THREE APPROACHES TO CHOW`S THEOREM 1. Statement and
... with the standard simplex in some complex affine space. The dimension of an analytic simplex is the dimension of the standard simplex in question. One should think of an analytic simplex as an element of a triangulation of an analytic set; it is an easy theorem that every analytic set in projective ...
... with the standard simplex in some complex affine space. The dimension of an analytic simplex is the dimension of the standard simplex in question. One should think of an analytic simplex as an element of a triangulation of an analytic set; it is an easy theorem that every analytic set in projective ...
7.1. Sheaves and sheafification. The first thing we have to do to
... on every affine open subset U = Spec R of X. Almost all sheaves that we will consider are of this form. This reduces local computations regarding these sheaves to computations in commutative algebra. A quasi-coherent sheaf on X is called locally free of rank r if it is locally isomorphic to OX⊕r . L ...
... on every affine open subset U = Spec R of X. Almost all sheaves that we will consider are of this form. This reduces local computations regarding these sheaves to computations in commutative algebra. A quasi-coherent sheaf on X is called locally free of rank r if it is locally isomorphic to OX⊕r . L ...
MATH 8253 ALGEBRAIC GEOMETRY HOMEWORK 1 1.2.10. Let A
... subschemes form a base for the topology of X. Proof. Let x ∈ X and U be a neighborhood of x in X. By assumption, x has a Noetherian open neighborhood V in X. So by Proposition 3.46 part (a), U ∩ V is a Noetherian neighborhood of x which is contained in U . This verifies the condition for being a bas ...
... subschemes form a base for the topology of X. Proof. Let x ∈ X and U be a neighborhood of x in X. By assumption, x has a Noetherian open neighborhood V in X. So by Proposition 3.46 part (a), U ∩ V is a Noetherian neighborhood of x which is contained in U . This verifies the condition for being a bas ...
rings without a gorenstein analogue of the govorov–lazard theorem
... Abstract It was proved by Beligiannis and Krause that over certain Artin algebras, there are Gorenstein flat modules which are not direct limits of finitely generated Gorenstein projective modules. That is, these algebras have no Gorenstein analogue of the Govorov–Lazard theorem. We show that, in fa ...
... Abstract It was proved by Beligiannis and Krause that over certain Artin algebras, there are Gorenstein flat modules which are not direct limits of finitely generated Gorenstein projective modules. That is, these algebras have no Gorenstein analogue of the Govorov–Lazard theorem. We show that, in fa ...
The Group Structure of Elliptic Curves Defined over Finite Fields
... The coordinate ring k[V ] of polynomial functions f : V − → k is so called since every polynomial function on V is a k-linear combination of products of the coordinate functions xi : V − → k whose value at a point P ∈ V is the ith coordinate of P . To see how k[V ] relates to the dimension of V , le ...
... The coordinate ring k[V ] of polynomial functions f : V − → k is so called since every polynomial function on V is a k-linear combination of products of the coordinate functions xi : V − → k whose value at a point P ∈ V is the ith coordinate of P . To see how k[V ] relates to the dimension of V , le ...
Curves of given p-rank with trivial automorphism group
... p-rank exactly f [7, Thm. 2.3]. By Theorem 2.3, Aut(Cη ) = 1. The sheaf Aut(C) is constructible on Γ0 , but there are only finitely many possibilities for the automorphism group of a curve of genus g. Therefore, there is a nonempty open subspace U ⊂ Γ0 such that, for each s ∈ U (k), Cs has p-rank f ...
... p-rank exactly f [7, Thm. 2.3]. By Theorem 2.3, Aut(Cη ) = 1. The sheaf Aut(C) is constructible on Γ0 , but there are only finitely many possibilities for the automorphism group of a curve of genus g. Therefore, there is a nonempty open subspace U ⊂ Γ0 such that, for each s ∈ U (k), Cs has p-rank f ...
