# INTERSECTION THEORY CLASS 19

Size: px
Start display at page:

Transcription

1 INTERSECTION THEORY CLASS 19 RAVI VAKIL CONTENTS 1. Recap of Last day New facts 2 2. Statement of the theorem GRR for a special case of closed immersions f : X Y = P(N 1) GRR for closed immersions in general 6 Today I m going to try to finish the proof of Grothendieck-Riemann-Roch in the case of projective morphisms from smooth varieties to smooth varieties. We ll see that we re essentially going to prove it more generally for projective lci morphisms. 1. RECAP OF LAST DAY Recall the definition of the Chern character and Todd class. Suppose F is a coherent sheaf. Let α 1,..., α n be the Chern roots of the vector bundle, so α α n = c 1 (F), etc. Define ch(f) = r i=1 exp(α i) This is additive on exact sequences. For vector bundles, we have ch(e E ) = ch(e) ch(e ). The Todd class td(e) of a vector bundle is defined by td(e) = r i=1 Q(α i) where x Q(x) = 1 e = x 2 x + ( 1) k 1 B k (2k)! x2k. It is multiplicative in exact sequences. We defined the Grothendieck groups K 0 X and K 0 X. They are vector bundles, respectively coherent sheaves, modulo the relation [E] = [E ] + [E ]. We have a pullback on K 0 : f : K 0 X K 0 Y. K 0 X is a ring: [E] [F] = [E F]. We have a pushforward on K 0 : f [F] = i 0 ( 1)i [R i f F]. We obviously have a homomorphism K 0 X K 0 X. K 0 X is a K 0 X-module: K 0 X K 0 X X is given by [E] [F] = [E F]. Unproved fact: If X is nonsingular and projective, the map K 0 X K 0 X is an isomorphism. (Reason: If X is nonsingular, then F has a finite resolution by locally free sheaves.) Date: Wednesday, November 24, k=1

2 The Chern character map descends to K(X): ch : K(X) A(X) Q. This does not commute with proper pushforward; Grothendieck-Riemann-Roch explains how to fix this New facts. Here are some useful facts, that I didn t mention last time. We have an excision exact sequence for K 0 : If Z X is a closed immersion, and U is the open complement, we have an excision exact sequence K 0 (Z) K 0 (X) K 0 (U) 0. The proof is similar to our proof for Chow; this is Hartshorne Exercise II.6.10(c). Similarly, we have K 0 (A 1 Y) = K(Y). Last time I showed: Lemma. The group K 0 (P m ) is generated by the classes [O P m(n)], with 0 n m. (Incidentally, I mentioned an interesting algebraic problem coming out of my previous proof. Joe gave a nice proof of it. If I have time, I ll type it up and put it in the posted notes.) I d like to do it differently today. [O P m( n)], with 0 n m. Instead, I ll show it is generated by the classes Using the excision exact sequence for K-theory, and P m = A 0 A 1 A m, we get inductively: K 0 (P m ) is generated by n + 1 things: [O P 0], [O P 1],..., [O P m]. I ll now express these in terms of O P m(n) s. From 0 O P m( 1) O P m O P m 1 0 shows [O P m 1] = [O P m] [O P m( 1)]. Similarly, [O P m 2] = [O P m 1] [O P m 1( 1)] = ([O P m] [O P m( 1)]) ([O P m( 1)] [O P m( 2)]) = [O P m] 2[O P m( 1)] [O P m( 2)] and you see the pattern (established by the obvious induction). Important philosophy behind Riemann-Roch: K(P m ) and A (P m ) are both m-dimensional vector spaces; Chern character provides an isomorphism between them. Multiplying by the Todd class provides a better isomorphism between them. More generally, the identical proof shows that for any Y, K(Y) K(P m ) K(P m Y) is surjective: cut up P m Y into Y A 1 Y A m Y, and proceed as before. 2

