Toroidal Embeddings and Desingularization
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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.
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