# COMPLEX ALGEBRAIC SURFACES CLASS 9

Size: px
Start display at page:

Transcription

1 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 of Singularities, and de Jong s Alternation Theorem.) Recap of last time. Proof of the universal property of blowing up. Exercise. The blow-up Bl p S S has a universal property in the other direction : every morphism f from Bl p S to a variety X to a variety X that contracts E to a point factors through S. (Hint, basically proof: reduce first to the case X affine, then X = A n, then to X = A 1, so we re talking about functions on S {p}. But every function on S {p} extends over S.) Minimal surfaces: those with no ( 1)-curves. To understand surfaces up to birational or biregular invariance, we should focus on minimal surfaces. Theorem (Castelnuovo s contractibility criterion). rational curve with E 2 = 1. Let S be a surface and E S a I won t prove this in its entirety, but will say the strategy. We ll produce a line bundle that will give a map to projective space that will preserve S E, but collapse E to a point. Then the harder part is to show that the point is a smooth point of the new surface. I ll omit that: the proofs involve some infinitesimal analysis beyond what I claimed are the prerequisites of the course. 1. CONSTRUCTION OF CASTELNUOVO S CONTRACTION MAP Choose a very ample divisor on S. By taking an appropriate multiple, we may assume that H 1 (S, O(H)) = 0 by Serre vanishing. Suppose k = H E, and consider the line bundle O(H + ke). Note that (H + ke) E = 0; this is the motivation for taking this divisor. Date: Wednesday, October 30. 1

2 We ll now show that O(H + ke) is basepoint free (and hence gives a map to projective space), and that it just collapses E and keeps the rest of the surface the way it was. First, we ll show that H 1 (X, O(H + ke)) = 0. Fix a section t of O S (E) vanishing along E. Take the exact sequence 0 O S ( E) t O S O E 0. The morphism labeled t means multiply by t. Twist this by H + ie to get 0 O S (H + (i 1)E) t O S (H + ie) O E (k i) 0. By Serre duality, H 1 (E, O E (r)) = H 0 (E, O E ( 2 r)), which is 0 for r 0, so taking the long exact sequence we get: 0 H 0 (S, O S (H + (i 1)E)) t H 0 (S, O S (H + ie)) H 0 (E, O E (k i)) H 1 (S, O S (H + (i 1)E)) t H 1 (S, O S (H + ie)) 0. for 1 i k. By induction on i, H 1 (S, O S (H + ie)) = 0 for 1 i k, so we have: 0 H 0 (S, O S (H + (i 1)E)) t H 0 (S, O S (H + ie)) H 0 (E, O E (k i)) 0. So we can build up H 0 (S, O S (H + ke)) inductively. Choose a basis s 0,..., s n of H 0 (S, O S (H)). Then using i = 1, the sections of H 0 (S, O S (H + E)) correspond to ts 0,..., ts n, plus k terms coming from H 0 (E, O E (k 1)). Using i = 2, the sections of H 0 (S, O S (H + 2E)) correspond to t 2 s 0,..., t 2 s n, plus k terms t, plus k 1 terms coming from H 0 (E, O E (k 2)). Keep doing this k times to get: t k s 0,..., t k s n, plus k terms ( t k 1 ) plus k 1 terms ( t k 2 ) plus plus 1 term ( t 0 ) By looking just at the first n + 1 terms, we see that there are no basepoints away from (t = 0) = E. For a point on E, all the terms are non-zero except for the last one. Thus we have a basepoint free map, that sends E to [0; ; 0; 1]. By looking at the first n + 1 terms, we see that it is an embedding everywhere else: it separates points and tangent vectors. That completes the construction of the map. However, we can see the blow-up as the resolution of the projection from [0; ; 0; 1]... The proof uses infinitesimal analysis, and I ll omit it. (If time, give them a sketch. The image surface is smooth at the key point p = [0;... ; 0; 1], if m/m 2 has dimension 2 as a k-vector space, i.e. O/m 2 has dimension 3. It is singular if the dimension is bigger than 3. We can pull back this scheme in the target, by intersecting x 2 0 = 0,..., x 2 n 1 = 0. We can pull back these equations to our original surface, to get as an equation the square of the ideal E. This has a 3-dimensional space of sections. All that needs to be checked is that the pullback of the first-order neighborhood of p really is the dimension of the pre-image scheme; this requires infinitesimal analysis.) 2

