256B Algebraic Geometry
... Thus isomorphism classes of n-dimensional vector bundles on P1 can be naturally identified with double cosets GLn (C[t−1 ])\GLn (C[t, t−1 ])/GLn (C[t]). The rest of the proof consists of the following exercise. Exercise 1.5. (Birkhoff factorization) Each double coset has a unique representative wit ...
... Thus isomorphism classes of n-dimensional vector bundles on P1 can be naturally identified with double cosets GLn (C[t−1 ])\GLn (C[t, t−1 ])/GLn (C[t]). The rest of the proof consists of the following exercise. Exercise 1.5. (Birkhoff factorization) Each double coset has a unique representative wit ...
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27
... Hence we can get a bunch of invertible sheaves, by taking differences of these two. In fact we “usually get them all”! It is very hard to describe an invertible sheaf on a finite type k-scheme that is not describable in such a way. For example, we will see soon that there are none if the scheme is n ...
... Hence we can get a bunch of invertible sheaves, by taking differences of these two. In fact we “usually get them all”! It is very hard to describe an invertible sheaf on a finite type k-scheme that is not describable in such a way. For example, we will see soon that there are none if the scheme is n ...
ON THE IRREDUCIBILITY OF SECANT CONES, AND
... Examples 11. By Corollary 3, any smooth 3-fold in P5 has the ISC property. A specific example is provided by the Segre variety Y = P1 × P2 ⊂ P5 (cf. [SR]). Any P1 × P1 is embedded as a quadric in a solid, and there is precisely one such solid through a general point Q ∈ P5 . Projection from Q maps Y ...
... Examples 11. By Corollary 3, any smooth 3-fold in P5 has the ISC property. A specific example is provided by the Segre variety Y = P1 × P2 ⊂ P5 (cf. [SR]). Any P1 × P1 is embedded as a quadric in a solid, and there is precisely one such solid through a general point Q ∈ P5 . Projection from Q maps Y ...
LINE BUNDLES AND DIVISORS IN ALGEBRAIC GEOMETRY
... we call the Picard group of X, henceforth denoted Pic X. Since only the rank one locally free sheaves are invertible, we get the following definition: Definition 2.10. A locally free sheaf of rank one is called an invertible sheaf. The obvious example of an invertible sheaf is OX itself. Other examp ...
... we call the Picard group of X, henceforth denoted Pic X. Since only the rank one locally free sheaves are invertible, we get the following definition: Definition 2.10. A locally free sheaf of rank one is called an invertible sheaf. The obvious example of an invertible sheaf is OX itself. Other examp ...
Usha - IIT Guwahati
... It is harder to prove that every maximal ideal has the form Ma . Let M be a maximal ideal, and let F denote the field K[x1 , . . . , xn ]/M . We restrict the canonical projection map π : K[x1 , . . . , xn ] → F to the subring K[x1 ] of polynomials in the first variable, obtaining a homomorphism φ1 : ...
... It is harder to prove that every maximal ideal has the form Ma . Let M be a maximal ideal, and let F denote the field K[x1 , . . . , xn ]/M . We restrict the canonical projection map π : K[x1 , . . . , xn ] → F to the subring K[x1 ] of polynomials in the first variable, obtaining a homomorphism φ1 : ...
INTERSECTION THEORY IN ALGEBRAIC GEOMETRY: COUNTING
... Note that the sum of the subscripts is 4 minus the dimension of the Schubert cycle. If it is necessary to indicate which flag these are based on, the relevant point, line, or plane will be indicated in brackets: Σ1 (L2 ) would be lines meeting the line L2 . 3. Intersection Theory and the Chow Ring T ...
... Note that the sum of the subscripts is 4 minus the dimension of the Schubert cycle. If it is necessary to indicate which flag these are based on, the relevant point, line, or plane will be indicated in brackets: Σ1 (L2 ) would be lines meeting the line L2 . 3. Intersection Theory and the Chow Ring T ...
STRUCTURE THEOREMS OVER POLYNOMIAL RINGS 1
... This concept originates in work of Karagueuzian and the author [7, 8], where it is shown that if S = k[x1 , . . . , xn ], k is finite, Un is the group of n × n upper triangular matrices acting in the natural way on S and R is the ring of invariants then S has a structure theorem over RUn . In this p ...
... This concept originates in work of Karagueuzian and the author [7, 8], where it is shown that if S = k[x1 , . . . , xn ], k is finite, Un is the group of n × n upper triangular matrices acting in the natural way on S and R is the ring of invariants then S has a structure theorem over RUn . In this p ...
Toric Varieties
... Combinatorially, a normal toric variety is determined by a fan; the cones in the fan yield affine varieties and the intersection of cones provides gluing data needed to assemble these affine pieces together. Algebraically, an embedded toric variety corresponds to a prime binomial ideal in a polynomi ...