REAL ALGEBRAIC GEOMETRY. A FEW BASICS. DRAFT FOR A
... (i) the ideal I(X) has real generators f1 , . . . , fk ∈ R[x0 , . . . , xn ]. (ii) X = X. Definition 1.2. The set of real points VR (f1 , . . . , fk ) where fi ∈ R[x0 , . . . , xn ] is called a real affine algebraic variety. In equivalent way, after Prop. 1.1, the conjugation operator acts on a comp ...
... (i) the ideal I(X) has real generators f1 , . . . , fk ∈ R[x0 , . . . , xn ]. (ii) X = X. Definition 1.2. The set of real points VR (f1 , . . . , fk ) where fi ∈ R[x0 , . . . , xn ] is called a real affine algebraic variety. In equivalent way, after Prop. 1.1, the conjugation operator acts on a comp ...
VARIATIONS ON A QUESTION OF LARSEN AND LUNTS 1
... We denote by Z[sb] the free abelian group generated by the stable birational equivalence classes of connected smooth projective k-varieties. Theorem 2.2 (Larsen-Lunts, [5]; see also [1]). Let k be a field of characteristic zero. There exists a unique group morphism SB : K0 (V ark ) → Z[sb], sending t ...
... We denote by Z[sb] the free abelian group generated by the stable birational equivalence classes of connected smooth projective k-varieties. Theorem 2.2 (Larsen-Lunts, [5]; see also [1]). Let k be a field of characteristic zero. There exists a unique group morphism SB : K0 (V ark ) → Z[sb], sending t ...
Degrees of curves in abelian varieties
... Let (X, A) be a principally polarized abelian variety of dimension n defined over an algebraically closed field k. The degree of a curve C contained in X is d = C • X. The first question we are interested in is to find what numbers can be degrees of irreducibles curves C. When C generates X, we prov ...
... Let (X, A) be a principally polarized abelian variety of dimension n defined over an algebraically closed field k. The degree of a curve C contained in X is d = C • X. The first question we are interested in is to find what numbers can be degrees of irreducibles curves C. When C generates X, we prov ...
the usual castelnuovo s argument and special subhomaloidal
... satisfying the hypothesis of the proposition are scheme theoretic intersection of the quadrics through them. Let X ⊂ Pr be a smooth non-degenerate variety of degree d = 3, then X is either a cubic hypersurface or s = codim(X ) = 2 (remember that for a non-degenerate variety d ≥ s + 1). In the last c ...
... satisfying the hypothesis of the proposition are scheme theoretic intersection of the quadrics through them. Let X ⊂ Pr be a smooth non-degenerate variety of degree d = 3, then X is either a cubic hypersurface or s = codim(X ) = 2 (remember that for a non-degenerate variety d ≥ s + 1). In the last c ...
Picard Groups of Affine Curves Victor I. Piercey University of Arizona Math 518
... on the one hand, while clearly uI = (1 · u)I ⊂ (Au)I. Therefore (Au)I = uI for any Au ∈ P C(A). Accordingly, it suffices to show that if I, J ∈ C(A) and I ∼ = J, then J = uI for some u ∈ K × . But this is immediate from the third statement in Theorem 2.8, since any isomorphism ψ : I → J is an elemen ...
... on the one hand, while clearly uI = (1 · u)I ⊂ (Au)I. Therefore (Au)I = uI for any Au ∈ P C(A). Accordingly, it suffices to show that if I, J ∈ C(A) and I ∼ = J, then J = uI for some u ∈ K × . But this is immediate from the third statement in Theorem 2.8, since any isomorphism ψ : I → J is an elemen ...
Bertini irreducibility theorems over finite fields
... Reduction to surfaces The most difficult case is that of (possibly singular) surfaces. Good news however: we have resolution of singularities in that case, so these cause less of a problem. Idea of the reduction: start with X ⊂ Pnk , smooth, irreducible, dim X = m ≥ 3. Find a single irreducible hyp ...
... Reduction to surfaces The most difficult case is that of (possibly singular) surfaces. Good news however: we have resolution of singularities in that case, so these cause less of a problem. Idea of the reduction: start with X ⊂ Pnk , smooth, irreducible, dim X = m ≥ 3. Find a single irreducible hyp ...