3 2. RULED SURFACES A surface is ruled if it is birationally equivalent to C P 1, where C is a smooth curve. (More generally, a variety is ruled if through every curve, there is a P 1.) Examples: 1. C P P a P 1 -bundle over C (i.e. something that Zariski-locally on C is isomorphic to P 1 C). 4. Special case of above: if E is a rank 2 vector bundle over C, then its projectivization is a ruled surface over C. Informal description. In fact, we ll see that this is the same as (3) above. Precise definition: F is a rank 2 locally free sheaf. Then the corresponding ruled surface is Proj C i=0 Sym i F. Question over C: Suppose you had a surface S C such that the fibers are all isomorphic to CP 1. (This is called geometrically ruled.) Is S necessarily a P 1 -bundle over C? Those of type (3) above are geometrically ruled. An important example are the following rational (geometrically) ruled surfaces over P 1, sometimes called Hirzebruch surfaces: the projectivization of the locally free sheaves O P 1 O P 1(n), where n 0. Called F n. In fancy notation F n = P P 1(O P 1 O P 1(n)). We ll see in a class or so that these are all different, and these are all the geometrically ruled surfaces over P 1. You may naively think that geometrically ruled is the same as (3) (P 1 -bundles). And you would be right, but it is not obvious. Here is an example showing that your intuition may be wrong. Example. There is a map from a threefold to a surface that is geometrically ruled but not of form (3). Instead, I ll give an example a sixfold fibered over a fivefold. Consider the P 5 parametrizing plane conics, with point [a x 2; a xy ; a y 2; a xz ; a yz ; a z 2] corresponding to conic a x 2x a z 2z 2 = 0 in P 2. There is a universal conic: U = {([x; y; z], [a x 2; ; a z 2]) : a x 2x a z 2z 2 = 0} P 2 P 5 P 5 Throw out the locus in P 5 corresponding to reducible conics: U U P 2 P 5 P 5 P 5 Fact: This isn t a trivial P 1 -bundle over any Zariski-open set. Argument is short, and if you re curious, ask me about it. Key idea: if it were, then there would be a divisor intersecting the class of a fiber with multiplicity 1. (Describe why.) We can work out the 3

4 topology of U explicitly, and show that any divisor meets the class of a fiber with even multiplicity. More strongly: if you restrict to the generic point, i.e. to the function field of P 5, then you don t get P 1 over that field. Arithmetic version of that same comment. Those of you who are more arithmetically inclined will be less surprised by this example. (Skip in class.) For example, any conic in the complex plane is isomorphic to the projective line, but over a trickier field such as Q this isn t true. For example, x 2 + y 2 + z 2 = 0 in P 2 R is not isomorphic to P1 R. This is just an artifact of working over a field that isn t algebraically closed. And this example shows that even if you think you only care about algebraically closed fields, such as C, then you may still come across non-algebraically closed fields, such as function fields of curves, or varieties in general. Noether-Enriques Theorem. Suppose π : S C is geometrically ruled. Then S is of type (3) above, i.e. it is the projectivization of some rank 2 invertible sheaf / vector bundle. Slightly more generally: Suppose π : S C, and x C such π is smooth over C and π 1 (x) is isomorphic to P 1. Then there is a Zariski-open subset U C containing x and a commutative diagram π 1 (U) U P 1 π U. Proof. Three-step proof. The key step is 2, where we produce a divisor H of S meeting the class of a fiber with multiplicity 1. Step 1: H 2 (S, O S ) = 0. Let F be the class of π 1 (x). F 2 = 0. By the genus formula, 2 = F K. Suppose otherwise that H 2 (S, O S ) > 0, so by Serre duality K has a section. Let K be the divisor associated to the zero set of that section. Then K = nf + K, where K has no F -component. Finally, 2 = K F = (K + nf ) F 0, contradiction. Step 2: There is a divisor H of S such that H F = 1. Here we work over C. There is another proof in Hartshorne Chapter V which works over any field. Recall that from the long exact sequence for 0 Z O S O S 1, we have Pic(S) H2 (S, Z) H 2 (S, O S ) = 0 using Step 1. Thus it suffices to find a class h H 2 (S, Z) with h f = 1, where f is the image of F in H 2 (S, Z). Consider the map of Z-modules H 2 (S, Z) Z given by a a f. The image is an ideal of Z, say dz for d 0. Clearly non-zero: take a very ample. We want to show that d = 1. The map a (a f)/d is a linear form on H 2 (S, Z). By Poincare duality, H 2 (S, Z) H 2 (S, Z) H 4 (S, Z) = Z is a duality, i.e. H 2 (S, Z) Hom(H 2 (S, Z), Z) is a surjective (with kernel equal to the torsion subgroup). Thus there is an element f H 2 (S, Z) such that a f = (a f)/d for all a H 2 (S, Z). If d > 1, this class is a bit hard to imagine; let s get a contradiction. 4

