example: given the equation X^2 + Y^2 + Z^2, one can asociate the set of points in complex projective 2 space which make this equation = 0. this is the old fashioned approach. I.e. an algebraic variety is a point set defined by polynomial equations.
Or one can view this equation as a functor from fields to sets, that associastes to each field K, the subset of K^3 that satisfies this equation. this is a more modern approach. i.e. an algebraic variety is a certain functor of rings.
8300 Day One Fall 2003. MWF 10:10 am. room 410.
Introduction to algebraic geometry: some fundamental problems, and connections between algebra and topology
Basic Questions:
What do solution sets of polynomial equations "look like"?
a. Are the solution sets empty or non empty?
b. If there are solutions, when are there infinitely many?
c. When can we parametrize the infinite solution set?
d. When are there a finite number of solutions, and then how many are there?
Since one application of algebraic geometry of interest to many of us, is to number theory, we look at a few examples of how geometry, as well as topology and analysis can impact number theory. If f is a polynomial in one or more variables, with integral or rational coefficients, we can look for integral, rational, real, or complex solutions of f(X) = 0. The complex solution set has a lot of classical geometry, for instance it is a CW complex and usually a complex manifold, while the set of integral or rational solutions belong more to number theory. The beautiful phenomenon is that there are relations between these different sets of solutions. E.g. the geometry of the set of complex solutions can affect the nature of the rational solutions. Here is a simple but important example.
Rational parametrizations:
Recall that in linear algebra we can always parametrize the solutions of a system of linear equations, i.e. there is a linear map from a linear parameter space onto the set of solutions, so that any choice of parameter yields a solution of the system of equations. Suppose we could do that for the solutions of a polynomial equation. I.e. suppose f(X,Y) has rational coefficients and there are non constant rational functions x(t), y(t) with rational coefficients, such that for all t, f(x(t),y(t)) = 0. Then we could produce as many rational solutions of f(X,Y) = 0 as we like, since for any rational value of t, then (x(t),y(t)) is a rational solution of f(X,Y) = 0. It turns out this sort of parametrization is usually not possible for purely topological reasons.
I.e. for each irreducible polynomial f(X,Y), the associated complex solutions of {f(X,Y) = 0} form a topological space which is a complex one manifold except at a finite set of points. One can then modify this space, by compactifying it with the addition of a finite number of points, then desingularize it by doing a simple surgery at another finite set of points, and obtain a complex one manifold called the associated Riemann surface of the given polynomial. The assignment of a Riemann surface to an irreducible plane curve is a "functor", i.e. it is natural in the sense that a rational map between two curves induces a holomorphic map of their Riemann surfaces. Hence a rational parametrization of a curve f(X,Y) = 0 over Q, which is a rational map from the rational line Q to the curve {f=0}, yields also a rational map from C to the complex points of f = 0, and a holomorphic map from the Riemann surface of Q to the Riemann surface of {f=0}. Now the Riemann surface of Q is the one point compactification of C, namely the Riemann sphere.
So a rational parametrization of {f=0} yields a surjective holomorphic map from the Riemann sphere to the Riemann surface of {f=0}, which is some compact topological surface, with a complex structure. Now a surjective holomorphic map allows us to pull back non zero differential forms from the target, to non zero differential forms on the source, but the Riemann sphere has no nonzero differential forms. So this holomorphic map cannot exist if the Riemann surface of {f=0} has any non zero holomorphic forms. The existence of such forms on a Riemann surface is equivalent to the Euler characteristic çtop being ? 0, so a rational parametrization cannot exist for a curve whose Riemann surface is a compact surface of genus g = 1 - (1/2)çtop ? 1.
Intuitively, a surface is a doughnut possibly with holes, the genus is the number of holes, and this number can be seen from a model of the curve made from lines. I.e. the genus, being an integer, is invariant under deformation, so to compute it we may assume our curve is a union of lines. A triangle obviously has one hole hence we claim a "non singular" curve of degree three has genus one, whereas a union of 4 lines has three holes, hence a non singular curve of degree 4 has genus 3, etc. Thus if the degree of the curve is ? 3, and there are no singular points of either the complex curve or its projectivization (compactification), then the genus is ? 1, so there can be no rational parametrization.
We can compute the genus without using this deformation argument, by projecting our surface onto the complex x - axis in the x,y plane and computing the branching behavior. For example the cubic curve y2 = x(x-1)(x+1) projects 2:1 onto the x axis, branched over the three points x = 0,1,-1. I.e. over each of these three points, there is only one preimage point instead of two. This projection induces a holomorphic map of the associated Riemann surfaces as before.
The (compact) Riemann surface of the x axis is the one point compactification of C, namely the Riemann sphere, and the Riemann surface of the curve y^2 = x(x-1)(x+1) is obtained by adding either one or two points over the point at infinity on the Riemann sphere. If it were the case that we add two points then neither is a branch point, but it follows from Hurwitz’ formula that such a map of compact surfaces must have an even number of branch points, so we in fact must add one point at infinity.
