What Are the Key Foundations of Algebraic Geometry?

[mathwonk]

advertisement for an algebraic geometry course

ALGEBRAIC GEOMETRY: MATH 8300: A GRAD. COURSE TO BE OFFERED FALL 2001.

Just as vector spaces are the geometric side of matrices, so algebraic varieties are the geometric aspect of polynomials. Since polynomials occur everywhere their geometry is fundamental. More explicitly, a linear subspace of R^n, or C^n, is the solution set of a finite system of linear equations, and an algebraic subvariety of R^n, or C^n, is the solution set of a finite system of polynomial equations. Even more than with vector spaces, the notion of algebraic variety permits geometric intuition to be brought to bear on a wide variety of problems, from pure algebra, ring theory, and number theory, to topology, real and complex analysis, differential equations and mathematical physics. Conversely, these subjects illumine and provide tools for algebraic geometry.

Algebraic varieties are an unusually rich source of interesting examples. The study of four-manifolds in topology has long been concerned with those which occur as complex algebraic surfaces, especially since the work of Simon Donaldson (Fields medalist in 1986). Historically the fundamental result on compact Riemann surfaces, is that every compact complex one-manifold is the Riemann surface of some "algebraic function", i.e. of some algebraic curve in P^2.

(Example: The fact that every compact complex one-manifold M of genus one has the form C/L for some lattice L in C is rather deep, but if we assume this we can represent M as a plane cubic curve using "elliptic functions" from complex analysis, as follows: the famous differential equation (P' )^2=4 P^3 - g2 P - g3 , for the Weierstrass P function (cf. Ahlfors) implies that the complex torus C/L is mapped by the pair of meromorphic functions ( P, P' ) to (the projectivization of) the non singular algebraic curve with equation y^2=4x^3-g2 x-g3. Since P has degree two in the period parallogram and P' is odd, this is an injection, hence an isomorphism.)

If a single equation in two variables can give rise to every compact complex one-manifold, just imagine how rich is the field of examples provided by arbitrary systems of equations in n variables! Furthermore, Grothendieck in the 50's and 60's generalized this classical setting enormously, to one in which every commutative ring can be considered the ring of regular functions on some abstract algebraic variety! From Grothendieck's point of view, commutative ring theory and algebraic number theory are special cases of algebraic geometry. Today some beginnings are being made also in non-commutative algebraic geometry, especially its links with the representation theory of groups and algebras.

Research into classification of classical algebraic varieties is most advanced in (but is not restricted to) the cases of one, two, and three dimensions, with the one dimensional case highly evolved but not at all completely understood, the two dimensional case still appearing to offer many unsolved problems, and the three dimensional case only recently beginning to emerge from the category of mostly uncharted territory, with the work of Mori, Kollar, and others.

For example the deceptively simple (but decades old) question of whether the two Fermat equations x^3+y^3+z^3+w^3=0 and x+y+z+w=0 have essentially isomorphic solution sets was only settled (negatively) in 1972, by C.H.Clemens and P.A.Griffiths in a famous 90 page Annals paper ! Over non -algebraically -closed fields the questions are even harder.

The famous Mordell conjecture in number theory, solved in the 80's by Faltings, was to decide whether a non singular algebraic plane curve over Q could have an infinite number of points when its Riemann surface over C has genus > 1. The answer, which required the refinement to the "arithmetic case" of much of the machinery of modern algebraic geometry, is "no", (as Mordell had conjectured).

The "Last Theorem" of Fermat (recently proved) is the assertion that in the special case of the curve x^n+y^n+z^n=0, whose Riemann surface has genus g=(1/2)(n-1)(n-2), that the finite number of points is actually zero (even when g>0, i.e. n>2).

One of the most fascinating topics in algebraic geometry, and one of my own favorites, is the study of "moduli spaces", a field initiated by Riemann. This topic investigates the consequences of the surprising and powerful point of view that in many cases the set of isomorphism classes of algebraic varieties, with fixed topological invariants, can itself be given the structure of an algebraic variety ! The best known case of this phenomenon is probably the familiar fact from complex analysis that the set of isomorphism classes of one dimensional compact complex tori correspond naturally, via the "J-invariant", to the set C of complex numbers.