3 2. STATEMENT OF THE THEOREM Grothendieck-Riemann-Roch Theorem. Suppose f : X Y is a proper morphism of smooth varieties. Then for any α K(X), ch(f α) td(t Y ) = f (ch(α) td(t X )). Interesting exercise: how do you make sense of this when X and Y are singular? For example, what if X Y is a smooth morphism, we get ch(f α) = f (ch(α) td(t X/Y )) where X/Y is the relative tangent bundle. As another example, what if X Y is a complete intersection? Then T X and T Y don t make sense, but N X/Y is a vector bundle, and then ch(f α) td(n X/Y ) = f (ch(α)). Combining these two, you can now make sense of GRR in the case when f is an lci morphism (i.e. closed immersion followed by a smooth morphism). The theorem may be interpreted to say that the homomorphism τ X : K(X) A(X) Q given by τ X (α) = ch(α) td(t X ) commutes with proper pushforward: f τ X = τ Y f. Last time we showed that this implies Lemma. Given X f Z g Y. Suppose GRR holds for f and g. Then it holds for g f. Hence the strategy is now to show GRR for Y P m Y, and for closed immersions. We ll use this interpretation of the theorem to show Theorem. GRR is true for P m Y Y. Proof. We showed last time that this is true in the case where Y is a point. Consider the following diagram. K(Y) K(P m ) τ Y τ P m A(Y) Q A(P m ) Q K(Y P m ) τ Y τ P m A(Y P m ) Q f K(Y) τ Y f A(Y) Q (I won t be using anything special about P m now.) We want to show that the bottom square commutes. Note that the top square commutes. Reason: T Y P m = p 1 T Y p 2 T P m (where p 1 and p 2 are the projections) from which td(t Y P m) = td(p 1 T Y) td(p 2 T P m). Moreover the upper left vertical arrow is surjective. So it suffices to show that the big rectangle commutes. But it does because we ve already shown that GRR holds for P m pt. 3

4 2.1. GRR for a special case of closed immersions f : X Y = P(N 1). Suppose f is a closed immersion into a projective completion of a normal bundle. Let d = rankn. We want to prove GRR for a vector bundle E. As the vector bundles generate K(X), this will suffice. This example comes the closest to telling me why the Todd class wants to be what it is. Let p : Y = P(N 1) X be the projection. Let O Y ( 1) be the tautological line bundle on Y = P ( N 1). Then as in previous lectures we have a tautological exact sequence of vector bundles on Y: 0 O Y ( 1) p (N 1) Q 0 where Q is the universal quotient bundle. (Recall that f Q = N X/Y.) Here is something you have to think through, although we ve implicitly used it before. We have a natural section of p (Q 1), the 1. This gives a section s of Q. This section vanishes precisely (scheme-theoretically) along X. In particular, for any α A(Y), f (f α) = c d (Q) α. (This was one of our results about the top Chern class. f f will knock the degree down by d, and we found that this operator was the same as capping with the top Chern class.) Lemma. We can resolve the sheaf f O X on Y by (1) 0 d Q 2 Q Q s O Y f O X 0. Note that everything except f O X is a vector bundle on Y. Proof. Rather than proving this precisely, I ll do a special case, to get across the main idea. This in fact becomes a proof, once the naturality of my argument is established. Suppose Y = Spec k[x 1,..., x n ], so O Y = k[x 1,..., x n ] (a bit sloppily) and X = 0 Y. Then let s build a resolution of O X. We start with O Y O X 0. We have a big kernel obviously: the ideal sheaf of O X. So our next step is: O Y x 1 O Y x 2 O Y x n O Y O X 0. We still have a kernel; ( x 2 x 1, x 1 x 2, 0,, 0) is in the kernel, for example. So our next step is: O Y x 1 x 2 O Y x n 1 x n (We need to check that we ve surjected onto the kernel! But that s not hard; you can try to prove that yourself.) And the pattern continues. We get: 0 O Y x 1 x n n i=1o Y x 1 ^x i x n n i=1o Y x i O Y O X 0. And this is what we wanted (in this special case). (All that is missing for this to be a proof is to realize that n i=1 O Yx i O Y is canonically Q.) 4

5 If E is a vector bundle on X, then we have an explicit resolution of f E, by tensoring (1) with p E: 0 d Q p E Q p E s p E f E 0. (Tensoring with a vector bundle is exact, and (p E) O X = f E.) Therefore ch f [E] = ( 1) p ch( p Q ) ch(p E). Lemma. ( 1) p ch( p Q ) = c d (Q) td(q) 1. This tells you why the Todd class is what it is! Proof. This is remarkably easy. Let α 1,..., α d be the Chern roots of Q. Then the Chern roots of p Q are α i1 α ip. Hence ch( p Q ) = e α i1 α ip from which ( 1) p ch( p Q ) = ( 1) p e α i 1 e α ip d = (1 e α i ) i=1 = (α 1 α d ) d 1 e α i α i i=1 = c d (Q) td(q) 1. Hence ch f [E] = c d (Q) td(q) 1 ch(p E) as desired! = f (f td(q) 1 f ch(p E)) (using c d (Q) β = f (f β), see 1st par of Section 2) = f (td(n X/Y ) 1 ch(e)) (using f Q = N X/Y, f p E = E) = f (td(t X )f td(t Y ) 1 ch(e)) = td(t Y )f (td(t X ) ch(e)) (projection formula) This ends the proof of GRR for a closed immersion of X into the projective completion of a normal bundle. 5

