BEZOUT S THEOREM CHRISTIAN KLEVDAL


 Rose Thornton
 3 years ago
 Views:
Transcription
1 BEZOUT S THEOREM CHRISTIAN KLEVDAL A weaker version of Bézout s theorem states that if C, D are projective plane curves of degrees c and d that intersect transversally, then C D = cd. The goal of this talk is to give a deformation theoretic proof of Bézout s theorem. Many of the tools used in this proof are overkill for proving something as simple as Bézout s theorem (especially since we consider only transverse intersection) but the important part is the techniques used, not so much the result. For example, a modified version of this argument can be used to show there are 27 lines on a cubic. Throughout the notes, k will be a fixed algebraically closed field. 1. Parameter Space of Curves Projective plane curves are defined by homogenous polynomials in k[x, y, z], which are in turn determined by their coefficients. To consider the parameter space of polynomials of a given degree, we should think of the coefficients as being coordinates. Let k[x, y, z] d denote the homogenous polynomials of degree d. As a k vector space it has dimension ( ) d+2 2. Note that if λ C is non zero and f k[x, y, z] d then λf and f determine the same planar curve, so the parameter space of curves of degree d should be (C (d+2 2 ) {0})/C = P (d+2 2 ) 1. Letting N = ( ) ( d e+2 ) 2 2, points of P N are pairs (f, g) where f is a polynomial of degree d and g a polynomial of degree e. Consider the projection map: π : P N P 2 P N Points of P N P 2 are triples (f, g, p) with f, g as above and p a point in the projective plane. The condition that f(p) g(p) = 0 (i.e. that p is a point on V (f) V (g)) is a polynomial condition, hence X = {(f, g, p) p V (f) V (g)} P N P 2 is a closed subvariety, and π restricts to the projection map X P N. The fiber X (f,g) over a point (f, g) P N is then just the intersection V (f) V (g) so in this context Bézout s theorem states that if V (f) and V (g) intersect transversally then X (f,g) = de. The rough idea of the proof is to show that the subset U P N of points with transverse intersection is open and that π 1 U U is a covering space. Then we show that the cardinality of the fiber is constant over points in U and show for one point (f, g) U that X (f,g) = de. Of course π 1 U U is not a covering space in the topological sense because the Zariski topology is too coarse so we first need to come up with the algebraic analog of a covering space. 2. Algebraic Covers If p: Y X is a covering space (is locally trivial), then for any x, x X there is a bijection p 1 (x) p 1 (x ). This can be constructed by taking a path γ with γ(0) = x 1
2 2 CHRISTIAN KLEVDAL and γ(1) = x, and a point y p 1 (x) is sent to γ(1) where γ is a lift of γ to Y starting at x. In a nutshell, Existence of path lifts defines the map. Local homeomorphism gives that it is injective (different lifts end at different points). Uniqueness of path lifts allows for the inverse to be defined. So we see that the two important properties that covering spaces satisfy for this are: (1) They are local homeomorphisms. (2) They have unique lifting of paths. The locally trivial definition does not work well for varieties, but the above definitions can be transferred to varieties. We will see that local homeomorphisms correspond to étale maps and unique path lifting corresponds to valuatively proper maps. Using these definitions, we can modify the argument that shows covering spaces have constant fiber size to show that valuatively proper étale maps have constant fiber size Étale maps. First, we start out with an example to show the Zariski topology is too coarse to detect maps which should be considered local homeomorphisms. Example 2.1. The map C C induced by the algebra map C[x, x 1 ] C[x, x 1 ] sending x x n is not a local homeomorphism in the Zariski topology, but it is a topological covering space in the analytic topology. A more useful idea comes from differential geometry. It is a consequence of the inverse function theorem that a smooth map f : M N is a local diffeomorphism at p M if and only if the map on tangent spaces D p f : T p M T f(p) N is an isomorphism. Definition 2.1. A map of schemes Z Z is an infinitesimal thickening if it is a closed embedding and the sheaf of ideals I Z/Z is nilpotent. Example 2.2. The important examples of an infinitesimal thickening are Spec k[t]/(t n ) Spec k[t]/(t n+1 ) induced by the quotient k[t]/(t n+1 ) k[t]/(t n ). We write k[ε] = k[t]/(t 2 ) which is called the ring of dual numbers. For n = 1 we have k[t]/(t) = k and Spec k is a point while Spec k[ε] is a point with a tangent vector. This can be seen by looking at maps out of Spec k and Spec k[ε]. If X is a variety over k, then a map Spec k X is the same as picking a point and a map Spec k[ε] X is picking a point and a tangent vector. Note that the map on the underlying topological spaces is a homeomorphism, which holds in general for infinitesimal thickenings. Definition 2.2. A map of schemes Y X is formally étale if any commutative diagram of solid lines where Z Z is an infinitesimal thickening, can be completed with a unique dashed line: Z Y Z X We call such a map a lift of Z X.