Several outstanding algebraic geometers have considered, over the last decade, the problem of determining the "Kodaira" dimension of the moduli spaces Mg of curves of genus g, and the spaces An of "principally polarized" complex tori of dimension n. Still, I believe most cases of curves of genera 13< g < 24, and tori of dimension n = 6, remain open.

A very exciting and beautiful recent development is a revolution in the classical subject of enumerative geometry wrought by inputs from physics. Nineteenth century geometers knew well that a smooth complex projective cubic surface carries exactly 27 lines and even that a general quintic threefold carries 2875 lines, but related questions such as how many conics and rational cubic curves lie on the quintic, were resolved only recently, with much effort.

Then physicists produced a link between “Calabi Yau” manifolds such as the quintic theefold, and quantum field theories, deducing a formula containing the expected numbers Nd of rational curves of degrees d for all d, which allows one to solve recursively for Nd in terms of the Ne for smaller values e < d. It remains to understand the physicists answer and determine whether it is really correct, since they fearlessly give a candidate number even in cases where the true number of curves is unknown even to be finite! This problem is still open for all d ≥ 10.

The course beginning in the fall will be an introductory one to the fascinating field of algebraic geometry, with the excellent book Basic Algebraic Geometry by Shafarevich, as primary text. My hope is to cover roughly part I, "Algebraic Varieties in a Projective Space" which treats the semi-modern (circa 1940-50) general theory of algebraic sets in the classical setting of projective space, supplemented by examples from the book of Joe Harris. Parts II and III of Shafarevich contain a very rudimentary account of the idea of an abstract variety, and a lovely treatment of varieties over the complex numbers with relations to complex analysis.

The yellow covered “redbook” by Mumford covers abstract algebraic varieties and more advanced topics such as schemes. Although we will likely not get into that book, I recommend interested students buy it anyway before it goes out of print. Mumford is one of the 20th century’s best masters of the subject and there is no adequate substitute for his writings among more recent books.

The technical prerecquisite for Math 846-7-8 is only some knowledge of rings, fields, and modules, but sometimes we might mention connections with topology and complex analysis if appropriate. The 8000 level algebra sequence is plenty of preparation, and for many topics the 6000 level sequence suffices. The student who wants to start reading in algebraic geometry at an elementary level can find the elements of the theory of curves, in a modern language, in the lovely book Algebraic Curves by Fulton. An even more elementary introduction is the book of the same title by R.J.Walker. I especially like chapter III of Walker.


Bibliography on Algebraic Geometry:
The classic, and singularly readable, 1949 work Introduction to Algebraic Geometry by Semple and Roth is an introduction to the subject as it was about 1940, just before the modern period began. Cohomology on algebraic varieties is treated in Serre's famous 1957 Annals paper "Faisceaux Algebriques Coherents", and also in recent Johns Hopkins course notes by George Kempf.

Joe Harris' Harvard course notes focus on examples of varieties. The Red Book of Varieties and Schemes by Mumford is an excellent introduction to the language of schemes, and Algebraic Geometry by Hartshorne includes schemes and cohomology.

Excellent books on complex algebraic geometry include Griffiths' Introduction to Algebraic Curves; Clemens' A Scrapbook of Complex Curve Theory; Mumford's Algebraic Geometry I; Geometry of Algebraic Curves by Arbarello, Cornalba, Griffiths, Harris; Beauville's Complex Algebraic Surfaces, and Principles of Algbraic Geometry by Griffiths-Harris.

It is particularly enlightening to follow the evolution of algebraic geometry in contemporary accounts by some of its prime movers: Hilbert, Castelnuovo, Zariski, Weil, Serre, Kodaira, Segre, Severi, Van der Waerden, Grothendieck, Hironaka, Mumford, Deligne, Fulton, Harris, MacPherson, Arbarello, Clemens, Faltings (Fields medal talk by Mazur), and Mori, in their ICM talks in 1900, 1928, 1950, 1954, 1958, 1962, 1970, 1974, 1982(3), 1986, and 1990. [Note that in 1950, Zariski and Weil, were placed in "algebra".]

 

[mathwonk]

riemann on curves 1) where modern algebraic geometry began

Math 8320 Spring 2004, Riemann’s view of plane curves

Riemann’s idea was to classify all complex holomorphic functions of one variable.

1) Method: Consider a convergent power series as representing a holomorphic function in an open disc, and consider two power series as representing the same function if one is an analytic continuation of the other.

