LECTURE 15-16: PROPER ACTIONS AND ORBIT SPACES

LECTURE 15-16: PROPER ACTIONS AND ORBIT SPACES LECTURE 15-16: PROPER ACTIONS AND ORBIT SPACES 1. Proper actions Suppose G acts on M smoothly, and m M. Then the orbit of G through m is G m = {g m g G}. If m, m lies in the same orbit, i.e. m = g m for

More information

Igusa fibre integrals over local fields

Igusa fibre integrals over local fields March 25, 2017 1 2 Standard definitions The map ord S The map ac S 3 Toroidal constructible functions Main Theorem Some conjectures 4 Toroidalization Key Theorem Proof of Key Theorem Igusa fibre integrals

More information

B 1 = {B(x, r) x = (x 1, x 2 ) H, 0 < r < x 2 }. (a) Show that B = B 1 B 2 is a basis for a topology on X.

B 1 = {B(x, r) x = (x 1, x 2 ) H, 0 < r < x 2 }. (a) Show that B = B 1 B 2 is a basis for a topology on X. Math 6342/7350: Topology and Geometry Sample Preliminary Exam Questions 1. For each of the following topological spaces X i, determine whether X i and X i X i are homeomorphic. (a) X 1 = [0, 1] (b) X 2

More information

The set of points at which a polynomial map is not proper

The set of points at which a polynomial map is not proper ANNALES POLONICI MATHEMATICI LVIII3 (1993) The set of points at which a polynomial map is not proper by Zbigniew Jelonek (Kraków) Abstract We describe the set of points over which a dominant polynomial

More information

Bredon, Introduction to compact transformation groups, Academic Press

Bredon, Introduction to compact transformation groups, Academic Press 1 Introduction Outline Section 3: Topology of 2-orbifolds: Compact group actions Compact group actions Orbit spaces. Tubes and slices. Path-lifting, covering homotopy Locally smooth actions Smooth actions

More information

On Mordell-Lang in Algebraic Groups of Unipotent Rank 1

On 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 information

GAUGED LINEAR SIGMA MODEL SPACES

GAUGED LINEAR SIGMA MODEL SPACES GAUGED LINEAR SIGMA MODEL SPACES FELIPE CASTELLANO-MACIAS ADVISOR: FELIX JANDA Abstract. The gauged linear sigma model (GLSM) originated in physics but it has recently made it into mathematics as an enumerative

More information

3. Lecture 3. Y Z[1/p]Hom (Sch/k) (Y, X).

3. Lecture 3. Y Z[1/p]Hom (Sch/k) (Y, X). 3. Lecture 3 3.1. Freely generate qfh-sheaves. We recall that if F is a homotopy invariant presheaf with transfers in the sense of the last lecture, then we have a well defined pairing F(X) H 0 (X/S) F(S)

More information

HARTSHORNE EXERCISES

HARTSHORNE EXERCISES HARTSHORNE EXERCISES J. WARNER Hartshorne, Exercise I.5.6. Blowing Up Curve Singularities (a) Let Y be the cusp x 3 = y 2 + x 4 + y 4 or the node xy = x 6 + y 6. Show that the curve Ỹ obtained by blowing

More information

LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY

LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY WEIMIN CHEN, UMASS, SPRING 07 1. Blowing up and symplectic cutting In complex geometry the blowing-up operation amounts to replace a point in

More information

Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra

Course 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 information

MONOID RINGS AND STRONGLY TWO- GENERATED IDEALS

MONOID RINGS AND STRONGLY TWO- GENERATED IDEALS California State University, San Bernardino CSUSB ScholarWorks Electronic Theses, Projects, and Dissertations Office of Graduate Studies 6-2014 MONOID RINGS AND STRONGLY TWO- GENERATED IDEALS Brittney

More information

Representations and Linear Actions

Representations 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 information

Demushkin s Theorem in Codimension One

Demushkin 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 information

Minimal surfaces in quaternionic symmetric spaces

Minimal surfaces in quaternionic symmetric spaces From: "Geometry of low-dimensional manifolds: 1", C.U.P. (1990), pp. 231--235 Minimal surfaces in quaternionic symmetric spaces F.E. BURSTALL University of Bath We describe some birational correspondences