This gives genus one for the curve y^2 = x(x-1)(x+1) as follows. Recall that the Euler characteristic of a compact surface is equal to V-E+F, where V is the number of vertices, E the number of edges, and F the number of faces in a triangulation of the surface. If we triangulate the Riemann sphere into small triangles and make sure each branch point is a vertex, then a branched 2:1 cover pulls back each triangle on the sphere to two triangles on the curve y2 = x(x-1)(x+1), and each edge to two edges, and each vertex either to two vertices, or only one if the vertex is a branch point.
Thus, if V,E,F are the numbers associated to our triangulation of the sphere, then the Euler characteristic of the Riemann surface of the curve y^2 = x(x-1)(x+1), equals (2V-b) - 2E + 2F = 2(V-E+F) - b. Since for the sphere V-E+F = 2, this gives 0 for the Euler characteristic of the curve y^2 = x(x-1)(x+1), which thus has genus 1.
The computation shows why there is always an even number of branch points for such a map as well, since the branch order is the difference between the Euler characteristic of the source, and a multiple of the Euler characteristic of the target, and the difference of these even numbers must be even.
Notice in this calculation that we are using a map from our curve onto a sphere to prove there can be no map from a sphere onto our curve. Given that parametrization maps do not often exist, the existence of these maps in the other direction is a fundamental tool for studying algebraic sets. I.e. we will show fairly early that there are always "finite" maps from any affine algebraic set of dimension n onto the affine space kn of dimension n, and eventually show that rational maps in the other direction, i.e. from affine space onto our algebraic set, exist only in special cases.
Guided by this geometric, topological, and analytical reasoning, we may conjecture that a plane curve of degree ? 3 over any field k, cannot be rationally parametrized if there are no "singular" points of the given curve over the algebraic closure of k. This is in fact true, and can be proved in a completely algebraic way, involving an algebraic version of differential forms, namely formal derivatives. We give a direct proof in the notes from 8300 in Fall 2001 (for the Fermat cubic in characteristic > 3 ?) as follows:
Non rationality of Fermat cubic curve
Assume that (x/z)^3 + (y/z)^3 = 1, where x, y,z are polynomial functions of t with no common factors. (since neither x/z nor y/z is constant, neither x nor y is zero.) Multiply through to obtain (1): x^3 + y^3 = z^3, and differentiate to obtain (2): x^2x' + y^2y' = z^2z', where ' denotes differentiation w.r.t. t. Now we want to eliminate the z terms. multiply (1) by z' and (2) by z, and subtract, to obtain z'x^3 + z'y^3 = x'x^2z + y'y^2z. Collecting terms, and factoring gives x^2( xz'-zx') = y^2 ( y'z - yz'). If either ( y'z - yz')=0, or ( xz'-zx') = 0, then both do, and hence by the quotient rule for derivatives, we would have x/z and y/z constant. Since x,y are rel prime, x^2 divides (yz'-zy'), in C[t], and thus 2degree(x) (< or =) deg(y)+deg(z)-1. Repeating the argument for each of the other two variables, i.e. eliminating the x and y terms, leads to the same inequality with the variables permuted. Adding the 3 inequalities gives 2[deg(x)+deg(y)+deg(z)] (< or =) 2[deg(x)+deg(y)+deg(z)] - 3, a contradiction. QED.
Thus this result is true over Q and C, for topological and analytic reasons, and the topological and analytic invariants involved have algebraic incarnations, which lead to a conceptual proof in characteristic p > 0 as well, as we will see later. I.e. we will give a purely algebraic definition of the genus, and prove that a non constant rational map cannot exist if the target has larger genus than the source.
There are also sufficient criteria for existence of parametrizations, based on topological criteria as well. I.e. if the Riemann surface of a curve has genus zero then the curve has a rational parametrization over C, but in some cases the parametrization also exists over Q. Now a curve with a rational parametrization has infinitely many rational points, so the curve X^2 + Y^2 + 1 = 0, which has no real points at all, hence no rational points, cannot be rationally parametrizable over Q. In fact the complex points of this curve do have genus zero, so there is a parametrization over C. To get a parametrization we need at least one rational point.
Then we can prove that an irreducible plane curve of degree 4 with four rational points, three of which are singular on the associated Riemann surface, is parametrizable over Q. I.e. a smooth curve of degree 4 has genus 3, and three singular points imply that three homology cycles on the riemann surface have been shrunk each to a point, collapsing the three holes and resulting in a Riemann surface of genus zero.
The fourth rational point then allows the parametrization. I.e. we consider all conics passing through all 4 rational points, and a general one of these meets the curve in one further point, for which we can solve rationally in terms of the others and the coefficients of the two intersecting curves. Hence this residual intersection point is also rational. Considered as a rational map from the quartic to a plane curve, the target is a conic, and the 4th points maps to a rational point of this conic.
Then the result follows from the fact that a conic with one rational point is parametrizable. In particular, it has an infinite number of rational points. technically we have found a map of degree one from our curve to a parametrizable curve, but we can also use conics to map the parametrizable curve to our curve, thus obtaining a parametrization of our curve.