6 2.2. GRR for closed immersions in general. Suppose f : X Y is a closed immersion. We ll prove GRR in this case; again, we need only to consider a generator of K(X), a vector bundle E on X. We ll show GRR by deformation to the normal cone. Let M = Bl X { } Y P 1. (Draw picture.) Recall that the fiber over is M = Bl X Y P(N 1). X f Y = M 0 X P 1 F M = Bl X 0 Y P 1 X M = Bl X Y P(N 1) {0} P 1 { } Define F (above), p : X P 1 X. Resolve p E on M: 0 G n G n 1 G 0 F (p E) 0. Both X P 1 and M are flat over P 1 (recall that dominant morphisms from irreducible varieties to a smooth curve are always flat), so restriction of these exact sequences to the fibers M 0 and M (also known as tensoring with the structure sheaves of the fibers) preserves exactness. Let j 0 : Y = M 0 M, j : Bl X Y P(N 1) = M M, k : P(N 1) M, l : Bl X Y M. Now j 0 G resolves f on Y = M 0. So j 0 (ch(f E)) = j 0 ch(j 0 G ) = ch(g ) j 0 [Y] (proj. formula) = ch(g ) j [M ] (pulling back rat l equivalence 0 P 1 ) = ch(g ) (k [P(N 1)] + l [Bl X Y]) Now G is exact away from X P 1, so it is exact on Bl X Y, so the Chern character of the complex (the alternating sums of the Chern characters of the terms) is 0. Hence: Using the projection formula again: = ch(g ) (k [P(N 1)]) = k (ch(f E) [P(N 1)]) (where f is the map X P(N 1)). (We re writing this as k (ch(f E).) So now we re dealing with the case X P(N 1)! We ve already calculated that this is f (td(n) 1 ch(e)). As [N] = [f T Y ] [T X ]: and we re done! address: ch(f E) td(t Y ) = f (ch(e) td(t X )) 6

### INTERSECTION THEORY CLASS 16

INTERSECTION THEORY CLASS 16 RAVI VAKIL CONTENTS 1. Where we are 1 1.1. Deformation to the normal cone 1 1.2. Gysin pullback for local complete intersections 2 1.3. Intersection products on smooth varieties

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24

FOUNDATIONS OF ALGEBRAIC GEOMETR CLASS 24 RAVI VAKIL CONTENTS 1. Normalization, continued 1 2. Sheaf Spec 3 3. Sheaf Proj 4 Last day: Fibers of morphisms. Properties preserved by base change: open immersions,

### INTERSECTION THEORY CLASS 7

INTERSECTION THEORY CLASS 7 RAVI VAKIL CONTENTS 1. Intersecting with a pseudodivisor 1 2. The first Chern class of a line bundle 3 3. Gysin pullback 4 4. Towards the proof of the big theorem 4 4.1. Crash

### INTERSECTION THEORY CLASS 12

INTERSECTION THEORY CLASS 12 RAVI VAKIL CONTENTS 1. Rational equivalence on bundles 1 1.1. Intersecting with the zero-section of a vector bundle 2 2. Cones and Segre classes of subvarieties 3 2.1. Introduction

### COMPLEX 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

### INTERSECTION 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. Pseudo-divisors 3 2. Intersecting with divisors 4 2.1.