4. Morphisms
... (d) If f : A2 → A2 is an isomorphism then f is affine linear, i. e. it is of the form f (x) = Ax + b for some A ∈ Mat(2 × 2, K) and b ∈ K 2 . Construction 4.15 (Affine varieties from finitely generated K-algebras). Corollary 4.8 allows us to construct affine varieties up to isomorphisms from finitel ...
... (d) If f : A2 → A2 is an isomorphism then f is affine linear, i. e. it is of the form f (x) = Ax + b for some A ∈ Mat(2 × 2, K) and b ∈ K 2 . Construction 4.15 (Affine varieties from finitely generated K-algebras). Corollary 4.8 allows us to construct affine varieties up to isomorphisms from finitel ...
Variations on Belyi`s theorem - Universidad Autónoma de Madrid
... Gal(C/k) the group of all field automorphisms of C which fix the elements in k. For k = Q we simply write Gal(C/Q)= Gal(C). For given σ ∈ Gal(C) and a ∈ C, we shall write aσ instead of σ(a). We shall employ the same rule to denote the obvious action induced by σ on the projective space Pn (C), the rin ...
... Gal(C/k) the group of all field automorphisms of C which fix the elements in k. For k = Q we simply write Gal(C/Q)= Gal(C). For given σ ∈ Gal(C) and a ∈ C, we shall write aσ instead of σ(a). We shall employ the same rule to denote the obvious action induced by σ on the projective space Pn (C), the rin ...
WHICH ARE THE SIMPLEST ALGEBRAIC VARIETIES? Contents 1
... Already in the curve case we saw that it is convenient to throw in some points at infinity. This can be done for any variety X, though the resulting compactification is not unique in dimensions ≥ 2. For now this ambiguity does not matter; any of these will be denoted by X̄. Theorem 14. Let X = (f1 ( ...
... Already in the curve case we saw that it is convenient to throw in some points at infinity. This can be done for any variety X, though the resulting compactification is not unique in dimensions ≥ 2. For now this ambiguity does not matter; any of these will be denoted by X̄. Theorem 14. Let X = (f1 ( ...
Around cubic hypersurfaces
... R2 Q3 = S 2 P (P − Q)(P − λQ) so that S 2 | Q3 and Q3 | S 2 , hence S 2 = Q3 (after adjusting the scalars) and R2 = P (P − Q)(P − λQ). Since the factors in the right-hand side are mutually coprime, we obtain, by unique factorization in K[T ], that P , P − Q, and P − λQ are all squares, and so is Q s ...
... R2 Q3 = S 2 P (P − Q)(P − λQ) so that S 2 | Q3 and Q3 | S 2 , hence S 2 = Q3 (after adjusting the scalars) and R2 = P (P − Q)(P − λQ). Since the factors in the right-hand side are mutually coprime, we obtain, by unique factorization in K[T ], that P , P − Q, and P − λQ are all squares, and so is Q s ...
SERRE DUALITY FOR NONCOMMUTATIVE PROJECTIVE
... satisfies χ and cd( proj A) ≤ Kdim(AA ) − 1 < ∞ where Kdim is Krull (RentschlerGabriel) dimension. Familiar examples of such rings are noetherian PI rings and quantum matrix algebras. A bimodule is called noetherian if it is both left and right noetherian. Theorem 3.1. Let A be a graded left and rig ...
... satisfies χ and cd( proj A) ≤ Kdim(AA ) − 1 < ∞ where Kdim is Krull (RentschlerGabriel) dimension. Familiar examples of such rings are noetherian PI rings and quantum matrix algebras. A bimodule is called noetherian if it is both left and right noetherian. Theorem 3.1. Let A be a graded left and rig ...
Chapter 1 PLANE CURVES
... If p = (x0 , x1 , x2 ) is a point of P2 , and if x0 6= 0, we may normalize the first entry to 1 without changing the point: (x0 , x1 , x2 ) ∼ (1, u1 , u2 ), where ui = xi /x0 . We did this for P1 above. The representative vector (1, u1 , u2 ) is uniquely determined by p, so points with x0 6= 0 corre ...
