SIMPLEST SINGULARITY IN NON-ALGEBRAIC

... A classical question in complex analytic geometry is to understand when a given analytic space is algebraic (i.e. analytification of an algebraic scheme). A necessary condition for this to hold is that the transcendence degree of the field of global meromorphic functions must be equal to the dimensi ...

... A classical question in complex analytic geometry is to understand when a given analytic space is algebraic (i.e. analytification of an algebraic scheme). A necessary condition for this to hold is that the transcendence degree of the field of global meromorphic functions must be equal to the dimensi ...

University of Leeds.

... single stable light source, taking into account features such as shade curves, cast shadows, surfaces markings, creases, corners and apparent contours. I shall describe a couple of examples showing how the necessary classifications are carried out. In many cases topological equivalence is needed to ...

... single stable light source, taking into account features such as shade curves, cast shadows, surfaces markings, creases, corners and apparent contours. I shall describe a couple of examples showing how the necessary classifications are carried out. In many cases topological equivalence is needed to ...

LOCAL CLASS GROUPS All rings considered here are commutative

... We have used this to compute Picard groups of singular surfaces. The hard part is that the Local Class Groups Cl OX,p are usually very difficult to compute. On the other hand, b for any normal local ring (R, m) there is an injective group homomorphism Cl R ,→ Cl R, so we may work in an appropriate p ...

... We have used this to compute Picard groups of singular surfaces. The hard part is that the Local Class Groups Cl OX,p are usually very difficult to compute. On the other hand, b for any normal local ring (R, m) there is an injective group homomorphism Cl R ,→ Cl R, so we may work in an appropriate p ...

INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 18 Contents

... B ⊂ R as well. Let n = mR ∩ B. Then n is a maximal ideal of B. (Hence it corresponds to a point of Y mapping to zero.) Also, Bn ⊂ R. And also nBn ⊂ mR . Now Bn is also a DVR of K/k, so Bn = R by the Commutative Algebra Lemma. Hence if y is in mR , then y is in n. To say that y is in n, means that y, ...

... B ⊂ R as well. Let n = mR ∩ B. Then n is a maximal ideal of B. (Hence it corresponds to a point of Y mapping to zero.) Also, Bn ⊂ R. And also nBn ⊂ mR . Now Bn is also a DVR of K/k, so Bn = R by the Commutative Algebra Lemma. Hence if y is in mR , then y is in n. To say that y is in n, means that y, ...

Lecture Notes - Mathematics

... called the analytic index of singularities or the log canonical threshold of f , and is an important invariant in algebraic geometry and complex analysis of several variables. Resolution of singularities is a huge topic with a classical history that has been re-interpreted many times over into the p ...

... called the analytic index of singularities or the log canonical threshold of f , and is an important invariant in algebraic geometry and complex analysis of several variables. Resolution of singularities is a huge topic with a classical history that has been re-interpreted many times over into the p ...

Some results on the syzygies of finite sets and algebraic

... Conjecture. If X ~ Pr fails to satisfy (Np ), then there is an integer s r, and a subset Y 9 X consisting of at least 2s + 2 - p points, such that Y is contained in a linear subspace ps ç Pr in which (Np ) fails for Y. instance, the conjecture predicts that the homogeneous ideal of six in points P3 ...

... Conjecture. If X ~ Pr fails to satisfy (Np ), then there is an integer s r, and a subset Y 9 X consisting of at least 2s + 2 - p points, such that Y is contained in a linear subspace ps ç Pr in which (Np ) fails for Y. instance, the conjecture predicts that the homogeneous ideal of six in points P3 ...

Homework sheet 1

... Please complete all the questions. For each question (even question 2, if you can see how) please provide examples/graphs/pictures illustrating the ideas behind the question and your answer. 1. Suppose that the field k is algebraically closed. Prove that an affine conic (i.e. a degree 2 curve in the ...

... Please complete all the questions. For each question (even question 2, if you can see how) please provide examples/graphs/pictures illustrating the ideas behind the question and your answer. 1. Suppose that the field k is algebraically closed. Prove that an affine conic (i.e. a degree 2 curve in the ...

ALGEBRAIC GEOMETRY - University of Chicago Math

... Please complete all the questions. For each question (even question 2, if you can see how) please provide examples/graphs/pictures illustrating the ideas behind the question and your answer. 1. Suppose that the field k is algebraically closed. Prove that an affine conic (i.e. a degree 2 curve in the ...

... Please complete all the questions. For each question (even question 2, if you can see how) please provide examples/graphs/pictures illustrating the ideas behind the question and your answer. 1. Suppose that the field k is algebraically closed. Prove that an affine conic (i.e. a degree 2 curve in the ...