3 BEZOUT S THEOREM 3 Example 2.3. The map in example 2.1 is formally étale. diagram of the form C G C[x, x 1 ] H We will check that for any F C[ε] C[x, x 1 ] that H exists and is unique. Note G sends x to λ n for some λ C, so by commutativity F (x) = λ n + aε H(x) = λ + bε for some a, b C. Commutativity of the diagram also forces b = (a/n) 1 n. This is a well defined map and is unique. Note that this shows that k k given by x x n is étale as long as the characteristic of k does not divide n. This definition categorizes étale maps as those for which infinitesimal motion lifts uniquely, which is similar to the criterion for local diffeomorphisms in differential geometry. Take Z Z to be the map Spec k Spec k[ε] from example 2.2. If f : Y X is étale we consider the map on Zariski tangent spaces D y f : T y Y T f(y) X. Interpreting maps from Spec k[ε] as picking out a point and a tangent vector, the existence of a lift shows D y f is surjective, while uniqueness ensures that D y is injective Proper maps. The algebraic analog of unique path lifting is the valuative criterion of properness. Definition 2.3. A map of schemes Y X is said to be valuatively proper if for any discrete valuation ring R with field of fractions K and any commutative diagram of solid lines, there exists a unique dashed line completing the diagram: Spec K Y Spec R X Again, such a map is called a lift of Spec R X. Example 2.4. The map P n k Spec k is proper, closed embeddings are proper. Proposition 2.1. Proper maps are stable under pullback, meaning if Y X is proper and Z X any map then Y X Z Z is proper. Proof. Consider the diagram Spec K Z X Y Y Spec R Z X
4 4 CHRISTIAN KLEVDAL The dashed arrow exists and is unique since Y X is proper, and the dotted arrow exists and is unique by the universal property of the fiber product. The dotted arrow is the required lift. Considering the maps P n Spec k and P m Spec k we get the following corollary: Corollary 2.1. Projection maps π : P n P m P n are proper. 3. Transverse Intersections Recall the earlier set up, where X is the space whose point are triples (f, g, p) where f, g are of degree d, e respectively and p V (f) V (g), and π : X P N is the projection map. The following proposition relates the intersection theory of V (f) and V (g) with properties of the map π. Proposition 3.1. Curves V (f) and V (g) intersect transversally if and only if the map π : X P N is formally étale at (f, g). Proof. The intersection V (f) V (g) = X (f,g) is transverse it is dimension 0 and reduced no infinitesimal motion is possible in the fiber π is étale at (f, g). Proposition 3.2. The set U P N over which π is étale is nonempty and open. Consequently π 1 U U is proper étale morphism. Proof. The fact that π 1 U U is proper follows from the stability of proper maps under base change. To show U is open, we need to use a different criterion for étale maps, in particular that (nice enought) maps of schemes are étale if they are flat and the sheaf of relative differentials vanish. Generic flatness says that the locus V P N over which π 1 V is flat is open and dense. The support Z X of Ω X/P N is closed so π(z) is closed since π is proper. Thus U = V (P N \ Z) which is open. 4. Proof of Bézout s Theorem We write V = π 1 U for U as above. Then π : V U is both proper and étale. Suppose a 1,..., a d k and b 1,..., b e k are sets of distinct points and let f = (x a 1 ) (x a d ), g = (y b 1 ) (y b e ). The pair (f, g) represent a point in U and the fiber X (f,g) = Spec k[x, y]/(f, g). Note MaxSpec k[x, y]/(f, g) = {(x a i, y b j ) 1 i d, 1 j e} so X (f,g) = de. The number of intersection points X (f,g) is the number of krational points (maps Spec k X (f,g), denoted by X (f,g) (k)), since these correspond to intersection points. The only part left is to show that for any points q 1 = (f 1, g 1 ), q 2 = (f 2, g 2 ) U that X q1 (k) = X q2 (k).