### Chern classes à la Grothendieck

Chern classes à la Grothendieck Theo Raedschelders October 16, 2014 Abstract In this note we introduce Chern classes based on Grothendieck s 1958 paper [4]. His approach is completely formal and he deduces

### FOUNDATIONS 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

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 47 AND 48

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 47 AND 48 RAVI VAKIL CONTENTS 1. The local criterion for flatness 1 2. Base-point-free, ample, very ample 2 3. Every ample on a proper has a tensor power that

### INTERSECTION THEORY CLASS 2

INTERSECTION THEORY CLASS 2 RAVI VAKIL CONTENTS 1. Last day 1 2. Zeros and poles 2 3. The Chow group 4 4. Proper pushforwards 4 The webpage http://math.stanford.edu/ vakil/245/ is up, and has last day

### FOUNDATIONS 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

### DESCENT THEORY (JOE RABINOFF S EXPOSITION)

DESCENT THEORY (JOE RABINOFF S EXPOSITION) RAVI VAKIL 1. FEBRUARY 21 Background: EGA IV.2. Descent theory = notions that are local in the fpqc topology. (Remark: we aren t assuming finite presentation,

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24 RAVI VAKIL CONTENTS 1. Vector bundles and locally free sheaves 1 2. Toward quasicoherent sheaves: the distinguished affine base 5 Quasicoherent and coherent sheaves

### FOUNDATIONS 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

### Math 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 so-called cohomology and base change theorem: Theorem 1.1 (Grothendieck).

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 41

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 41 RAVI VAKIL CONTENTS 1. Normalization 1 2. Extending maps to projective schemes over smooth codimension one points: the clear denominators theorem 5 Welcome back!

### THE GROTHENDIECK-RIEMANN-ROCH THEOREM FOR VARIETIES

THE GROTHENDIECK-RIEMANN-ROCH THEOREM FOR VARIETIES PETER XU ABSTRACT. We give an exposition of the Grothendieck-Riemann-Roch theorem for algebraic varieties. Our proof follows Borel and Serre [3] and

### Algebraic 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

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 48

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 48 RAVI VAKIL CONTENTS 1. A little more about cubic plane curves 1 2. Line bundles of degree 4, and Poncelet s Porism 1 3. Fun counterexamples using elliptic curves

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 43 AND 44

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 43 AND 44 RAVI VAKIL CONTENTS 1. Flat implies constant Euler characteristic 1 2. Proof of Important Theorem on constancy of Euler characteristic in flat families

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2 RAVI VAKIL CONTENTS 1. Where we were 1 2. Yoneda s lemma 2 3. Limits and colimits 6 4. Adjoints 8 First, some bureaucratic details. We will move to 380-F for Monday

### Algebraic 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

### INTERSECTION THEORY CLASS 1

INTERSECTION THEORY CLASS 1 RAVI VAKIL CONTENTS 1. Welcome! 1 2. Examples 2 3. Strategy 5 1. WELCOME! Hi everyone welcome to Math 245, Introduction to intersection theory in algebraic geometry. Today I

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27 RAVI VAKIL CONTENTS 1. Proper morphisms 1 2. Scheme-theoretic closure, and scheme-theoretic image 2 3. Rational maps 3 4. Examples of rational maps 5 Last day:

### Porteous s Formula for Maps between Coherent Sheaves

Michigan Math. J. 52 (2004) Porteous s Formula for Maps between Coherent Sheaves Steven P. Diaz 1. Introduction Recall what the Thom Porteous formula for vector bundles tells us (see [2, Sec. 14.4] for

### COMPLEX ALGEBRAIC SURFACES CLASS 6

COMPLEX ALGEBRAIC SURFACES CLASS 6 RAVI VAKIL CONTENTS 1. The intersection form 1.1. The Neron-Severi group 3 1.. Aside: The Hodge diamond of a complex projective surface 3. Riemann-Roch for surfaces 4

### 12. Linear systems Theorem Let X be a scheme over a ring A. (1) If φ: X P n A is an A-morphism 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 A-morphism then L = φ O P n A (1) is an invertible sheaf on X, which is generated by the global sections s 0, s

### FOUNDATIONS 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:

### the 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 k-cycles on X is Z k (X) = fz linear combinations of subvarieties of dimension kg:

### INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 23

INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 23 RAVI VAKIL Contents 1. More background on invertible sheaves 1 1.1. Operations on invertible sheaves 1 1.2. Maps to projective space correspond to a vector

### NOTES 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

### FOUNDATIONS 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

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37 RAVI VAKIL CONTENTS 1. Application of cohomology: Hilbert polynomials and functions, Riemann- Roch, degrees, and arithmetic genus 1 1. APPLICATION OF COHOMOLOGY:

### MATH 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)

### APPENDIX 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

### DEFINITION OF ABELIAN VARIETIES AND THE THEOREM OF THE CUBE

DEFINITION OF ABELIAN VARIETIES AND THE THEOREM OF THE CUBE ANGELA ORTEGA (NOTES BY B. BAKKER) Throughout k is a field (not necessarily closed), and all varieties are over k. For a variety X/k, by a basepoint

### 3. 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

### The Riemann-Roch Theorem

The Riemann-Roch Theorem TIFR Mumbai, India Paul Baum Penn State 7 August, 2015 Five lectures: 1. Dirac operator 2. Atiyah-Singer revisited 3. What is K-homology? 4. Beyond ellipticity 5. The Riemann-Roch

### Lecture 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

### INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 14

INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 14 RAVI VAKIL Contents 1. Dimension 1 1.1. Last time 1 1.2. An algebraic definition of dimension. 3 1.3. Other facts that are not hard to prove 4 2. Non-singularity:

### HARTSHORNE 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

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2 RAVI VAKIL CONTENTS 1. Some constructions using universal properties, old and new 1 2. Adjoint functors 3 3. Sheaves 6 Last day: What is algebraic geometry? Crash

### Recall for an n n matrix A = (a ij ), its trace is defined by. a jj. It has properties: In particular, if B is non-singular n n matrix,

Chern characters Recall for an n n matrix A = (a ij ), its trace is defined by tr(a) = n a jj. j=1 It has properties: tr(a + B) = tr(a) + tr(b), tr(ab) = tr(ba). In particular, if B is non-singular n n

### COMPLEX ALGEBRAIC SURFACES CLASS 15

COMPLEX ALGEBRAIC SURFACES CLASS 15 RAVI VAKIL CONTENTS 1. Every smooth cubic has 27 lines, and is the blow-up of P 2 at 6 points 1 1.1. An alternate approach 4 2. Castelnuovo s Theorem 5 We now know that

### Factorization of birational maps for qe schemes in characteristic 0

Factorization of birational maps for qe schemes in characteristic 0 AMS special session on Algebraic Geometry joint work with M. Temkin (Hebrew University) Dan Abramovich Brown University October 24, 2014

### Factorization of birational maps on steroids

Factorization of birational maps on steroids IAS, April 14, 2015 Dan Abramovich Brown University April 14, 2015 This is work with Michael Temkin (Jerusalem) Abramovich (Brown) Factorization of birational

### Lecture 21: Crystalline cohomology and the de Rham-Witt complex

Lecture 21: Crystalline cohomology and the de Rham-Witt complex Paul VanKoughnett November 12, 2014 As we ve been saying, to understand K3 surfaces in characteristic p, and in particular to rediscover

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 25

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 25 RAVI VAKIL CONTENTS 1. Quasicoherent sheaves 1 2. Quasicoherent sheaves form an abelian category 5 We began by recalling the distinguished affine base. Definition.

### COMPLEX 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

### Algebraic 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

### Fourier Mukai transforms II Orlov s criterion

Fourier Mukai transforms II Orlov s criterion Gregor Bruns 07.01.2015 1 Orlov s criterion In this note we re going to rely heavily on the projection formula, discussed earlier in Rostislav s talk) and

### Algebraic 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

### Theta divisors and the Frobenius morphism

Theta divisors and the Frobenius morphism David A. Madore Abstract We introduce theta divisors for vector bundles and relate them to the ordinariness of curves in characteristic p > 0. We prove, following

### CHEAT 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

### Special cubic fourfolds

Special cubic fourfolds 1 Hodge diamonds Let X be a cubic fourfold, h H 2 (X, Z) be the (Poincaré dual to the) hyperplane class. We have h 4 = deg(x) = 3. By the Lefschetz hyperplane theorem, one knows

### LECTURES ON SINGULARITIES AND ADJOINT LINEAR SYSTEMS

LECTURES ON SINGULARITIES AND ADJOINT LINEAR SYSTEMS LAWRENCE EIN Abstract. 1. Singularities of Surfaces Let (X, o) be an isolated normal surfaces singularity. The basic philosophy is to replace the singularity

### Algebraic 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

### Topology of Toric Varieties, Part II

Topology of Toric Varieties, Part II Daniel Chupin April 2, 2018 Abstract Notes for a talk leading up to a discussion of the Hirzebruch-Riemann-Roch (HRR) theorem for toric varieties, and some consequences