Universal spaces in birational geometry

... Universal spaces in birational geometry — Fedor Bogomolov, October 8, 2010 I want to discuss our joint results with Yuri Tschinkel. The Bloch-Kato conjecture implies that cohomology elements with finite constant coefficients of an algebraic variety can be induced from abelian quotient of the fundame ...

... Universal spaces in birational geometry — Fedor Bogomolov, October 8, 2010 I want to discuss our joint results with Yuri Tschinkel. The Bloch-Kato conjecture implies that cohomology elements with finite constant coefficients of an algebraic variety can be induced from abelian quotient of the fundame ...

Natural Homogeneous Coordinates

... Natural Homogeneous Coordinates In projective geometry parallel lines intersect at a point. • The point at infinity is called an ideal point. • There is an ideal point for every slope. • The collection of ideal points is called an ideal line. • We might think of the line as a circle. ...

... Natural Homogeneous Coordinates In projective geometry parallel lines intersect at a point. • The point at infinity is called an ideal point. • There is an ideal point for every slope. • The collection of ideal points is called an ideal line. • We might think of the line as a circle. ...

Picard groups and class groups of algebraic varieties

... (a) Taking X = C, the prime divisors are simply points a ∈ C. Such a divisor is Cartier because it is defined (globally) by the vanishing of g(x) = x − a. (b) Let X = Z(y 2 − (x3 + x2 )) ⊂ C2 . The variety X is not smooth, it has a node at the origin. In this case the prime divisors are still points ...

... (a) Taking X = C, the prime divisors are simply points a ∈ C. Such a divisor is Cartier because it is defined (globally) by the vanishing of g(x) = x − a. (b) Let X = Z(y 2 − (x3 + x2 )) ⊂ C2 . The variety X is not smooth, it has a node at the origin. In this case the prime divisors are still points ...

Spatial resolution of the HRRT PET scanner using 3D

... reconstruction. The improvement is 43% ±6% at all distances when comparing the radial and tangential FWHM for the two reconstructions (top and bottom point series in Fig. 2). The FWHM of both the radial and tangential axis is 1.44 mm ± 0.09mm (n = 46) for r < 60mm using the PSF reconstruction. There ...

... reconstruction. The improvement is 43% ±6% at all distances when comparing the radial and tangential FWHM for the two reconstructions (top and bottom point series in Fig. 2). The FWHM of both the radial and tangential axis is 1.44 mm ± 0.09mm (n = 46) for r < 60mm using the PSF reconstruction. There ...

2 - arXiv

... mi [Ui ] is an algebraic cycle • g is proper • Every component of U maps onto an irreducible component of Z, and every fibre is either n-dimensional or empty • A final technical condition must be satisfied: we omit description of it as it is automatically satisfied when the base is normal. There are ...

... mi [Ui ] is an algebraic cycle • g is proper • Every component of U maps onto an irreducible component of Z, and every fibre is either n-dimensional or empty • A final technical condition must be satisfied: we omit description of it as it is automatically satisfied when the base is normal. There are ...

Tuesday, Lecture 2, Ronald van Luijk, Please let

... points is indeed a del Pezzo surface (see [4], Thm. 24.5). For r = 6 we get the famous cubic surfaces in P3 , see [3], section V.4. For r = 7 or r = 8, life gets significantly more complicated because then the anticanonical sheaf is no longer very ample, just ample (c.f. [4], Rem. 26.3). For a proof ...

... points is indeed a del Pezzo surface (see [4], Thm. 24.5). For r = 6 we get the famous cubic surfaces in P3 , see [3], section V.4. For r = 7 or r = 8, life gets significantly more complicated because then the anticanonical sheaf is no longer very ample, just ample (c.f. [4], Rem. 26.3). For a proof ...

1.5.4 Every abelian variety is a quotient of a Jacobian

... there are only finitely many primes of bad reduction). Thus we may assume that R is a discrete valuation ring. The next step is to pass to the “strict henselization” R0 of R. A local ring R with maximal ideal ℘ is henselian if “every simple root lifts uniquely”; more precisely, if whenever f (x) ∈ R ...

... there are only finitely many primes of bad reduction). Thus we may assume that R is a discrete valuation ring. The next step is to pass to the “strict henselization” R0 of R. A local ring R with maximal ideal ℘ is henselian if “every simple root lifts uniquely”; more precisely, if whenever f (x) ∈ R ...

Spencer Bloch: The proof of the Mordell Conjecture

... on several c o u n t s . For o n e t h i n g , the s o l u t i o n set m i s s e s " p o i n t s at i n f i n i t y " . To avoid h a v i n g s o m e Probably most mathematicians w o u l d have agreed fiend stash all the goodies out at infinity where we with Weil (certainly I would have), until earli ...

... on several c o u n t s . For o n e t h i n g , the s o l u t i o n set m i s s e s " p o i n t s at i n f i n i t y " . To avoid h a v i n g s o m e Probably most mathematicians w o u l d have agreed fiend stash all the goodies out at infinity where we with Weil (certainly I would have), until earli ...