... Combinatorially, a normal toric variety is determined by a fan; the cones in the fan yield affine varieties and the intersection of cones provides gluing data needed to assemble these affine pieces together. Algebraically, an embedded toric variety corresponds to a prime binomial ideal in a polynomi ...
geometric congruence
... preserved by affine transformations, the class of specific geometric configurations is wider than that of the class of the same geometric configuration under Euclidean congruence. For example, all triangles are affine congruent, whereas Euclidean congruent triangles are confined only to those that a ...
... preserved by affine transformations, the class of specific geometric configurations is wider than that of the class of the same geometric configuration under Euclidean congruence. For example, all triangles are affine congruent, whereas Euclidean congruent triangles are confined only to those that a ...
THE GEOMETRY OF FORMAL VARIETIES IN ALGEBRAIC
... many interesting computations in topology, and certain formal varieties called commutative, one-dimensional formal groups give the best global picture of stable homotopy theory currently available. I will give as friendly an introduction to these ideas as can be managed; in particular, I will not as ...
... many interesting computations in topology, and certain formal varieties called commutative, one-dimensional formal groups give the best global picture of stable homotopy theory currently available. I will give as friendly an introduction to these ideas as can be managed; in particular, I will not as ...
Affine Varieties
... V1 ⊆ V2 ⇔ I(V1 ) ⊇ I(V2 ) as is easily checked from the definition. Definition: A closed set Z ⊆ Cn is irreducible if there is no pair of proper closed subsets Z1 , Z2 ⊂ Z with the property that Z = Z1 ∪ Z2 . Proposition 3.2: Features of the Zariski topology include the following: (a) Every descendi ...
... V1 ⊆ V2 ⇔ I(V1 ) ⊇ I(V2 ) as is easily checked from the definition. Definition: A closed set Z ⊆ Cn is irreducible if there is no pair of proper closed subsets Z1 , Z2 ⊂ Z with the property that Z = Z1 ∪ Z2 . Proposition 3.2: Features of the Zariski topology include the following: (a) Every descendi ...
LINE BUNDLES OVER FLAG VARIETIES Contents 1. Introduction 1
... 2.1. Varieties. The idea of a variety is a geometric object that locally looks like the locus cut out by the vanishing of a collection of polynomials. A thorough treatment of varieties can be found in [1, Chapter 1]. We first define 2 classes of varieties, affine and projective varieties. Then we wi ...
... 2.1. Varieties. The idea of a variety is a geometric object that locally looks like the locus cut out by the vanishing of a collection of polynomials. A thorough treatment of varieties can be found in [1, Chapter 1]. We first define 2 classes of varieties, affine and projective varieties. Then we wi ...
The structure of Coh(P1) 1 Coherent sheaves
... Proposition 2.4. If X is additionally a noetherian separated scheme, then H i (C p (U, F)) = H i (X, F). (X must be separated so that the intersections UI are all affine.) Roughly, the idea is that this Cech resolution is taking the place of the injective resolution as the fibrant replacement for F. ...
... Proposition 2.4. If X is additionally a noetherian separated scheme, then H i (C p (U, F)) = H i (X, F). (X must be separated so that the intersections UI are all affine.) Roughly, the idea is that this Cech resolution is taking the place of the injective resolution as the fibrant replacement for F. ...
Néron Models of Elliptic Curves.
... roots and R[x] is factorial, this implies that 2g1 and 2g2 are in R[x]. Since 2 is invertible, we have g1 , g2 ∈ R[x], hence g = g1 + g2 y ∈ A. ...
... roots and R[x] is factorial, this implies that 2g1 and 2g2 are in R[x]. Since 2 is invertible, we have g1 , g2 ∈ R[x], hence g = g1 + g2 y ∈ A. ...
Solutions Sheet 7
... g : Y ,→ X. Also f [ factors as R f [ (R) ,→ OZ (Z) and thus f factors as g Z → Y → X. Consider any other closed subscheme Y 0 of X through which f factors as Z → Y 0 → X. Since Y 0 is a closed subscheme of an affine scheme, it is affine and in fact Y 0 = Spec(R/I) for some ideal I ⊂ R. Thus f [ f ...
... g : Y ,→ X. Also f [ factors as R f [ (R) ,→ OZ (Z) and thus f factors as g Z → Y → X. Consider any other closed subscheme Y 0 of X through which f factors as Z → Y 0 → X. Since Y 0 is a closed subscheme of an affine scheme, it is affine and in fact Y 0 = Spec(R/I) for some ideal I ⊂ R. Thus f [ f ...
Solutions Sheet 8
... is non-zero; hence so is its subring Sg,0 , which therefore possesses a prime ideal, which in turn corresponds to a point in the standard open subset Dg ∼ = Spec Sg,0 of Proj S; hence Proj S 6= ∅. (b) Suppose there is an integer k such that Sn = 0 for all n > k. Then for any f ∈ S+ , each homogeneou ...
... is non-zero; hence so is its subring Sg,0 , which therefore possesses a prime ideal, which in turn corresponds to a point in the standard open subset Dg ∼ = Spec Sg,0 of Proj S; hence Proj S 6= ∅. (b) Suppose there is an integer k such that Sn = 0 for all n > k. Then for any f ∈ S+ , each homogeneou ...