5 BEZOUT S THEOREM 5 First we find a hyperplane H P N so that q 1, q 2 H. Then P N \ H A N. Consider the closed linear embedding γ : A 1 A N t (1 t)q 1 + tq 2. Our strategy will be to show that X γ(0) = X L = X γ(1) where L is the generic point of imγ. The inverse image γ 1 U will be the complement of V (h) Spec k[t] for some h k[t]. Since γ(0), γ(1) U it follows that h(0) 0 h(1), and from this we get that the image of h under the quotient maps k[t] k[t]/(t n ) is a unit. Hence there is a well defined morphism k[t, h 1 ] lim k[t]/(t n ) = k[[t]]. Thus we have defined a map Spec k[[t]] γ 1 U A 1 U and we know that Spec k[[t]] = {(0), (t)} sends the generic point (0) to L. Thus we have defined a map Start with a k rational point Spec k X γ(0). From this, we get the following diagram π 1 U Spec k Spec k[t]/(t n ) Spec k[[t]] U The dashed lines can be filled in uniquely since π 1 U U is étale and Spec k Spec k[t]/(t n ) is an infinitesimal thickening. The dotted line comes from the universal property of inverse limits. Restricting the dotted line to the generic point, we obtain a morphism Spec k(t) π 1 (U) which is a point in X L, i.e. we have constructed a map Ψ: X γ(0) (k) X L (k(t)). It is injective because the map Spec k[[t]] π 1 U is the unique extension of the map Spec k π 1 U that we started with. We will show that Ψ is surjective. Starting with a point of X L, i.e. a lift of L, is the same thing as giving a commutative diagram of solid lines Spec k((t)) π 1 U Spec k[[t]] U The dashed line exists and is unique by the valuative criterion of properness, and using the composition Spec k Spec k[[t]] we get a point in X γ(0) that maps to the given point in X L. Therefore Ψ: X γ(0) (k) X L (k(t)) is bijective. The exact same argument exhibits a bijection X γ(1) (k) X L (k(t)) hence we conclude that X γ(0) (k) = X γ(1) (k). 5. Deformation Theory A key part of the proof of Bézout s theorem is to show that the solutions (number of points in a fiber) are constant under small perturbations. Deformation theory studies how spaces behave under these small perturbations.
6 6 CHRISTIAN KLEVDAL Definition 5.1. If X 0 is a variety over k, an infinitesimal deformation of X 0 is a scheme X, flat over Spec A and a commutative diagram X 0 X Spec k Spec A such that A is a local artinian kalgebra and X 0 X Spec A Spec k. In our case, A was taken to be k[t]/(t n ).
SPACES OF RATIONAL CURVES IN COMPLETE INTERSECTIONS
SPACES OF RATIONAL CURVES IN COMPLETE INTERSECTIONS ROYA BEHESHTI AND N. MOHAN KUMAR Abstract. We prove that the space of smooth rational curves of degree e in a general complete intersection of multidegree
More informationSPACES OF RATIONAL CURVES ON COMPLETE INTERSECTIONS
SPACES OF RATIONAL CURVES ON COMPLETE INTERSECTIONS ROYA BEHESHTI AND N. MOHAN KUMAR Abstract. We prove that the space of smooth rational curves of degree e on a general complete intersection of multidegree
More informationCHEAT SHEET: PROPERTIES OF MORPHISMS OF SCHEMES
CHEAT SHEET: PROPERTIES OF MORPHISMS OF SCHEMES BRIAN OSSERMAN The purpose of this cheat sheet is to provide an easy reference for definitions of various properties of morphisms of schemes, and basic results
More informationMath 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 informationLogarithmic 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 2428, 2015 Abramovich (Brown) Logarithmic geometry
More information4. Images of Varieties Given a morphism f : X Y of quasiprojective varieties, a basic question might be to ask what is the image of a closed subset
4. Images of Varieties Given a morphism f : X Y of quasiprojective varieties, a basic question might be to ask what is the image of a closed subset Z X. Replacing X by Z we might as well assume that Z
More informationHARTSHORNE 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 informationExercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo. Alex Massarenti
Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo Alex Massarenti SISSA, VIA BONOMEA 265, 34136 TRIESTE, ITALY Email address: alex.massarenti@sissa.it These notes collect a series of
More informationABSTRACT NONSINGULAR CURVES
ABSTRACT NONSINGULAR CURVES Affine Varieties Notation. Let k be a field, such as the rational numbers Q or the complex numbers C. We call affine nspace the collection A n k of points P = a 1, a,..., a
