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Algebraic Geometry

 

 

 

 

 

Algebraic geometry is one of the oldest and vastest

branches of mathematics. Besides being an active field of

research for many centuries, it plays a central role in

number theory, differential geometry, group theory,

mathematical physics and other branches of science.

           

In McGill, algebraic geometry is represented by Peter Russell and Eyal Goren

and is very much connected to the interests of other members of the department:

Henri Darmon, Jacques Husrtubise and Niky Kamran. We are running a joint

seminar with CICMA, Université de Montreal and UQAM. In addition there

are courses given in algebraic geometry and related areas. Check it out!

 

 

 

 

 

 

 

 

 

 

What is Algebraic Geometry?

 

Initially, algebraic geometry was concerned with the study of curves in the plane

and really started boosting up with the discovery of projective geometry. In this

geometry, the line k (k any field) is completed by adding one point at infinity

resulting in the projective line P1, the plane k2 is completed by adding a projective

line at infinity to yield the projective plane P2. Thus P2= k2 ÈP1 =  k2 Èk È{pt}.

 

On a more conceptual level, the projective space Pn over a field k is the parameter

space of lines in kn+1. Such a line is determined by a non-zero point on it

(a0, …, an), where one is allowed to replace (a0, …, an) by (ta0, …, tan) for any t ¹ 0.

We denote such a class of points by (a0:…: an). Thus Pn is the collection of (a0:…: an).

The points with a1 ¹ 0 may be normalized so that a1 = 1. Then a1, …, an in

(1: a1:…: an) are determined uniquely and are in one to one correspondence with kn

by the map (1: a1:…: an) ® (a1,…, an). The complement is the points (0: a1:…: an),

which are precisely Pn -1 . Thus we get an inductive dissection

Pn  = kn È kn-1 È … È k È {*} where * is the point (0:…:0:1).

 

One of the nicest theorems is

 

Bezout’s theorem: Two curves C and D in P2, defined by equations of degree M and N

intersect at MN points, counting multiplicity.

 

 

 

 

 

 

 

 

 

 

 

 

 

 


In particular, every two lines intersect at a unique point. This last fact was in fact one of

the drives to develop projective geometry. It came from the invention of perspective in

art and architecture. Invention whose roots are in the world around us:

 

 

 

 

 

 

 

 

 

 

For long time geometry was the study of projective geometry. By that one means the

study of sets defined as the set of solutions for a given system of homogeneous

polynomials. Such sets look like a set Y Í kn such that the points of Y are the common

solutions to a system of polynomials {f1, …, fr) in n variables, and where one adds to Y

its limit points at infinity.

 

 

 

 

 

 

 

 

 

 

 

One is interested in intrinsic properties of such sets, properties not depending on metric

and coordinates. Properties like dimension, singular points and others. In fact,

a characteristic research problem is finding such intrinsic properties.

 

 

Much effort went into freeing algebraic geometry from the necessity of dealing with sets

defined by a particular collection of equations. This was to serve not only for solving

some technical problems at the time, but also to signal out precisely what are the objects

of study to themselves before we incorporate them in any particular way into a space.

This is very similar to the way manifolds are studied today via the definition of atlas.

 

Another problem was to find a way of actually saying what is geometry. One can think

about geometry as a collection of objects, where for any two objects it is known whether

“one lies on the other”, and where it is known what a regular function on such a structure

is.

 

Yet another problem is putting on a firm ground the “feeling” that a family of algebraic

sets (say a family of curves) varying smoothly in time is “like” a curve defined using

equations with integer coefficients where for every prime number one can consider the

curve modulo this prime. Hence the curve varies over the parameter space whose points

are primes.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

All these problems, and many other problems coming from the attempt to generalize

known results in complex geometry to results in algebraic geometry, lead to the

development of modern algebraic topology. The chief inventor and force behind which

was Grothendieck with many deep contributions by Dieudonné, Serre, Deligne and

many others.

In this theory of schemes, the basic objects are composed of rings. The objects of the

geometry are the prime ideals of the ring and the notion of “lying on” is given by

“containing”. The functions are given by elements of the ring itself. Mysterious as it

seems, this makes for a totally coherent geometric theory, where such spaces (called

“spectra”) can be glued together, and where any classical geometric notion can be

defined.