2) Problem: Two power series may be analytic continuations of each other and yet not determine the same function on the same open disc in the complex plane, so a family of such power series does not actually define a function.

Solution: Construct the Riemann surface S on which they do give a well defined holomorphic function, by considering all pairs (U,f) where U is an open disc, f is a convergent power series in U, and f is an analytic continuation of some fixed power series f0. Then take the disjoint union of all the discs U, subject to the identification that on their overlaps the discs are identified if and only if the (overlap is non empty and the) functions they define agree there.

Then S is a connected real 2 manifold, with a holomorphic structure and a holomorphic projection S-->C mapping S to the union (not disjoint union) of the discs U, and f is a well defined holomorphic function on S.

3) Enlarging the surface: If we include also points where f is meromorphic, and allow discs U which are open neighborhoods of the point at infinity on the complex line, then we get a holomorphic projection S-->P^1 = C union {p}, and f is also a holomorphic function
S-->P^1.

4) This poses a new 2 part problem:
(i) Classify all the holomorphic surfaces S.
(ii) Given a surface S, classify all the meromorphic functions on S.

5) The fundamental example
Given a polynomial F(z,w) of two complex variables, for each solution pair F(a,b) = 0, such that ∂F/∂w (a,b) ≠ 0, there is by the implicit function theorem, a neighborhood U of a, and a nbhd V of b, and a holomorphic function w = f(z) defined in U such that for all z in U, we have f(z) = w if and only if w is in V and F(z,w) = 0. I.e. we say F determines w = f(z) as an “implicit” function. If F is irreducible, then any two different implicit functions determined by F are analytic continuations of each other. For instance if F(z,w) = z-w^2, then there are for each a ≠ 0, two holomorphic functions w(z) defined near a, the two square roots of z.

In this example, the surface S determined by F is “essentially” equal to the closure of the plane curve X: {F(z,w) = 0}, in the projective plane P^2. More precisely, S is constructed by removing and then adding back a finite number of points to X as follows.

Consider the open set of X where either ∂F/∂w (a,b) ≠ 0 or
∂F/∂z (a,b) ≠ 0. These are the non singular points of X. To these we wish to add some points in place of the singular points of X. I.e. the set of non singular points is a non compact manifold and we wish to compactify it.

Consider an omitted i.e. a singular point p of X. These are always isolated, and projection of X onto an axis, either the z or w axis, is in the neighborhood of p, a finite covering space of the punctured disc U* centered at the z or w coordinate of p. All such connected covering spaces are of form t-->t^r for some r ≥ 1, and hence the domain of the covering map, which need not be connected, is a finite disjoint union of copies of U*. Then we can enlarge this space by simply adding in a separate center for each disc, making a larger 2 manifold.

Doing this on an open cover of X in P2, by copies of the plane C2, we eventually get the surface S, which is in fact compact, and comes equipped with a holomorphic map S-->X, which is an isomorphism over the non singular points of X. S is thus a “desingularization” of X. For example if X crosses itself with two transverse branches at p, then S has two points lying over p, one for each branch or direction. If X has a cusp, or pinch point at p, but a punctured neighborhood of p is still connected, there is only one point of S over p, but it is not pinched.


Theorem: (i) The Riemann surface S constructed above from an irreducible polynomial F is compact and connected, and conversely, any compact connected Riemann surface arises in this way.
(ii) The field of meromorphic functions M(S) on S is isomorphic to the field of rational functions k(C) on the plane curve C, i.e. to the field generated by the rational functions z and w on C.

I.e. this example precisely exhausts all the compact examples of Riemann surfaces.

Corollary: The study of compact Riemann surfaces and meromorphic functions on them is equivalent to the study of algebraic plane curves and rational functions on them.


6) Analyzing the meromorphic function field M(S).

If S is any compact R.S. then M(S) = C(f,g) is a finitely generated field extension of C of transcendence degree one, hence by the primitive element theorem, can be generated by two elements, and any two such elements define a holomorphic map S-->X in P^2 of degree one onto an irreducible plane algebraic curve, such that k(X) = M(S).

Question: (i) Is it possible to embed S isomorphically onto an algebraic curve, either one in P^2 or in some larger space P^n?
(ii) More generally, try to classify all holomorphic mappings S-->P^n and decide which ones are embeddings.