More informationwhere 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 informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #15 10/29/2013
18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #15 10/29/2013 As usual, k is a perfect field and k is a fixed algebraic closure of k. Recall that an affine (resp. projective) variety is an
More informationLecture 3: Flat Morphisms
Lecture 3: Flat Morphisms September 29, 2014 1 A crash course on Properties of Schemes For more details on these properties, see [Hartshorne, II, 15]. 1.1 Open and Closed Subschemes If (X, O X ) is a
More information10. 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 informationGeometric motivic integration
Université Lille 1 Modnet Workshop 2008 Introduction Motivation: padic integration Kontsevich invented motivic integration to strengthen the following result by Batyrev. Theorem (Batyrev) If two complex
More informationMATH 221 NOTES BRENT HO. Date: January 3, 2009.
MATH 22 NOTES BRENT HO Date: January 3, 2009. 0 Table of Contents. Localizations......................................................................... 2 2. Zariski Topology......................................................................
More informationMATH 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 informationAlgebraic Geometry. Question: What regular polygons can be inscribed in an ellipse?
Algebraic Geometry Question: What regular polygons can be inscribed in an ellipse? 1. Varieties, Ideals, Nullstellensatz Let K be a field. We shall work over K, meaning, our coefficients of polynomials
More informationAlgebraic Geometry. Andreas Gathmann. Class Notes TU Kaiserslautern 2014
Algebraic Geometry Andreas Gathmann Class Notes TU Kaiserslautern 2014 Contents 0. Introduction......................... 3 1. Affine Varieties........................ 9 2. The Zariski Topology......................
More informationNOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY
NOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY RUNE HAUGSENG The aim of these notes is to define flat and faithfully flat morphisms and review some of their important properties, and to define the fpqc
More informationAlgebraic varieties. Chapter A ne varieties
Chapter 4 Algebraic varieties 4.1 A ne varieties Let k be a field. A ne nspace A n = A n k = kn. It s coordinate ring is simply the ring R = k[x 1,...,x n ]. Any polynomial can be evaluated at a point
More informationAN INTRODUCTION TO AFFINE SCHEMES
AN INTRODUCTION TO AFFINE SCHEMES BROOKE ULLERY Abstract. This paper gives a basic introduction to modern algebraic geometry. The goal of this paper is to present the basic concepts of algebraic geometry,
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43 RAVI VAKIL CONTENTS 1. Facts we ll soon know about curves 1 1. FACTS WE LL SOON KNOW ABOUT CURVES We almost know enough to say a lot of interesting things about
More informationAlgebraic 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 informationALGEBRAIC GROUPS JEROEN SIJSLING
ALGEBRAIC GROUPS JEROEN SIJSLING The goal of these notes is to introduce and motivate some notions from the theory of group schemes. For the sake of simplicity, we restrict to algebraic groups (as defined
More information1. 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 quasiprojective varieties over a field k Affine Varieties 1.
More information3. The Sheaf of Regular Functions
24 Andreas Gathmann 3. The Sheaf of Regular Functions After having defined affine varieties, our next goal must be to say what kind of maps between them we want to consider as morphisms, i. e. as nice
More informationALGEBRAIC GROUPS. Disclaimer: There are millions of errors in these notes!
ALGEBRAIC GROUPS Disclaimer: There are millions of errors in these notes! 1. Some algebraic geometry The subject of algebraic groups depends on the interaction between algebraic geometry and group theory.
More information12. Linear systems Theorem Let X be a scheme over a ring A. (1) If φ: X P n A is an Amorphism then L = φ O P n
12. Linear systems Theorem 12.1. Let X be a scheme over a ring A. (1) If φ: X P n A is an Amorphism then L = φ O P n A (1) is an invertible sheaf on X, which is generated by the global sections s 0, s
More informationALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES.
ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES. ANDREW SALCH 1. Affine schemes. About notation: I am in the habit of writing f (U) instead of f 1 (U) for the preimage of a subset
More informationVector bundles in Algebraic Geometry Enrique Arrondo. 1. The notion of vector bundle
Vector bundles in Algebraic Geometry Enrique Arrondo Notes(* prepared for the First Summer School on Complex Geometry (Villarrica, Chile 79 December 2010 1 The notion of vector bundle In affine geometry,
More informationwhere Σ is a finite discrete Gal(K sep /K)set unramified along U and F s is a finite Gal(k(s) sep /k(s))subset
Classification of quasifinite étale separated schemes As we saw in lecture, Zariski s Main Theorem provides a very visual picture of quasifinite étale separated schemes X over a henselian local ring
More informationCOMPLEX MULTIPLICATION: LECTURE 13
COMPLEX MULTIPLICATION: LECTURE 13 Example 0.1. If we let C = P 1, then k(c) = k(t) = k(c (q) ) and the φ (t) = t q, thus the extension k(c)/φ (k(c)) is of the form k(t 1/q )/k(t) which as you may recall
More informationNONSINGULAR CURVES BRIAN OSSERMAN
NONSINGULAR CURVES BRIAN OSSERMAN The primary goal of this note is to prove that every abstract nonsingular curve can be realized as an open subset of a (unique) nonsingular projective curve. Note that
More informationDMATH Algebraic Geometry FS 2018 Prof. Emmanuel Kowalski. Solutions Sheet 1. Classical Varieties
DMATH 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 informationALGEBRAIC GEOMETRY (NMAG401) Contents. 2. Polynomial and rational maps 9 3. Hilbert s Nullstellensatz and consequences 23 References 30
ALGEBRAIC GEOMETRY (NMAG401) JAN ŠŤOVÍČEK Contents 1. Affine varieties 1 2. Polynomial and rational maps 9 3. Hilbert s Nullstellensatz and consequences 23 References 30 1. Affine varieties The basic objects
More informationProjective 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 informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationVector Bundles on Algebraic Varieties
Vector Bundles on Algebraic Varieties Aaron Pribadi December 14, 2010 Informally, a vector bundle associates a vector space with each point of another space. Vector bundles may be constructed over general
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37 RAVI VAKIL CONTENTS 1. Motivation and game plan 1 2. The affine case: three definitions 2 Welcome back to the third quarter! The theme for this quarter, insofar
More informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #17 11/05/2013
18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #17 11/05/2013 Throughout this lecture k denotes an algebraically closed field. 17.1 Tangent spaces and hypersurfaces For any polynomial f k[x
More informationSCHEMES. David Harari. Tsinghua, FebruaryMarch 2005
SCHEMES David Harari Tsinghua, FebruaryMarch 2005 Contents 1. Basic notions on schemes 2 1.1. First definitions and examples.................. 2 1.2. Morphisms of schemes : first properties.............
More informationLecture 7: Etale Fundamental Group  Examples
Lecture 7: Etale Fundamental Group  Examples October 15, 2014 In this lecture our only goal is to give lots of examples of etale fundamental groups so that the reader gets some feel for them. Some of
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 51 AND 52
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 51 AND 52 RAVI VAKIL CONTENTS 1. Smooth, étale, unramified 1 2. Harder facts 5 3. Generic smoothness in characteristic 0 7 4. Formal interpretations 11 1. SMOOTH,
More informationPICARD GROUPS OF MODULI PROBLEMS II
PICARD GROUPS OF MODULI PROBLEMS II DANIEL LI 1. Recap Let s briefly recall what we did last time. I discussed the stack BG m, as classifying line bundles by analyzing the sense in which line bundles may
More informationSummer Algebraic Geometry Seminar
Summer Algebraic Geometry Seminar Lectures by Bart Snapp About This Document These lectures are based on Chapters 1 and 2 of An Invitation to Algebraic Geometry by Karen Smith et al. 1 Affine Varieties
More informationElliptic curves, Néron models, and duality
Elliptic curves, Néron models, and duality Jean Gillibert Durham, Pure Maths Colloquium 26th February 2007 1 Elliptic curves and Weierstrass equations Let K be a field Definition: An elliptic curve over
More informationAPPENDIX 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 informationAn Atlas For Bun r (X)
An Atlas For Bun r (X) As told by Dennis Gaitsgory to Nir Avni October 28, 2009 1 Bun r (X) Is Not Of Finite Type The goal of this lecture is to find a smooth atlas locally of finite type for the stack
More informationPROBLEMS, MATH 214A. Affine and quasiaffine varieties
PROBLEMS, MATH 214A k is an algebraically closed field Basic notions Affine and quasiaffine 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 information12. 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 informationMA 206 notes: introduction to resolution of singularities
MA 206 notes: introduction to resolution of singularities Dan Abramovich Brown University March 4, 2018 Abramovich Introduction to resolution of singularities 1 / 31 Resolution of singularities Let k be
More informationLEVINE S CHOW S MOVING LEMMA
LEVINE S CHOW S MOVING LEMMA DENIS NARDIN The main result of this note is proven in [4], using results from [2]. What is here is essentially a simplified version of the proof (at the expense of some generality).