5 Let k be the image of K in H 2 (S, Z). I claim that for any b H 2 (S, Z), F (b) = b 2 + b k is always even. Proof: it is always an integer. It s even for all irreducible curves. And it is additive modulo 2, i.e. F (b 1 + b 2 ) F (b 1 ) + F (b 2 ) (mod 2). Thus the following number is even: f f + f k = f f/4 + f k/2 = 2/d, and we re done. Interesting consequence we ll us later: there are no multiple fibers. (Draw picture.) (A multiple fiber is a multiple of an irreducible F by an integer greater than 1.) Reason: there is a divisor class meeting it with multiplicity 1 Step 3: next day. 5

### 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,

### 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

### 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 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

### INTERSECTION THEORY CLASS 19

INTERSECTION THEORY CLASS 19 RAVI VAKIL CONTENTS 1. Recap of Last day 1 1.1. New facts 2 2. Statement of the theorem 3 2.1. GRR for a special case of closed immersions f : X Y = P(N 1) 4 2.2. GRR for closed

### 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 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 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:

### 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 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 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

### 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

### 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!

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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.

### 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

### 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

### 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:

### 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

### 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

### 3. Lecture 3. Y Z[1/p]Hom (Sch/k) (Y, X).

3. Lecture 3 3.1. Freely generate qfh-sheaves. We recall that if F is a homotopy invariant presheaf with transfers in the sense of the last lecture, then we have a well defined pairing F(X) H 0 (X/S) F(S)

### Homework 4: Mayer-Vietoris Sequence and CW complexes

Homework 4: Mayer-Vietoris Sequence and CW complexes Due date: Friday, October 4th. 0. Goals and Prerequisites The goal of this homework assignment is to begin using the Mayer-Vietoris sequence and cellular

### MA 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

### Resolution 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.

### 14. Rational maps It is often the case that we are given a variety X and a morphism defined on an open subset U of X. As open sets in the Zariski

14. Rational maps It is often the case that we are given a variety X and a morphism defined on an open subset U of X. As open sets in the Zariski topology are very large, it is natural to view this as

### 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 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

### Introduction 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

### 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,

### 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

### Non-uniruledness results for spaces of rational curves in hypersurfaces

Non-uniruledness results for spaces of rational curves in hypersurfaces Roya Beheshti Abstract We prove that the sweeping components of the space of smooth rational curves in a smooth hypersurface of degree

### ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 4: MORE ABOUT VARIETIES AND REGULAR FUNCTIONS.

ALGERAIC GEOMETRY COURSE NOTES, LECTURE 4: MORE AOUT VARIETIES AND REGULAR FUNCTIONS. ANDREW SALCH. More about some claims from the last lecture. Perhaps you have noticed by now that the Zariski topology

### 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

### The Cone Theorem. Stefano Filipazzi. February 10, 2016

The Cone Theorem Stefano Filipazzi February 10, 2016 These notes are supposed to be a handout for the student seminar in algebraic geometry at the University of Utah. In this seminar, we will give an overview

### 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

### 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:

### Algebraic 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,

### 7. Classification of Surfaces The key to the classification of surfaces is the behaviour of the canonical

7. Classification of Surfaces The key to the classification of surfaces is the behaviour of the canonical divisor. Definition 7.1. We say that a smooth projective surface is minimal if K S is nef. Warning:

### 0. Introduction 1 0. INTRODUCTION

0. Introduction 1 0. INTRODUCTION In a very rough sketch we explain what algebraic geometry is about and what it can be used for. We stress the many correlations with other fields of research, such as

### 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

### PROBLEMS FOR VIASM MINICOURSE: GEOMETRY OF MODULI SPACES LAST UPDATED: DEC 25, 2013

PROBLEMS FOR VIASM MINICOURSE: GEOMETRY OF MODULI SPACES LAST UPDATED: DEC 25, 2013 1. Problems on moduli spaces The main text for this material is Harris & Morrison Moduli of curves. (There are djvu files

### ALGORITHMS FOR ALGEBRAIC CURVES

ALGORITHMS FOR ALGEBRAIC CURVES SUMMARY OF LECTURE 7 I consider the problem of computing in Pic 0 (X) where X is a curve (absolutely integral, projective, smooth) over a field K. Typically K is a finite