### (1) is an invertible sheaf on X, which is generated by the global sections

7. Linear systems First a word about the base scheme. We would lie to wor in enough generality to cover the general case. On the other hand, it taes some wor to state properly the general results if one

### Hodge Theory of Maps

Hodge Theory of Maps Migliorini and de Cataldo June 24, 2010 1 Migliorini 1 - Hodge Theory of Maps The existence of a Kähler form give strong topological constraints via Hodge theory. Can we get similar

### Algebraic 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

### 18.727, Topics in Algebraic Geometry (rigid analytic geometry) Kiran S. Kedlaya, fall 2004 Kiehl s finiteness theorems

18.727, Topics in Algebraic Geometry (rigid analytic geometry) Kiran S. Kedlaya, fall 2004 Kiehl s finiteness theorems References: [FvdP, Chapter 4]. Again, Kiehl s original papers (in German) are: Der

### Smooth morphisms. Peter Bruin 21 February 2007

Smooth morphisms Peter Bruin 21 February 2007 Introduction The goal of this talk is to define smooth morphisms of schemes, which are one of the main ingredients in Néron s fundamental theorem [BLR, 1.3,

### Summer 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

### Cohomology and Base Change

Cohomology and Base Change Let A and B be abelian categories and T : A B and additive functor. We say T is half-exact if whenever 0 M M M 0 is an exact sequence of A-modules, the sequence T (M ) T (M)

### MODULI SPACES AND DEFORMATION THEORY, CLASS 1. Contents 1. Preliminaries 1 2. Motivation for moduli spaces Deeper into that example 4

MODULI SPACES AND DEFORMATION THEORY, CLASS 1 RAVI VAKIL Contents 1. Preliminaries 1 2. Motivation for moduli spaces 3 2.1. Deeper into that example 4 1. Preliminaries On the off chance that any of you

### where 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 quasi-finite étale separated schemes As we saw in lecture, Zariski s Main Theorem provides a very visual picture of quasi-finite étale separated schemes X over a henselian local ring

### Chapter 17 Computation of Some Hodge Numbers

Chapter 17 Computation of Some Hodge Numbers The Hodge numbers of a smooth projective algebraic variety are very useful invariants. By Hodge theory, these determine the Betti numbers. In this chapter,

### K-theory, Chow groups and Riemann-Roch

K-theory, Chow groups and Riemann-Roch N. Mohan Kumar October 26, 2004 We will work over a quasi-projective variety over a field, though many statements will work for arbitrary Noetherian schemes. The

### CHAPTER 0 PRELIMINARY MATERIAL. Paul Vojta. University of California, Berkeley. 18 February 1998

CHAPTER 0 PRELIMINARY MATERIAL Paul Vojta University of California, Berkeley 18 February 1998 This chapter gives some preliminary material on number theory and algebraic geometry. Section 1 gives basic

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 21

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 21 RAVI VAKIL CONTENTS 1. Integral extensions, the going-up theorem, Noether normalization, and a proof of the big dimension theorem (that transcendence degree =

### BEZOUT S THEOREM CHRISTIAN KLEVDAL

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

### Algebraic 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

### p,q H (X), H (Y ) ), where the index p has the same meaning as the

There are two Eilenberg-Moore spectral sequences that we shall consider, one for homology and the other for cohomology. In contrast with the situation for the Serre spectral sequence, for the Eilenberg-Moore

### ON A THEOREM OF CAMPANA AND PĂUN

ON A THEOREM OF CAMPANA AND PĂUN CHRISTIAN SCHNELL Abstract. Let X be a smooth projective variety over the complex numbers, and X a reduced divisor with normal crossings. We present a slightly simplified

### Preliminary Exam Topics Sarah Mayes

Preliminary Exam Topics Sarah Mayes 1. Sheaves Definition of a sheaf Definition of stalks of a sheaf Definition and universal property of sheaf associated to a presheaf [Hartshorne, II.1.2] Definition

### Algebraic Cobordism Lecture 1: Complex cobordism and algebraic cobordism

Algebraic Cobordism Lecture 1: Complex cobordism and algebraic cobordism UWO January 25, 2005 Marc Levine Prelude: From homotopy theory to A 1 -homotopy theory A basic object in homotopy theory is a generalized

### Math 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

### 10. 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

### Synopsis of material from EGA Chapter II, 4. Proposition (4.1.6). The canonical homomorphism ( ) is surjective [(3.2.4)].