More information

SUBJECT I. TOROIDAL PARTIAL COMPACTIFICATIONS AND MODULI OF POLARIZED LOGARITHMIC HODGE STRUCTURES

SUBJECT I. TOROIDAL PARTIAL COMPACTIFICATIONS AND MODULI OF POLARIZED LOGARITHMIC HODGE STRUCTURES Introduction This book is the full detailed version of the paper [KU1]. In [G1], Griffiths defined and studied the classifying space D of polarized Hodge structures of fixed weight w and fixed Hodge numbers

More information

MODULI SPACES OF CURVES

MODULI SPACES OF CURVES MODULI SPACES OF CURVES SCOTT NOLLET Abstract. My goal is to introduce vocabulary and present examples that will help graduate students to better follow lectures at TAGS 2018. Assuming some background

More information

Trace fields of knots

Trace fields of knots JT Lyczak, February 2016 Trace fields of knots These are the knotes from the seminar on knot theory in Leiden in the spring of 2016 The website and all the knotes for this seminar can be found at http://pubmathleidenunivnl/

More information

The Grothendieck-Katz Conjecture for certain locally symmetric varieties

The Grothendieck-Katz Conjecture for certain locally symmetric varieties The Grothendieck-Katz Conjecture for certain locally symmetric varieties Benson Farb and Mark Kisin August 20, 2008 Abstract Using Margulis results on lattices in semi-simple Lie groups, we prove the Grothendieck-

More information

M4P52 Manifolds, 2016 Problem Sheet 1

M4P52 Manifolds, 2016 Problem Sheet 1 Problem Sheet. Let X and Y be n-dimensional topological manifolds. Prove that the disjoint union X Y is an n-dimensional topological manifold. Is S S 2 a topological manifold? 2. Recall that that the discrete

More information

ALGORITHMS FOR ALGEBRAIC CURVES

ALGORITHMS FOR ALGEBRAIC CURVES ALGORITHMS FOR ALGEBRAIC CURVES SUMMARY OF LECTURE 7 I consider the problem of computing in Pic 0 (X) where X is a curve (absolutely integral, projective, smooth) over a field K. Typically K is a finite

More information

ne varieties (continued)

ne varieties (continued) Chapter 2 A ne varieties (continued) 2.1 Products For some problems its not very natural to restrict to irreducible varieties. So we broaden the previous story. Given an a ne algebraic set X A n k, we

More information

LECTURE 11: SYMPLECTIC TORIC MANIFOLDS. Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8

LECTURE 11: SYMPLECTIC TORIC MANIFOLDS. Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8 LECTURE 11: SYMPLECTIC TORIC MANIFOLDS Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8 1. Symplectic toric manifolds Orbit of torus actions. Recall that in lecture 9

More information

Part II. Riemann Surfaces. Year

Part II. Riemann Surfaces. Year Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 96 Paper 2, Section II 23F State the uniformisation theorem. List without proof the Riemann surfaces which are uniformised

More information

Math 418 Algebraic Geometry Notes

Math 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 information

arxiv: v2 [math.dg] 6 Nov 2014

arxiv: v2 [math.dg] 6 Nov 2014 A SHARP CUSP COUNT FOR COMPLEX HYPERBOLIC SURFACES AND RELATED RESULTS GABRIELE DI CERBO AND LUCA F. DI CERBO arxiv:1312.5368v2 [math.dg] 6 Nov 2014 Abstract. We derive a sharp cusp count for finite volume

More information

9. Birational Maps and Blowing Up

9. Birational Maps and Blowing Up 72 Andreas Gathmann 9. Birational Maps and Blowing Up In the course of this class we have already seen many examples of varieties that are almost the same in the sense that they contain isomorphic dense

More information

Affine Geometry and Discrete Legendre Transform

Affine Geometry and Discrete Legendre Transform Affine Geometry and Discrete Legendre Transform Andrey Novoseltsev April 24, 2008 Abstract The combinatorial duality used in Gross-Siebert approach to mirror symmetry is the discrete Legendre transform