### On Mordell-Lang in Algebraic Groups of Unipotent Rank 1

On Mordell-Lang in Algebraic Groups of Unipotent Rank 1 Paul Vojta University of California, Berkeley and ICERM (work in progress) Abstract. In the previous ICERM workshop, Tom Scanlon raised the question

### NONSINGULAR 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

### 12. 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

### Algebraic 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......................

### Pacific Journal of Mathematics

Pacific Journal of Mathematics STABLE REFLEXIVE SHEAVES ON SMOOTH PROJECTIVE 3-FOLDS PETER VERMEIRE Volume 219 No. 2 April 2005 PACIFIC JOURNAL OF MATHEMATICS Vol. 219, No. 2, 2005 STABLE REFLEXIVE SHEAVES

### 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:

### 9. Birational Maps and Blowing Up

72 Andreas Gathmann 9. Birational Maps and Blowing Up In the course of this class we have already seen many examples of varieties that are almost the same in the sense that they contain isomorphic dense

### 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

### Topics in Algebraic Geometry

Topics in Algebraic Geometry Nikitas Nikandros, 3928675, Utrecht University n.nikandros@students.uu.nl March 2, 2016 1 Introduction and motivation In this talk i will give an incomplete and at sometimes

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 45

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 45 RAVI VAKIL CONTENTS 1. Hyperelliptic curves 1 2. Curves of genus 3 3 1. HYPERELLIPTIC CURVES A curve C of genus at least 2 is hyperelliptic if it admits a degree

### a 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 4-dimensional projective space

### Introduction to Arithmetic Geometry Fall 2013 Lecture #24 12/03/2013

18.78 Introduction to Arithmetic Geometry Fall 013 Lecture #4 1/03/013 4.1 Isogenies of elliptic curves Definition 4.1. Let E 1 /k and E /k be elliptic curves with distinguished rational points O 1 and

### 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 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

### ABSTRACT 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 n-space the collection A n k of points P = a 1, a,..., a

### Isogeny invariance of the BSD conjecture

Isogeny invariance of the BSD conjecture Akshay Venkatesh October 30, 2015 1 Examples The BSD conjecture predicts that for an elliptic curve E over Q with E(Q) of rank r 0, where L (r) (1, E) r! = ( p

### Geometric Class Field Theory

Geometric Class Field Theory Notes by Tony Feng for a talk by Bhargav Bhatt April 4, 2016 In the first half we will explain the unramified picture from the geometric point of view, and in the second half

### NOTES ON DIVISORS AND RIEMANN-ROCH

NOTES ON DIVISORS AND RIEMANN-ROCH NILAY KUMAR Recall that due to the maximum principle, there are no nonconstant holomorphic functions on a compact complex manifold. The next best objects to study, as

### 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,

### (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

### 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

### Oral exam practice problems: Algebraic Geometry

Oral exam practice problems: Algebraic Geometry Alberto García Raboso TP1. Let Q 1 and Q 2 be the quadric hypersurfaces in P n given by the equations f 1 x 2 0 + + x 2 n = 0 f 2 a 0 x 2 0 + + a n x 2 n

### Algebraic Curves and Riemann Surfaces

Algebraic Curves and Riemann Surfaces Rick Miranda Graduate Studies in Mathematics Volume 5 If American Mathematical Society Contents Preface xix Chapter I. Riemann Surfaces: Basic Definitions 1 1. Complex

### Part III Positivity in Algebraic Geometry

Part III Positivity in Algebraic Geometry Theorems with proof Based on lectures by S. Svaldi Notes taken by Dexter Chua Lent 2018 These notes are not endorsed by the lecturers, and I have modified them

### ALGEBRAIC GEOMETRY I, FALL 2016.

ALGEBRAIC GEOMETRY I, FALL 2016. DIVISORS. 1. Weil and Cartier divisors Let X be an algebraic variety. Define a Weil divisor on X as a formal (finite) linear combination of irreducible subvarieties of

### CHAPTER 1. TOPOLOGY OF ALGEBRAIC VARIETIES, HODGE DECOMPOSITION, AND APPLICATIONS. Contents

CHAPTER 1. TOPOLOGY OF ALGEBRAIC VARIETIES, HODGE DECOMPOSITION, AND APPLICATIONS Contents 1. The Lefschetz hyperplane theorem 1 2. The Hodge decomposition 4 3. Hodge numbers in smooth families 6 4. Birationally