 

Today, algebraic geometry is composed of many different branches. Some are closely

related to complex analysis (geometry over the complex numbers), some are closely

related to number theory (moduli spaces), some to mathematical physics (Calabi-Yau

manifolds and mirror symmetry), some to algebraic topology (elliptic cohomology, étale

cohomology, topoi) etc.

 

Apart from those, as any developed branch of science, algebraic geometry is troubled

with its own house keeping: How many curves of a degree 3 pass through 9 points in the

plane? How many lines are there on a cubic surface? How to parameterize all the

varieties in Pn of some given invariants? How to understand group actions on a given

space and existence of a quotient? What is the structure and properties of algebraic

groups? What do the automorphisms of an algebraic variety look like? etc… etc…

 

Members of the department studying algebraic geometry are Peter Russell and

Eyal Goren. Members whose interests are bordering with algebraic geometry are

Niky Kamran, Jacques Hurtubise, Henri Darmon and many of the members of CICMA.

 

 

 

 

 

 

Individual research interests:

 

 

Eyal Goren:  I am interested in questions where number theory is studied via geometry.

Specific key words are modular forms, moduli spaces, elliptic curves, abelian varieties,

p-divisible groups, L functions.

In the last couple of years I was working on the ambitious project of bringing our

knowledge of moduli spaces of abelian varieties and modular forms on them to the level

of our knowledge in the case of elliptic curves. Here we comment that an abelian variety

is a complete algebraic group

(over C it is topologically a torus) 

and the moduli space is a variety that

parameterize all those abelian varieties.

Every variety gives a point in the space and

a curve C in the moduli space gives a family of

abelian varieties parameterized by C.

In the case of elliptic curves this theory played a crucial role in the proof of Fermat’s last

theorem. It is therefore expected that much deep mathematics is to be discovered in the

study of such moduli spaces, connecting questions from number theory (in particular

Diophantine equations and modular forms) to questions on the geometry of these moduli

spaces.

Specific questions I have been working on together with E. Bachmat and F. Oort are the

geometry of Hilbert modular varieties in positive characteristic (stratification, variation of

Newton polygons, singularities), p-adic and mod p Hilbert modular forms and p-divisible

groups with real multiplication.

 

 

 

 

Peter Russell: I have long been interested in studying affine spaces and closely related varieties (e.g., over C, contractible affine varieties) as algebraic varieties in their own right. The famous cancellation problem, for instance, asks whether X is an affine space provided that X x C is an affine space. This is known if the dimension of X is at most 2.

Closely related is the quest to understand the automorphism group of affine space Cn, still a mystery for n bigger than 2. A quite recent result (obtained in collaboration with M. Koras and others) says that C* actions on C3 are linear in a suitable coordinate system. Again, this is not known for Cn, n >3.

Among my other interests are the study of embeddings of rational curves in the affine plane (generically rational curves, closed embeddings of C*, …); some special topics in positive characteristic geometry (purely inseparable forms, Frobenius sandwiches, …).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Activities in Algebraic Geometry

 

 

Courses (current and recent years):

1997-1998      “Commutative algebra and algebraic geometry” (Algebre 1). UdM. MAT 6608. Lecturer: B. Broer. Automn 97.

 

Topics in Algebra III: “Approaches to the Jacobian Problem”. McGill. 189-  722A. Lecturer: A. Sathaye.

 

1998-1999      “Algebraic surfaces”: A semester course leading to the study of systems of curves on an algebraic surface and the notion of Kodaira dimension for compact and non-compact surfaces.

 

“Algebraic Curves”. McGill. 189-612B. Lecturer: J. Hurtubise.

 

1999-2000      A year long course in algebraic geometry, from classical theory to schemes is offered by McGill. The course is aimed at the graduate, or advanced undergraduate level. It is called “Topics in Geometry and Topology 189-706A/707B”. Pre-requisites are some basic algebra and topology. For more details go here.

 

The graduate students seminar is devoted this year to Faltings’  Theorems. The proofs involve sophisticated tools from arithmetic algebraic geometry and algebraic number theory.

 

 

 

Seminars:

 

Algebraic Geometry Seminar.

 

Quebec-Vermont Number Theory Seminar.

 

graduate students seminar.

 

See also the activities of

CICMA