More information

APPENDIX 3: AN OVERVIEW OF CHOW GROUPS

APPENDIX 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 information

Siegel Moduli Space of Principally Polarized Abelian Manifolds

Siegel Moduli Space of Principally Polarized Abelian Manifolds Siegel Moduli Space of Principally Polarized Abelian Manifolds Andrea Anselli 8 february 2017 1 Recall Definition 11 A complex torus T of dimension d is given by V/Λ, where V is a C-vector space of dimension

More information

MATH 8253 ALGEBRAIC GEOMETRY WEEK 12

MATH 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 information

Birational geometry and deformations of nilpotent orbits

Birational 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 information

Algebraic 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 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 information

Boundary of Cohen-Macaulay cone and asymptotic behavior of system of ideals

Boundary of Cohen-Macaulay cone and asymptotic behavior of system of ideals Boundary of Cohen-Macaulay cone and asymptotic behavior of system of ideals Kazuhiko Kurano Meiji University 1 Introduction On a smooth projective variety, we can define the intersection number for a given

More information

Another proof of the global F -regularity of Schubert varieties

Another proof of the global F -regularity of Schubert varieties Another proof of the global F -regularity of Schubert varieties Mitsuyasu Hashimoto Abstract Recently, Lauritzen, Raben-Pedersen and Thomsen proved that Schubert varieties are globally F -regular. We give

More information

Toric Varieties and the Secondary Fan

Toric 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 information

AG NOTES MARK BLUNK. 3. Varieties

AG NOTES MARK BLUNK. 3. Varieties AG NOTES MARK BLUNK Abstract. Some rough notes to get you all started. 1. Introduction Here is a list of the main topics that are discussed and used in my research talk. The information is rough and brief,

More information

(1) The embedding theorem. It says that a Stein manifold can always be embedded into C n for sufficiently large n.

(1) The embedding theorem. It says that a Stein manifold can always be embedded into C n for sufficiently large n. Class 1. Overview Introduction. The subject of this course is complex manifolds. Recall that a smooth manifold is a space in which some neighborhood of every point is homeomorphic to an open subset of

More information

Holomorphic line bundles

Holomorphic line bundles Chapter 2 Holomorphic line bundles In the absence of non-constant holomorphic functions X! C on a compact complex manifold, we turn to the next best thing, holomorphic sections of line bundles (i.e., rank

More information

Projective Varieties. Chapter Projective Space and Algebraic Sets

Projective 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 information

12. Hilbert Polynomials and Bézout s Theorem

12. 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 information

Chapter 19 Clifford and the Number of Holes

Chapter 19 Clifford and the Number of Holes Chapter 19 Clifford and the Number of Holes We saw that Riemann denoted the genus by p, a notation which is still frequently used today, in particular for the generalizations of this notion in higher dimensions.

More information

= F (b) F (a) F (x i ) F (x i+1 ). a x 0 x 1 x n b i

= F (b) F (a) F (x i ) F (x i+1 ). a x 0 x 1 x n b i Real Analysis Problem 1. If F : R R is a monotone function, show that F T V ([a,b]) = F (b) F (a) for any interval [a, b], and that F has bounded variation on R if and only if it is bounded. Here F T V

More information

1 Notations and Statement of the Main Results

1 Notations and Statement of the Main Results An introduction to algebraic fundamental groups 1 Notations and Statement of the Main Results Throughout the talk, all schemes are locally Noetherian. All maps are of locally finite type. There two main

More information

Geometry 9: Serre-Swan theorem

Geometry 9: Serre-Swan theorem Geometry 9: Serre-Swan theorem Rules: You may choose to solve only hard exercises (marked with!, * and **) or ordinary ones (marked with! or unmarked), or both, if you want to have extra problems. To have

More information

THE EULER CHARACTERISTIC OF A LIE GROUP

THE EULER CHARACTERISTIC OF A LIE GROUP THE EULER CHARACTERISTIC OF A LIE GROUP JAY TAYLOR 1 Examples of Lie Groups The following is adapted from [2] We begin with the basic definition and some core examples Definition A Lie group is a smooth

More information

D-MATH Algebraic Geometry FS 2018 Prof. Emmanuel Kowalski. Solutions Sheet 1. Classical Varieties

D-MATH Algebraic Geometry FS 2018 Prof. Emmanuel Kowalski. Solutions Sheet 1. Classical Varieties D-MATH Algebraic Geometry FS 2018 Prof. Emmanuel Kowalski Solutions Sheet 1 Classical Varieties Let K be an algebraically closed field. All algebraic sets below are defined over K, unless specified otherwise.