### CORRESPONDENCE BETWEEN ELLIPTIC CURVES IN EDWARDS-BERNSTEIN AND WEIERSTRASS FORMS

CORRESPONDENCE BETWEEN ELLIPTIC CURVES IN EDWARDS-BERNSTEIN AND WEIERSTRASS FORMS DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF OTTAWA SUPERVISOR: PROFESSOR MONICA NEVINS STUDENT: DANG NGUYEN

### GEOMETRIC CLASS FIELD THEORY I

GEOMETRIC CLASS FIELD THEORY I TONY FENG 1. Classical class field theory 1.1. The Artin map. Let s start off by reviewing the classical origins of class field theory. The motivating problem is basically

### 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

### This is a closed subset of X Y, by Proposition 6.5(b), since it is equal to the inverse image of the diagonal under the regular map:

Math 6130 Notes. Fall 2002. 7. Basic Maps. Recall from 3 that a regular map of affine varieties is the same as a homomorphism of coordinate rings (going the other way). Here, we look at how algebraic properties

### 11. Dimension. 96 Andreas Gathmann

96 Andreas Gathmann 11. Dimension We have already met several situations in this course in which it seemed to be desirable to have a notion of dimension (of a variety, or more generally of a ring): for

### 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

### We can choose generators of this k-algebra: s i H 0 (X, L r i. H 0 (X, L mr )

MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 43 5.3. Linearisations. An abstract projective scheme X does not come with a pre-specified embedding in a projective space. However, an ample line bundle

### Math 145. Codimension

Math 145. Codimension 1. Main result and some interesting examples In class we have seen that the dimension theory of an affine variety (irreducible!) is linked to the structure of the function field in

### Corollary. Let X Y be a dominant map of varieties, with general fiber F. If Y and F are rationally connected, then X is.

1 Theorem. Let π : X B be a proper morphism of varieties, with B a smooth curve. If the general fiber F of f is rationally connected, then f has a section. Corollary. Let X Y be a dominant map of varieties,

### Math 797W Homework 4

Math 797W Homework 4 Paul Hacking December 5, 2016 We work over an algebraically closed field k. (1) Let F be a sheaf of abelian groups on a topological space X, and p X a point. Recall the definition

### AFFINE 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 C-linear. 1.

### Introduction to Arithmetic Geometry Fall 2013 Lecture #23 11/26/2013

18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #23 11/26/2013 As usual, a curve is a smooth projective (geometrically irreducible) variety of dimension one and k is a perfect field. 23.1

### Holomorphic line bundles

Chapter 2 Holomorphic line bundles In the absence of non-constant holomorphic functions X! C on a compact complex manifold, we turn to the next best thing, holomorphic sections of line bundles (i.e., rank

### Exercises for algebraic curves

Exercises for algebraic curves Christophe Ritzenthaler February 18, 2019 1 Exercise Lecture 1 1.1 Exercise Show that V = {(x, y) C 2 s.t. y = sin x} is not an algebraic set. Solutions. Let us assume that

### CHEVALLEY S THEOREM AND COMPLETE VARIETIES

CHEVALLEY S THEOREM AND COMPLETE VARIETIES BRIAN OSSERMAN In this note, we introduce the concept which plays the role of compactness for varieties completeness. We prove that completeness can be characterized

### ARITHMETICALLY COHEN-MACAULAY BUNDLES ON HYPERSURFACES

ARITHMETICALLY COHEN-MACAULAY BUNDLES ON HYPERSURFACES N. MOHAN KUMAR, A. P. RAO, AND G. V. RAVINDRA Abstract. We prove that any rank two arithmetically Cohen- Macaulay vector bundle on a general hypersurface

### LECTURE 6: THE ARTIN-MUMFORD EXAMPLE

LECTURE 6: THE ARTIN-MUMFORD EXAMPLE In this chapter we discuss the example of Artin and Mumford [AM72] of a complex unirational 3-fold which is not rational in fact, it is not even stably rational). As

### Proof of Langlands for GL(2), II

Proof of Langlands for GL(), II Notes by Tony Feng for a talk by Jochen Heinloth April 8, 016 1 Overview Let X/F q be a smooth, projective, geometrically connected curve. The aim is to show that if E is

### PICARD 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

### Exercises 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 E-mail address: alex.massarenti@sissa.it These notes collect a series of

### Mini-Course on Moduli Spaces

Mini-Course 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, One-dimensional

### Density of rational points on Enriques surfaces

Density of rational points on Enriques surfaces F. A. Bogomolov Courant Institute of Mathematical Sciences, N.Y.U. 251 Mercer str. New York, (NY) 10012, U.S.A. e-mail: bogomolo@cims.nyu.edu and Yu. Tschinkel

### 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