... If p = (x0 , x1 , x2 ) is a point of P2 , and if x0 6= 0, we may normalize the first entry to 1 without changing the point: (x0 , x1 , x2 ) ∼ (1, u1 , u2 ), where ui = xi /x0 . We did this for P1 above. The representative vector (1, u1 , u2 ) is uniquely determined by p, so points with x0 6= 0 corre ...
as a PDF
... Af(a, i V r) fails. But this implies that M(s V r, a) fails, and by the argument used above, there are atoms s' and ?' under a such that M(s V i, j' V i') fails. Let / = í' V t' and ffl=iV(. Since M(l, m) fails, / A m — 0. For some atom/» < w, we must have /» < (/» V /) A m; hence (/» V /) A m = m, ...
... Af(a, i V r) fails. But this implies that M(s V r, a) fails, and by the argument used above, there are atoms s' and ?' under a such that M(s V i, j' V i') fails. Let / = í' V t' and ffl=iV(. Since M(l, m) fails, / A m — 0. For some atom/» < w, we must have /» < (/» V /) A m; hence (/» V /) A m = m, ...
ABELIAN VARIETIES A canonical reference for the subject is
... Lemma 1.11. Let X, Y be complete varieties and let Z be a geometrically connected k-scheme of finite type. If x, y, z are k-points of X, Y, Z such that the restrictions of L to {x} × Y × Z, X × {y} × Z, and X × Y × {z} are trivial, then L is trivial. Proof. Let f : X × Y × Z → Z be the structure map ...
... Lemma 1.11. Let X, Y be complete varieties and let Z be a geometrically connected k-scheme of finite type. If x, y, z are k-points of X, Y, Z such that the restrictions of L to {x} × Y × Z, X × {y} × Z, and X × Y × {z} are trivial, then L is trivial. Proof. Let f : X × Y × Z → Z be the structure map ...
PROJECTIVITY AND FLATNESS OVER THE
... k be a commutative ring, H a Hopf algebra over k, and Λ a left H-module algebra. Then we can consider the smash product Λ#H and the subring of invariants ΛH . Then we can give necessary and sufficient conditions for the projectivity and flatness over ΛH of a left Λ#H-module P . The results from [9] ...
... k be a commutative ring, H a Hopf algebra over k, and Λ a left H-module algebra. Then we can consider the smash product Λ#H and the subring of invariants ΛH . Then we can give necessary and sufficient conditions for the projectivity and flatness over ΛH of a left Λ#H-module P . The results from [9] ...
A Problem Course on Projective Planes
... Incidence structures and configurations. The geometrical notion that we will focus on, to the exclusion of notions like distance and angle, is that of incidence, i.e. the relation of points being on lines or lines passing through points. Definition 1.1. An incidence structure is a triple (P, L, I), ...
... Incidence structures and configurations. The geometrical notion that we will focus on, to the exclusion of notions like distance and angle, is that of incidence, i.e. the relation of points being on lines or lines passing through points. Definition 1.1. An incidence structure is a triple (P, L, I), ...
Closed sets and the Zariski topology
... The Hilbert Basis Theorem, which we will prove below, says that every ideal in k[x] is finitely generated. It will follow that every Zariski closed subset of An has the form V (F) where F is finite. The Zariski topology is a coarse topology in the sense that it does not have many open sets. In fact ...
... The Hilbert Basis Theorem, which we will prove below, says that every ideal in k[x] is finitely generated. It will follow that every Zariski closed subset of An has the form V (F) where F is finite. The Zariski topology is a coarse topology in the sense that it does not have many open sets. In fact ...
COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY Classical
... Xan is Hausdorff, then X is a variety. However, this requires putting together some of the deeper results which we have developed. We can generalize Corollary 1.5 as follows: Corollary 2.7. Let X be a complex curve, or more generally an irreducible one-dimensional scheme of finite type over C. Then ...
... Xan is Hausdorff, then X is a variety. However, this requires putting together some of the deeper results which we have developed. We can generalize Corollary 1.5 as follows: Corollary 2.7. Let X be a complex curve, or more generally an irreducible one-dimensional scheme of finite type over C. Then ...