More information

Chern numbers and Hilbert Modular Varieties

Chern numbers and Hilbert Modular Varieties Chern numbers and Hilbert Modular Varieties Dylan Attwell-Duval Department of Mathematics and Statistics McGill University Montreal, Quebec attwellduval@math.mcgill.ca April 9, 2011 A Topological Point

More information

arxiv: v1 [quant-ph] 19 Jan 2010

arxiv: 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 information

Riemann s goal was to classify all complex holomorphic functions of one variable.

Riemann s goal was to classify all complex holomorphic functions of one variable. Math 8320 Spring 2004, Riemann s view of plane curves Riemann s goal was to classify all complex holomorphic functions of one variable. 1) The fundamental equivalence relation on power series: Consider

More information

NOTES ON ALGEBRAIC GEOMETRY MATH 202A. Contents Introduction Affine varieties 22

NOTES 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 information

Locally Symmetric Varieties

Locally Symmetric Varieties Locally Symmetric Varieties So far, we have only looked at Hermitian symmetric domains. These have no algebraic structure (e.g., think about the case of the unit disk D 1 ). This part of the course will

More information

VERY STABLE BUNDLES AND PROPERNESS OF THE HITCHIN MAP

VERY STABLE BUNDLES AND PROPERNESS OF THE HITCHIN MAP VERY STABLE BUNDLES AND PROPERNESS OF THE HITCHIN MAP CHRISTIAN PAULY AND ANA PEÓN-NIETO Abstract. Let X be a smooth complex projective curve of genus g 2 and let K be its canonical bundle. In this note

More information

Chapter 3. Riemannian Manifolds - I. The subject of this thesis is to extend the combinatorial curve reconstruction approach to curves

Chapter 3. Riemannian Manifolds - I. The subject of this thesis is to extend the combinatorial curve reconstruction approach to curves Chapter 3 Riemannian Manifolds - I The subject of this thesis is to extend the combinatorial curve reconstruction approach to curves embedded in Riemannian manifolds. A Riemannian manifold is an abstraction

More information

Rings and groups. Ya. Sysak

Rings 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 information

MAKSYM FEDORCHUK. n ) = z1 d 1 zn d 1.

MAKSYM FEDORCHUK. n ) = z1 d 1 zn d 1. DIRECT SUM DECOMPOSABILITY OF SMOOTH POLYNOMIALS AND FACTORIZATION OF ASSOCIATED FORMS MAKSYM FEDORCHUK Abstract. We prove an if-and-only-if criterion for direct sum decomposability of a smooth homogeneous

More information

A PROOF OF BOREL-WEIL-BOTT THEOREM

A PROOF OF BOREL-WEIL-BOTT THEOREM A PROOF OF BOREL-WEIL-BOTT THEOREM MAN SHUN JOHN MA 1. Introduction In this short note, we prove the Borel-Weil-Bott theorem. Let g be a complex semisimple Lie algebra. One basic question in representation

More information

Rigidity, locally symmetric varieties and the Grothendieck-Katz Conjecture

Rigidity, locally symmetric varieties and the Grothendieck-Katz Conjecture Rigidity, locally symmetric varieties and the Grothendieck-Katz Conjecture Benson Farb and Mark Kisin May 8, 2009 Abstract Using Margulis s results on lattices in semisimple Lie groups, we prove the Grothendieck-

More information

LECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL

LECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL LECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL In this lecture we discuss a criterion for non-stable-rationality based on the decomposition of the diagonal in the Chow group. This criterion

More information

Patrick Iglesias-Zemmour

Patrick Iglesias-Zemmour Mathematical Surveys and Monographs Volume 185 Diffeology Patrick Iglesias-Zemmour American Mathematical Society Contents Preface xvii Chapter 1. Diffeology and Diffeological Spaces 1 Linguistic Preliminaries

More information

Deformation groupoids and index theory

Deformation groupoids and index theory Deformation groupoids and index theory Karsten Bohlen Leibniz Universität Hannover GRK Klausurtagung, Goslar September 24, 2014 Contents 1 Groupoids 2 The tangent groupoid 3 The analytic and topological