Resolution - ALCom Lab

... • Induction on number of quantifiers: – For single ∃-variable it is just a usual resolution – For single ∀-variable, falsity of formula->there is at least one non-tautological clause, which can be universally reduced – Induction step for ∀-variable (a) will choose the value of a, which leads to UNSA ...

... • Induction on number of quantifiers: – For single ∃-variable it is just a usual resolution – For single ∀-variable, falsity of formula->there is at least one non-tautological clause, which can be universally reduced – Induction step for ∀-variable (a) will choose the value of a, which leads to UNSA ...

The Picard group

... • On X(C), we have all the tools available to study complex analytic varieties. For example, X(C) has cohomology groups which give much information about its geometry, and these come with Hodge decompositions. • It may not be obvious that much can be said about X(Q̄). However, a general idea known a ...

... • On X(C), we have all the tools available to study complex analytic varieties. For example, X(C) has cohomology groups which give much information about its geometry, and these come with Hodge decompositions. • It may not be obvious that much can be said about X(Q̄). However, a general idea known a ...

Homework sheet 6

... (DUE FRIDAY FEBRUARY 28) Please complete all the questions. 1. Let X → Y be a morphism of affine varieties over a field k. Show that the induced morphism k[Y ] → k[X] on rings of regular functions is injective if and only if the original morphism X → Y has dense image. 2. Prove that the Segre embedd ...

... (DUE FRIDAY FEBRUARY 28) Please complete all the questions. 1. Let X → Y be a morphism of affine varieties over a field k. Show that the induced morphism k[Y ] → k[X] on rings of regular functions is injective if and only if the original morphism X → Y has dense image. 2. Prove that the Segre embedd ...

Singularity surfaces

... http://www.irccyn.ecnantes.fr/~chablat/3RPR.html. The remaining singularity surfaces are defined by the quadratic equation in x used to calculate the position (x, y) once t is found by the characteristic polynomial, and its derivative with respect to x. Eliminating x from the quadratic equation and ...

... http://www.irccyn.ecnantes.fr/~chablat/3RPR.html. The remaining singularity surfaces are defined by the quadratic equation in x used to calculate the position (x, y) once t is found by the characteristic polynomial, and its derivative with respect to x. Eliminating x from the quadratic equation and ...

Cubics points on cubic curves and the Brauer

... to the Hasse principle on all curves of genus at least 2 and all smooth, proper, geometrically integral, rationally connected varieties over number fields (see [18] and [5], p. 3). For K3 surfaces, however, it is not at all clear whether the Brauer-Manin obstruction is the only one. Even if in gener ...

... to the Hasse principle on all curves of genus at least 2 and all smooth, proper, geometrically integral, rationally connected varieties over number fields (see [18] and [5], p. 3). For K3 surfaces, however, it is not at all clear whether the Brauer-Manin obstruction is the only one. Even if in gener ...

ON THE IRREDUCIBILITY OF SECANT CONES, AND

... W is normal, then so is a general fibre of h: this can be seen, e.g. using Serre’s criterion, or alternatively, use [G], Thm 12.2.4, which says, in the scheme-theoretic context, that the set N (h) of points t ∈ T such that h−1 (t) is normal is open; when W is normal, N (h) contains the generic point ...

... W is normal, then so is a general fibre of h: this can be seen, e.g. using Serre’s criterion, or alternatively, use [G], Thm 12.2.4, which says, in the scheme-theoretic context, that the set N (h) of points t ∈ T such that h−1 (t) is normal is open; when W is normal, N (h) contains the generic point ...

THE MOVING CURVE IDEAL AND THE REES

... In other words, the syzygy module Syz(a, b, c) is free with generators p, q of degree µ and n − µ. We assume µ ≤ n − µ, and we can regard p, q as moving lines p = Ax + By + Cz q = Ex + F y + Gz. On easily sees that p, q are the degree 1 generators of the moving curve ideal MC. Question What can we s ...

... In other words, the syzygy module Syz(a, b, c) is free with generators p, q of degree µ and n − µ. We assume µ ≤ n − µ, and we can regard p, q as moving lines p = Ax + By + Cz q = Ex + F y + Gz. On easily sees that p, q are the degree 1 generators of the moving curve ideal MC. Question What can we s ...

the isoperimetric problem on some singular surfaces

... best. For double discs and cylinders, we use spherical Schwarz symmetrization to limit complexity and then show that the minimizers have one component by creating illegal singularities in multi-component competitors. For general dimension double discs, we use Schwarz spherical symmetrization combine ...

... best. For double discs and cylinders, we use spherical Schwarz symmetrization to limit complexity and then show that the minimizers have one component by creating illegal singularities in multi-component competitors. For general dimension double discs, we use Schwarz spherical symmetrization combine ...