Intersection Theory course notes
... Motivation. Intersection theory had been developed mainly in order to give a rigorous foundation for methods of enumerative geometry. Here is a typical question considered in enumerative geometry. How many lines in 3-space intersect 4 given lines in general position? Here is Schubert’s solution. Ch ...
... Motivation. Intersection theory had been developed mainly in order to give a rigorous foundation for methods of enumerative geometry. Here is a typical question considered in enumerative geometry. How many lines in 3-space intersect 4 given lines in general position? Here is Schubert’s solution. Ch ...
LECTURES ON ALGEBRAIC VARIETIES OVER F1 Contents
... Let χ(ν, x) be the character of C(ν). From the lemma and Frobenius’ theorem we deduce that 1 X χ(ν, x)χ(µ, x) = δµ,ν , #G x∈G and, to get 2.1, it remains to check that χ(ν, 1) > 0. 2.2. The field F1 . In [10] Tits noticed that the analogy above extends to an analogy between the group G(Fq ) of point ...
... Let χ(ν, x) be the character of C(ν). From the lemma and Frobenius’ theorem we deduce that 1 X χ(ν, x)χ(µ, x) = δµ,ν , #G x∈G and, to get 2.1, it remains to check that χ(ν, 1) > 0. 2.2. The field F1 . In [10] Tits noticed that the analogy above extends to an analogy between the group G(Fq ) of point ...
Notes 1
... (1) Let f ∈ k[x1 , . . . , xn ] be a non-constant polynomial. Then V (f ) ⊂ An is called the hypersurface defined by f . If the degree of f is one, then the corresponding hypersurface is called a hyperplane. If the degree of f is two, then the corresponding hypersurface is called a quadric hypersurf ...
... (1) Let f ∈ k[x1 , . . . , xn ] be a non-constant polynomial. Then V (f ) ⊂ An is called the hypersurface defined by f . If the degree of f is one, then the corresponding hypersurface is called a hyperplane. If the degree of f is two, then the corresponding hypersurface is called a quadric hypersurf ...
A Lefschetz hyperplane theorem with an assigned base point
... V is a closed subset of Y . X is the quasi-projective variety Y n V . W is a Whitney stratification of Y such that V is a union of strata. A is a codimension 2 linear subspace of a fixed ambient projective space of Y . W jY nA is the Whitney stratification of Y n A obtained by restricting W . PA is ...
... V is a closed subset of Y . X is the quasi-projective variety Y n V . W is a Whitney stratification of Y such that V is a union of strata. A is a codimension 2 linear subspace of a fixed ambient projective space of Y . W jY nA is the Whitney stratification of Y n A obtained by restricting W . PA is ...
EXERCISES IN MA 510 : COMMUTATIVE ALGEBRA
... elements. Let m be a maximal ideal of R. Show that there are atmost n maximal ideals in S containing mS. (28) Let k be an algebraically closed field and R = k[x1 , x2 , . . . , xn ]. Let I be an ideal of R. Suppose that V (I) = {P1 , P2 , . . . , Pr }. Consider the map φ : R → k r defined by φ(f ) = ...
... elements. Let m be a maximal ideal of R. Show that there are atmost n maximal ideals in S containing mS. (28) Let k be an algebraically closed field and R = k[x1 , x2 , . . . , xn ]. Let I be an ideal of R. Suppose that V (I) = {P1 , P2 , . . . , Pr }. Consider the map φ : R → k r defined by φ(f ) = ...
Subgroup Complexes
... by a geometry, there is usually a simplicial complex involved, it is associated to a prime p, and the stabilizers of simplices are treated as analogues of parabolic subgroups. One can take the view that the most canonically defined nontrivial simplicial complex on which G acts, associated to the pri ...
... by a geometry, there is usually a simplicial complex involved, it is associated to a prime p, and the stabilizers of simplices are treated as analogues of parabolic subgroups. One can take the view that the most canonically defined nontrivial simplicial complex on which G acts, associated to the pri ...
Lecture 11
... Note that if (X, OX ) is a ringed space then there are potentially two different ways to take the right derived functors of Γ(X, F), if F is an OX -module. We could forget that X is a ringed space or we could work in the smaller category of OX -modules. We check that it does not matter in which cat ...
... Note that if (X, OX ) is a ringed space then there are potentially two different ways to take the right derived functors of Γ(X, F), if F is an OX -module. We could forget that X is a ringed space or we could work in the smaller category of OX -modules. We check that it does not matter in which cat ...
Problem set 3 - Math Berkeley
... we constructed the covering f : X → Y purely algebraically, there is no locally constant sheaf in the Zariski topology on Y that detects its monodromy. It is possible to define a purely algebraic notion of “sheaf in the étale topology,” which for complex algebraic varieties more closely resembles t ...
... we constructed the covering f : X → Y purely algebraically, there is no locally constant sheaf in the Zariski topology on Y that detects its monodromy. It is possible to define a purely algebraic notion of “sheaf in the étale topology,” which for complex algebraic varieties more closely resembles t ...