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 26
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 26 RAVI VAKIL CONTENTS 1. Proper morphisms 1 Last day: separatedness, definition of variety. Today: proper morphisms. I said a little more about separatedness of
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27 RAVI VAKIL CONTENTS 1. Proper morphisms 1 2. Schemetheoretic closure, and schemetheoretic image 2 3. Rational maps 3 4. Examples of rational maps 5 Last day:
More information2. Intersection Multiplicities
2. Intersection Multiplicities 11 2. Intersection Multiplicities Let us start our study of curves by introducing the concept of intersection multiplicity, which will be central throughout these notes.
More informationIntersecting valuation rings in the ZariskiRiemann space of a field
Intersecting valuation rings in the ZariskiRiemann space of a field Bruce Olberding Department of Mathematical Sciences New Mexico State University November, 2015 Motivation from Birational Algebra Problem:
More informationMath 203A, Solution Set 6.
Math 203A, Solution Set 6. Problem 1. (Finite maps.) Let f 0,..., f m be homogeneous polynomials of degree d > 0 without common zeros on X P n. Show that gives a finite morphism onto its image. f : X P
More informationIntroduction 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 informationSUMMER COURSE IN MOTIVIC HOMOTOPY THEORY
SUMMER COURSE IN MOTIVIC HOMOTOPY THEORY MARC LEVINE Contents 0. Introduction 1 1. The category of schemes 2 1.1. The spectrum of a commutative ring 2 1.2. Ringed spaces 5 1.3. Schemes 10 1.4. Schemes
More informationINTEGRATION OF ONEFORMS ON padic ANALYTIC SPACES
INTEGRATION OF ONEFORMS ON padic ANALYTIC SPACES VLADIMIR G. BERKOVICH Recall that there is a unique way to define for every comple manifold, every closed analytic oneform ω, and every continuous path
More information1 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 information0.1 Spec of a monoid
These notes were prepared to accompany the first lecture in a seminar on logarithmic geometry. As we shall see in later lectures, logarithmic geometry offers a natural approach to study semistable schemes.
More informationNotes on pdivisible Groups
Notes on pdivisible Groups March 24, 2006 This is a note for the talk in STAGE in MIT. The content is basically following the paper [T]. 1 Preliminaries and Notations Notation 1.1. Let R be a complete
More informationMath 248B. Applications of base change for coherent cohomology
Math 248B. Applications of base change for coherent cohomology 1. Motivation Recall the following fundamental general theorem, the socalled cohomology and base change theorem: Theorem 1.1 (Grothendieck).
More informationINTERSECTION THEORY CLASS 6
INTERSECTION THEORY CLASS 6 RAVI VAKIL CONTENTS 1. Divisors 2 1.1. Crash course in Cartier divisors and invertible sheaves (aka line bundles) 3 1.2. Pseudodivisors 3 2. Intersecting with divisors 4 2.1.