 

[mathwonk]

riemann on curves 2) the beginning of modern alg geom

Riemann’s intrinsic approach:

Given a holomorphic map ƒ:S-->Pn, with homogeneous coordinates z0,...,zn on P^n, the fractions zi/z0 pull back to meromorphic functions ƒ1,...,ƒn on S, which are holomorphic on S0 = ƒ-1(z0≠0), and these ƒi determine back the map ƒ. Indeed the ƒi determine the holomorphic map S0-->Cn = {z0≠0}in P^n.

Analyzing ƒ by the poles of the ƒi
Note that since the ƒi are holomorphic in ƒ-1(z0≠0), their poles are contained in the finite set ƒ-1(z0=0),and on that set the pole order cannot exceed the order of the zeroes of the function z0 at these points. I.e. the hyperplane divisor {z0 = 0}0 in P^n pulls back to a “divisor” ∑ njpj on S, and if ƒi = zi/z0 then the meromorphic function ƒi has divisor div(ƒi) = div(zi/z0) = div(zi) - div(z0) = ƒ*(Hi)-ƒ*(H0).
Hence div(ƒi) + ƒ*(H0) = ƒ*(Hi) ≥ 0, and this is also true for every linear combination of these functions.

I.e. the pole divisor of every ƒi is dominated by ƒ*(H0) = D0. Let's give a name to these functions whose pole divisor is dominated by D0.

Definition: L(D0) = {f in M(S): f = 0 or div(f) +D0 ≥ 0}.

Thus we see that a holomorphic map ƒ:S-->Pn is determined by a subspace of L(D0) where D0 = ƒ*(H0) is the divisor of the hyperplane section H0.

Theorem(Riemann): For any divisor D on S, the space L(D) is finite dimensional over C. Moreover, if g = genus(S) as a toplogical surface,
(i) deg(D) + 1 ≥ dimL(D) ≥ deg(D) +1 -g.
(ii) If there is a positive divisor D with dimL(D) = deg(D)+1, then S = P^1.
(iii) If deg(D) > 2g-2, then dimL(D) = deg(D)+1-g.

Corollary of (i): If deg(D) ≥ g then dim(L(D)) ≥ 1, and deg(D)≥g+1 implies dimL(D) ≥ 2, hence, there always exists a holomorphic branched cover S-->P1 of degree ≤ g+1.

Q: When does there exist such a cover of lower degree?

Definition: S is called hyperelliptic if there is such a cover of degree 2, if and only if M(S) is a quadratic extension of C(z).

Corollary of (iii): If deg(D) ≥ 2g+1, then L(D) defines an embedding S-->P^(d-g), in particular S always embeds in P^(g+1).

In fact S always embeds in P^3.
Question: Which S embed in P^2?

Remark: The stronger Riemann Roch theorem implies that if K is the divisor of zeroes of a holomorphic differential on S, then L(K) defines an embedding in P^(g-1), the “canonical embedding”, if and only if S is not hyperelliptic.

7) Classifying projective mappings
To classify all algebraic curves with Riemann surface S, we need to classify all holomorphic mappings S-->X in P^n to curves in projective space. We have asociated to each map ƒ:S-->P^n a divisor Do that determines ƒ, but the association is not a natural one, being an arbitrary choice of the hyperplane section by H0. We want to consider all hyperplane sections and ask what they have in common. If h: ∑cjz^j is any linear polynomial defining a hyperplane H, then h/z0 is a rational function f with div(f) = ƒ*(H)-ƒ*(H0) = D-D0, so we say

Definition: two divisors D,D0 on S are linearly equivalent and write D ≈ D0, if and only if there is a meromorphic function f on S with D-D0 = div(f), iff D = div(f)+D0.

In particular, D≈D0 implies that L(D) isom. L(D0) via multiplication by f. and L(D) defines an embedding iff L(D0) does so. Indeed from the isomorphism taking g to fg, we see that a basis in one space corresponds to a basis of the other defining the same map to P^n, i.e. (ƒ0,...,ƒn) and (fƒ0,...,fƒn) define the same map.

Thus to classify projective mappings of S, it suffices to classify divisors on S up to linear equivalence.

Definition: Pic(S) = set of linear equivalence classes of divisors on S.

Fact: The divisor of a meromorphic function on S has degree zero.