Synopsis of material from EGA Chapter II, 4 4.1. Definition of projective bundles. 4. Projective bundles. Ample sheaves Definition (4.1.1). Let S(E) be the symmetric algebra of a quasi-coherent O Y -module.

### NOTES ON THE DEGREE FORMULA

NOTES ON THE DEGREE FORMULA MARKUS ROST The major aim of this text is to provide a proof of Remark 10.4 in [1]. I am indebted to A. Suslin for helpful and encouraging comments. 1. The invariants ρ i Let

### AN 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,

### Reid 5.2. Describe the irreducible components of V (J) for J = (y 2 x 4, x 2 2x 3 x 2 y + 2xy + y 2 y) in k[x, y, z]. Here k is algebraically closed.

Reid 5.2. Describe the irreducible components of V (J) for J = (y 2 x 4, x 2 2x 3 x 2 y + 2xy + y 2 y) in k[x, y, z]. Here k is algebraically closed. Answer: Note that the first generator factors as (y

### 1 Flat, Smooth, Unramified, and Étale Morphisms

1 Flat, Smooth, Unramified, and Étale Morphisms 1.1 Flat morphisms Definition 1.1. An A-module M is flat if the (right-exact) functor A M is exact. It is faithfully flat if a complex of A-modules P N Q

### FOUNDATIONS 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,

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27 RAVI VAKIL CONTENTS 1. Quasicoherent sheaves of ideals, and closed subschemes 1 2. Invertible sheaves (line bundles) and divisors 2 3. Some line bundles on projective

### Vector 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

### BRIAN OSSERMAN. , let t be a coordinate for the line, and take θ = d. A differential form ω may be written as g(t)dt,

CONNECTIONS, CURVATURE, AND p-curvature BRIAN OSSERMAN 1. Classical theory We begin by describing the classical point of view on connections, their curvature, and p-curvature, in terms of maps of sheaves

### Matrix factorizations over projective schemes

Jesse Burke (joint with Mark E. Walker) Department of Mathematics University of California, Los Angeles January 11, 2013 Matrix factorizations Let Q be a commutative ring and f an element of Q. Matrix

### Equivariant Algebraic K-Theory

Equivariant Algebraic K-Theory Ryan Mickler E-mail: mickler.r@husky.neu.edu Abstract: Notes from lectures given during the MIT/NEU Graduate Seminar on Nakajima Quiver Varieties, Spring 2015 Contents 1

### FOUNDATIONS 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

### COMPLEX ALGEBRAIC SURFACES CLASS 4

COMPLEX ALGEBRAIC SURFACES CLASS 4 RAVI VAKIL CONTENTS 1. Serre duality and Riemann-Roch; back to curves 2 2. Applications of Riemann-Roch 2 2.1. Classification of genus 2 curves 3 2.2. A numerical criterion

### mult V f, where the sum ranges over prime divisor V X. We say that two divisors D 1 and D 2 are linearly equivalent, denoted by sending

2. The canonical divisor In this section we will introduce one of the most important invariants in the birational classification of varieties. Definition 2.1. Let X be a normal quasi-projective variety

### Introduction and preliminaries Wouter Zomervrucht, Februari 26, 2014

Introduction and preliminaries Wouter Zomervrucht, Februari 26, 204. Introduction Theorem. Serre duality). Let k be a field, X a smooth projective scheme over k of relative dimension n, and F a locally

### The Grothendieck Ring of Varieties

The Grothendieck Ring of Varieties Ziwen Zhu University of Utah October 25, 2016 These are supposed to be the notes for a talk of the student seminar in algebraic geometry. In the talk, We will first define

### Dot Products, Transposes, and Orthogonal Projections

Dot Products, Transposes, and Orthogonal Projections David Jekel November 13, 2015 Properties of Dot Products Recall that the dot product or standard inner product on R n is given by x y = x 1 y 1 + +

### Lecture 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, 1-5]. 1.1 Open and Closed Subschemes If (X, O X ) is a

### Algebraic Cobordism. 2nd German-Chinese Conference on Complex Geometry East China Normal University Shanghai-September 11-16, 2006.

Algebraic Cobordism 2nd German-Chinese Conference on Complex Geometry East China Normal University Shanghai-September 11-16, 2006 Marc Levine Outline: Describe the setting of oriented cohomology over a