More informationResolution 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 information14 Lecture 14: Basic generallities on adic spaces
14 Lecture 14: Basic generallities on adic spaces 14.1 Introduction The aim of this lecture and the next two is to address general adic spaces and their connection to rigid geometry. 14.2 Two open questions
More informationTunisian Journal of Mathematics an international publication organized by the Tunisian Mathematical Society
unisian Journal of Mathematics an international publication organized by the unisian Mathematical Society Ramification groups of coverings and valuations akeshi Saito 2019 vol. 1 no. 3 msp msp UNISIAN
More informationThe Hitchin map, local to global
The Hitchin map, local to global Andrei Negut Let X be a smooth projective curve of genus g > 1, a semisimple group and Bun = Bun (X) the moduli stack of principal bundles on X. In this talk, we will present
More information1 Flat, Smooth, Unramified, and Étale Morphisms
1 Flat, Smooth, Unramified, and Étale Morphisms 1.1 Flat morphisms Definition 1.1. An Amodule M is flat if the (rightexact) functor A M is exact. It is faithfully flat if a complex of Amodules P N Q
More informationLECTURES ON SYMPLECTIC REFLECTION ALGEBRAS
LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS IVAN LOSEV 16. Symplectic resolutions of µ 1 (0)//G and their deformations 16.1. GIT quotients. We will need to produce a resolution of singularities for C 2n
More informationAlgebraic 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 informationAlgebraic 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 informationVERY STABLE BUNDLES AND PROPERNESS OF THE HITCHIN MAP
VERY STABLE BUNDLES AND PROPERNESS OF THE HITCHIN MAP CHRISTIAN PAULY AND ANA PEÓNNIETO Abstract. Let X be a smooth complex projective curve of genus g 2 and let K be its canonical bundle. In this note
More informationLecture 21: Crystalline cohomology and the de RhamWitt complex
Lecture 21: Crystalline cohomology and the de RhamWitt complex Paul VanKoughnett November 12, 2014 As we ve been saying, to understand K3 surfaces in characteristic p, and in particular to rediscover
More informationLogarithmic geometry and moduli
Logarithmic geometry and moduli Lectures at the Sophus Lie Center Dan Abramovich Brown University June 1617, 2014 Abramovich (Brown) Logarithmic geometry and moduli June 1617, 2014 1 / 1 Heros: Olsson
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 18
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 18 CONTENTS 1. Invertible sheaves and divisors 1 2. Morphisms of schemes 6 3. Ringed spaces and their morphisms 6 4. Definition of morphisms of schemes 7 Last day:
More informationBoolean Algebras, Boolean Rings and Stone s Representation Theorem
Boolean Algebras, Boolean Rings and Stone s Representation Theorem Hongtaek Jung December 27, 2017 Abstract This is a part of a supplementary note for a Logic and Set Theory course. The main goal is to
More informationCHAPTER 1. AFFINE ALGEBRAIC VARIETIES
CHAPTER 1. AFFINE ALGEBRAIC VARIETIES During this first part of the course, we will establish a correspondence between various geometric notions and algebraic ones. Some references for this part of the
More informationdiv(f ) = D and deg(d) = deg(f ) = d i deg(f i ) (compare this with the definitions for smooth curves). Let:
Algebraic Curves/Fall 015 Aaron Bertram 4. Projective Plane Curves are hypersurfaces in the plane CP. When nonsingular, they are Riemann surfaces, but we will also consider plane curves with singularities.
More informationa double cover branched along the smooth quadratic line complex
QUADRATIC LINE COMPLEXES OLIVIER DEBARRE Abstract. In this talk, a quadratic line complex is the intersection, in its Plücker embedding, of the Grassmannian of lines in an 4dimensional projective space
More informationMATH 233B, FLATNESS AND SMOOTHNESS.
MATH 233B, FLATNESS AND SMOOTHNESS. The discussion of smooth morphisms is one place were Hartshorne doesn t do a very good job. Here s a summary of this week s material. I ll also insert some (optional)
More informationCalculating deformation rings
Calculating deformation rings Rebecca Bellovin 1 Introduction We are interested in computing local deformation rings away from p. That is, if L is a finite extension of Q l and is a 2dimensional representation
More informationV (f) :={[x] 2 P n ( ) f(x) =0}. If (x) ( x) thenf( x) =
20 KIYOSHI IGUSA BRANDEIS UNIVERSITY 2. Projective varieties For any field F, the standard definition of projective space P n (F ) is that it is the set of one dimensional F vector subspaces of F n+.