Corollary: Pic(S) = ∑ Pic^d(S) where d is the degree of the divisors classes in Pic^d(S).

Definition: Pic^0(S) = Jac(S) is called the Jacobian variety of S.

Definition: S^(d) = (Sx..xS)/Symd = dth symmetric product of S
= set of positive divisors of degree d on S.

Then there is a natural map S^(d)-->Pic^d(S), taking a positive divisor D to its linear equivalence class O(D), called the Abel map. [Actually the notation O(D) usually denotes another equivalent notion the locally free rank one sheaf determined by D.]

Remark: If L is a point of Pic^d(S) with d > 0, L = O(D) for some D>0 if and only if dimL(D) > 0.
Proof: If D > 0, then C is contained in L(D). And if dimL(D)>0, then there is an f ≠ 0 in L(D) hence D+div(f) ≥ 0, hence > 0.QED.

Corollary: The map S^(g)-->Pic^g (S) is surjective.
Proof: Riemann’s theorem showed that dimL(D)>0 if deg(D) ≥ g. QED.

It can be shown that Pic^g hence every Pic^d can be given the structure of algebraic variety of dimension g. In fact.
Theorem: (i) Pic^d(S) isom C^g/L, where L is a rank 2g lattice subgroup of C^g.
(ii) The image of the map S^(g-1)-->Pic^(g-1)(S) is a subvariety “theta” of codimension one, i.e. dimension g-1, called the “theta divisor”.
(iii) There is an embedding Pic^(g-1)-->P^N such that 3.theta is a hyperplane section divisor.
(iv) If O(D) = L in Pic^(g-1)(S) is any point, then dimL(D) = multL(theta).
(v) If g(S) ≥ 4, then g-3 ≥ dim(sing(theta)) ≥ g-4, and dim(sing(theta)) = g-3 iff S is hyperelliptic.
(vi) If g(S) ≥ 5 and S is not hyperelliptic, then rank 4 double points are dense in sing(theta), and the intersection in P(T0Pic^(g-1)(S)) isom P^(g-1), of the quadric tangent cones to theta at all such points, equals the canonically embedded model of S.
(vii) Given g,r,d ≥0, every S of genus g has a divisor D of degree d with dimL(D) ≥ r+1 iff g-(r+1)(g-d+r) ≥ 0.

 

[fopc]

I see your .pdf on "naive into. to algebraic geometry" on your home webpage.

It's looks like it contains posts 14 and 15 on this thread.
Do you plan to expand this material and leave it on your webpage as a .pdf?

What about posts 31-33 on this thread? Some interesing historical material and more. Could it also go somewhere in your new .pdf?

 

[mathwonk]

yes i was so inspired by the question here, i put the answer on my page. i do hope to expand it, it just takes time, as all such activity is "merely" for the greater good, as we get no credit from our jobs for it, so it goes slowly.

 

[mathwonk]

for those naturally daunted by the large amount of prerecquisites i laid out for alg geom, try miles reid's little book, undergraduate algebraic geometry. it requires very little background, probably only stuff many of you already have, and yet does some really good geometry.

 

[mathwonk]

by the way "my" birthday algebraic geometry conference at UGA was just an unbelievable high, with wonderful young geometers from all over the world, and old friends too. may you all have such good friends as the young people who put this on and did all the work, as well as the friends who came from distant places to wish us well at UGA, and help us with our program.

Our young grad students got to meet people from the best places in the world, and make friends and contacts that should help them for years. Those if you hoping to become amthematicians should take note and try to attend good conferences as soon as possible in your career.

In my case too, I realized I had been falling way behind the current trends and language in my own area by staying home too much and not interacting and listening to top people talk. So I got a tremendous boost and have been thinking about live questions all weekend and also since they left yesterday.

It is really hard to believe how generous these people are, since they all left home and families and traveled to be there, and some of them also had to miss the final four games!

You young kids who are wondering how in the heck you learn all this stuff, well you do it by listening to people talk who know the stuff well. This is much more efficient than trying to plow through books. An expert can tell you in in one minute the key idea behind a difficult topic, that would take you a very long time to get through alone.now i have to force myself to log off, quit lallygagging here, pleasant as it is, and actually think about a little question someone asked me yesterday.

 

[mathwonk]

ok i solved it.