More information

ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES

ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES HONGHAO GAO FEBRUARY 7, 2014 Quasi-coherent and coherent sheaves Let X Spec k be a scheme. A presheaf over X is a contravariant functor from the category of open

More information

Porteous s Formula for Maps between Coherent Sheaves

Porteous s Formula for Maps between Coherent Sheaves Michigan Math. J. 52 (2004) Porteous s Formula for Maps between Coherent Sheaves Steven P. Diaz 1. Introduction Recall what the Thom Porteous formula for vector bundles tells us (see [2, Sec. 14.4] for

More information

Math 429/581 (Advanced) Group Theory. Summary of Definitions, Examples, and Theorems by Stefan Gille

Math 429/581 (Advanced) Group Theory. Summary of Definitions, Examples, and Theorems by Stefan Gille Math 429/581 (Advanced) Group Theory Summary of Definitions, Examples, and Theorems by Stefan Gille 1 2 0. Group Operations 0.1. Definition. Let G be a group and X a set. A (left) operation of G on X is

More information

10. The subgroup subalgebra correspondence. Homogeneous spaces.

10. The subgroup subalgebra correspondence. Homogeneous spaces. 10. The subgroup subalgebra correspondence. Homogeneous spaces. 10.1. The concept of a Lie subgroup of a Lie group. We have seen that if G is a Lie group and H G a subgroup which is at the same time a

More information

K-stability and Kähler metrics, I

K-stability and Kähler metrics, I K-stability and Kähler metrics, I Gang Tian Beijing University and Princeton University Let M be a Kähler manifold. This means that M be a complex manifold together with a Kähler metric ω. In local coordinates

More information

LECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE

LECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE LECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE JOHANNES EBERT 1.1. October 11th. 1. Recapitulation from differential topology Definition 1.1. Let M m, N n, be two smooth manifolds

More information

Algebraic varieties and schemes over any scheme. Non singular varieties

Algebraic varieties and schemes over any scheme. Non singular varieties Algebraic varieties and schemes over any scheme. Non singular varieties Trang June 16, 2010 1 Lecture 1 Let k be a field and k[x 1,..., x n ] the polynomial ring with coefficients in k. Then we have two

More information

Elliptic Curves and Elliptic Functions

Elliptic Curves and Elliptic Functions Elliptic Curves and Elliptic Functions ARASH ISLAMI Professor: Dr. Chung Pang Mok McMaster University - Math 790 June 7, 01 Abstract Elliptic curves are algebraic curves of genus 1 which can be embedded

More information

1 Structures 2. 2 Framework of Riemann surfaces Basic configuration Holomorphic functions... 3

1 Structures 2. 2 Framework of Riemann surfaces Basic configuration Holomorphic functions... 3 Compact course notes Riemann surfaces Fall 2011 Professor: S. Lvovski transcribed by: J. Lazovskis Independent University of Moscow December 23, 2011 Contents 1 Structures 2 2 Framework of Riemann surfaces

More information

Logarithmic geometry and rational curves

Logarithmic geometry and rational curves Logarithmic geometry and rational curves Summer School 2015 of the IRTG Moduli and Automorphic Forms Siena, Italy Dan Abramovich Brown University August 24-28, 2015 Abramovich (Brown) Logarithmic geometry

More information

Introduction to Arithmetic Geometry Fall 2013 Lecture #18 11/07/2013

Introduction to Arithmetic Geometry Fall 2013 Lecture #18 11/07/2013 18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #18 11/07/2013 As usual, all the rings we consider are commutative rings with an identity element. 18.1 Regular local rings Consider a local

More information

Geometry and combinatorics of spherical varieties.

Geometry 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 information

QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday January 20, 2015 (Day 1)

QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday January 20, 2015 (Day 1) Tuesday January 20, 2015 (Day 1) 1. (AG) Let C P 2 be a smooth plane curve of degree d. (a) Let K C be the canonical bundle of C. For what integer n is it the case that K C = OC (n)? (b) Prove that if

More information

H(G(Q p )//G(Z p )) = C c (SL n (Z p )\ SL n (Q p )/ SL n (Z p )).

H(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 information