More informationthe complete linear series of D. Notice that D = PH 0 (X; O X (D)). Given any subvectorspace V H 0 (X; O X (D)) there is a rational map given by V : X
2. Preliminaries 2.1. Divisors and line bundles. Let X be an irreducible complex variety of dimension n. The group of kcycles on X is Z k (X) = fz linear combinations of subvarieties of dimension kg:
More informationSymplectic varieties and Poisson deformations
Symplectic varieties and Poisson deformations Yoshinori Namikawa A symplectic variety X is a normal algebraic variety (defined over C) which admits an everywhere nondegenerate dclosed 2form ω on the
More informationOn Maps Taking Lines to Plane Curves
Arnold Math J. (2016) 2:1 20 DOI 10.1007/s4059801500271 RESEARCH CONTRIBUTION On Maps Taking Lines to Plane Curves Vsevolod Petrushchenko 1 Vladlen Timorin 1 Received: 24 March 2015 / Accepted: 16 October
More informationSome remarks on Frobenius and Lefschetz in étale cohomology
Some remarks on obenius and Lefschetz in étale cohomology Gabriel Chênevert January 5, 2004 In this lecture I will discuss some more or less related issues revolving around the main idea relating (étale)
More informationh M (T ). The natural isomorphism η : M h M determines an element U = η 1
MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 7 2.3. Fine moduli spaces. The ideal situation is when there is a scheme that represents our given moduli functor. Definition 2.15. Let M : Sch Set be a moduli
More informationCOMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY
COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY BRIAN OSSERMAN Classical algebraic geometers studied algebraic varieties over the complex numbers. In this setting, they didn t have to worry about the Zariski
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 5
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 5 RAVI VAKIL CONTENTS 1. The inverse image sheaf 1 2. Recovering sheaves from a sheaf on a base 3 3. Toward schemes 5 4. The underlying set of affine schemes 6 Last
More informationDIVISORS ON NONSINGULAR CURVES
DIVISORS ON NONSINGULAR CURVES BRIAN OSSERMAN We now begin a closer study of the behavior of projective nonsingular curves, and morphisms between them, as well as to projective space. To this end, we introduce
More informationÉTALE π 1 OF A SMOOTH CURVE
ÉTALE π 1 OF A SMOOTH CURVE AKHIL MATHEW 1. Introduction One of the early achievements of Grothendieck s theory of schemes was the (partial) computation of the étale fundamental group of a smooth projective
More informationA course in. Algebraic Geometry. Taught by Prof. Xinwen Zhu. Fall 2011
A course in Algebraic Geometry Taught by Prof. Xinwen Zhu Fall 2011 1 Contents 1. September 1 3 2. September 6 6 3. September 8 11 4. September 20 16 5. September 22 21 6. September 27 25 7. September
More informationMiniCourse on Moduli Spaces
MiniCourse on Moduli Spaces Emily Clader June 2011 1 What is a Moduli Space? 1.1 What should a moduli space do? Suppose that we want to classify some kind of object, for example: Curves of genus g, Onedimensional
More informationAFFINE PUSHFORWARD AND SMOOTH PULLBACK FOR PERVERSE SHEAVES
AFFINE PUSHFORWARD AND SMOOTH PULLBACK FOR PERVERSE SHEAVES YEHAO ZHOU Conventions In this lecture note, a variety means a separated algebraic variety over complex numbers, and sheaves are Clinear. 1.
More informationCOMPLEX ALGEBRAIC SURFACES CLASS 9
COMPLEX ALGEBRAIC SURFACES CLASS 9 RAVI VAKIL CONTENTS 1. Construction of Castelnuovo s contraction map 1 2. Ruled surfaces 3 (At the end of last lecture I discussed the Weak Factorization Theorem, Resolution
More informationMODULI 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 informationK3 Surfaces and Lattice Theory
K3 Surfaces and Lattice Theory Ichiro Shimada Hiroshima University 2014 Aug Singapore 1 / 26 Example Consider two surfaces S + and S in C 3 defined by w 2 (G(x, y) ± 5 H(x, y)) = 1, where G(x, y) := 9
More informationTHE KEEL MORI THEOREM VIA STACKS
THE KEEL MORI THEOREM VIA STACKS BRIAN CONRAD 1. Introduction Let X be an Artin stack (always assumed to have quasicompact and separated diagonal over Spec Z; cf. [2, 1.3]). A coarse moduli space for
More information