CHAPTER O Introduction

0.1 What is algebraicgeometry

Algebraic geometry is the studyof geometrical objects defined by polynomialequations This is a veryold subject thinkofthe study of conics in theclassical antiquity which is related tomanyothersubjects

numbertheory

physics

combinatorics

topology

complexanalysis

Classically one of themain objects are algebraicvarieties One considers algebraicsubsets

DefinitionLet k be a field An algebraic subset of ATK le is a subset

defined by a finite familyof polynomial equations Pi Pr C KIK XD2 z ai xn C K PCR P z

But an algebraicvariety is more than just a at ofpoints first it is also a topological

space closed subsets beinggivenby algebraic subsets One need also toconsiderfunctions on thistopologicalspace the socalled regular functions defined locally

by polynomial equationsBesides one need also to allowmore objects tobe able toglue geometrical objects

together anddefine eg projective spaces

This classical approachworks over an algebraicallyclosed field such as butfailsover more general fields Moreover Hilbert's Nullstellensatz establishes a beautiful

correspondence between algebraandgeometry inthat case

Modern algebraic geometry as introduced by Grothendieck goesbeyond thislimitationand introducesthenotion ofscheme which turns any commutativeringinto ageometrical object

This generalization has become thecentralobject in algebraic geometryeven over an

algebraicallyclosed field formanyreasonsallows todogeometrywith non reduced ringsallows to do geometry over finite fields Weil conjectures ortheringofintegers Gir of a number field K Cie finiteextension ofQ

This brings together arithmetic and geometry

0.2 What is algebraicgeometrygood for

Manythingsactually By turning algebra into geometryand vice versa it providesnew

insights to solve problems It brings the geometrical viewpointin arithmetic and in

algebraic geometry the following become deeply connected algebra geometry topology

0.3 What is the goal ofthiscourse

In this course I hope togive an introduction to algebraicgeometry with a view

towards intersectiontheory Let's sketch part of the content of thiscourse

Let te be a fieldand X be a bi variety that is a separated k scheme

offinitetype The main objects in intersection theory are algebraiccycles2 X Z isomorphismdamesofclosed subschernes of X

that are integral and of codimensionr

One mainalso use a gradingbydimension

Zr X Z isomorphismdamesofclosed subschernes of Xthat are integralandofdimension r

Whyconsider a freeabeliangroup The purpose of intersection theory is to

understand what happenswhen closed subschernes intersect It turns out that

irreducible reducedclosed subschernes that is integral are the buildingblocks

and that multiplicities have to be considered To understandthis remember that

a polynomial Pot degreed has d roots in a bigenoughfield it counted with

multiplicities These roots are the intersection points with oftheline defined by x Oin the plane and the graph of P

The next step requires considering a quotient of the groupof cycles we mod

out by rational equivalence to define theChowgroupsCHEX and CHR X

One of the main results at the basis of intersection theory is the intersection

pairing If X is Smooth over te there is an intersection pairingCHP X CH9 X CHP x

which makes CH X into a ringBézout theorem on the intersection of planecurves is a numerical consequence of the

existence of thispairingHistorically this was constructed via studyof multiplicities andmovinglemmas

Fulton has developed an approach without movinglemmas which uses the déformation

to the normal cone a particular example of blowup

0.4What is intersectiontheorygood for

It is an essential tool in algebraic geometry since its beginning a tool sharpened

and improved alongthecenturies Like cohomology Chowgroups are a tool tounderstand

the geometry and topologyofvarieties It is alsoimportantfor encenerative geometryUnlike cohomology groups Chow groups ofvarieties are usually not finitedimensional

i e CHKX q CHR X zOh is not a finitedimensional vs They are usually

huge an their structure is subject to deep conjectures related to motives has

imagined by Grothendieck