In fact with suitable coordinates, both the map from our quartic to the parametrizable conic and the inverse map can be given by the same map, the standard quadratic transform (yz, xz, xy). It should follow in a similar way, that an irreducible curve of degree d with "d-1 choose 2" rational singular points, and one more rational point, at least in characteristic zero, is rationally parametrizable.
Now that we know the method of parametrization is inapplicable to smooth curves of genus > 0, we can ask whether there is some other reason for such curves over Q to have an infinite number of rational points. For genus one, i.e. smooth plane curves of degree 3, there is a method of producing more rational points from a given one, the "tangent method". I.e. the tangent line at a rational point intersects the curve in a second rational point. Then the tangent line at the second point meets the curve again at another rational point, etc...
So it is conceivable that a curve of genus one may have an infinite number of rational points, and in fact this can happen. Mordell conjectured that if the genus is > 1 however, then in fact there are never an infinite number of rational points, and this was proved by Deligne about 30 years ago.
As foreshadowed by the arguments above, some of our principal goals will be to define an irreducible variety (the geometric analog of irreducible polynomials), define dimension of an irreducible variety, then show that every irreducible n dimensional affine or projective variety can be mapped finitely onto either an n dimensional affine or an n dimensional projective space, and then develop the algebraic analog of differential calculus and use it to prove that a smooth n dimensional projective hypersurface of degree > n+1 cannot be rationally parametrized.
Along the way we will discuss the geometry of some hypersurfaces of degree > 2, in the plane, in 3 space, and in 4 space. We will also discuss the case of a non singular quadric in 5 space, which parametrizes the famous "Grassman variety", the parameter space or "moduli" variety for lines in three space. There are still many difficult open questions about which varieties have rational parametrizations, especially which varieties admit parametrizations of degree one. For example, a 2:1 rational parametrization of any non singular cubic threefold in 4 space was known classically, and it was suspected that parametrization of no degree one could exist. The proof of this non existence result was achieved in 1972 using deep and beautiful results from the theory of abelian varieties.
8300 2003 Day 2: Roots of polynomials in several variables
Algebraic geometry arises from the problem of solving systems of polynomial equations. In modern times, this study has become the theory of “schemes”. Many of us have seen the definition of an “affine scheme” as the set of prime ideals of a commutative ring with identity. To see what these have to do with one another, we want to retrace in the classical case of polynomials over a field, how prime ideals arise in looking for common solutions of systems of equations.
Let k be any field, and {fi} in k[T1,...,Tn] = k[T] a collection of polynomials in n variables over k. A “solution” or “common zero” of the system {fi=0} will be a vector, or point, p = (p1,...,pn) in E^n, where E is a field extension of k, such that for all i, fi(p) = 0, in E. The key to connecting a solutions vector p with a prime ideal of k[T] is to look at the associated evaluation map π:k[T]—>E, taking f(T) to f(p), and its kernel ker(π) = Ip. Then p is a common zero of the polynomials {fi} if and only if {fi} is in Ip. Moreover since E is a field, the ideal Ip is prime in k[T]. Thus a “solution” of the system {fi} in k[T], which might be thought of as a “point” of the algebraic variety defined by this system, corresponds to a prime ideal of k[T] containing {fi}, or equivalently, to a prime ideal of the quotient ring k[T]/({fi}), where ({fi}) is the ideal generated by the set {fi}.
Conversely, if {fi} is in I is in k[T] where I is a prime ideal, then there is an associated k algebra map π:k[T]—>k[T]/I is in E where E is the field of fractions of the domain k[T]/I. If we define the point p in E^n by setting pi = π(Ti) in E, then this map π is evaluation at p, in the sense above.
Thus prime ideals of the quotient ring k[T]/({fi}) correspond to common zeroes of the system {fi}, with components in extensions of k. This correspondence is not however one-one, since e.g. both i and -i in C define the same prime ideal in R[T], since R[T]/(T^2+1) has two isomorphisms with C, taking T to either i or -i. Thus considering solutions in a field extension of k involves some Galois theory over k. Indeed if E is an algebraic closure of k, then maximal prime ideals m Ã k[T] correspond to orbits of Gal(E/k) in E^n, parametrizing different embeddings of k[T]/m in E. (See Mumford, II.4., pp. 96 - 99.)
Natural questions:
1) Which prime ideals correspond to solutions in k^n, i.e. when do the solutions have coordinates in the original field k itself?
2) Which prime ideals correspond to solution vectors p = (p1,...,pn) with coordinates pi lying in some algebraic extension E of k?
3) What is the geometric meaning of solution vectors whose coordinates lie in transcendental extensions of k?
Question 1). Let p in k^n be a solution of the system {fi}. Then the evaluation map k[T1,...,Tn]—>k, is surjective since it equals the identity on k, hence its kernel is a maximal ideal m … {fi}. Thus p in kn corresponds to a maximal ideal m … {fi} such that the composition k—>k[T]/m—>k is an isomorphism, and since k[T]/m—>k is injective, equivalently such that
k—>k[T]/m is an isomorphism.
Conversely let {fi} in m in k[T] and assume the map k—>k[T]/m is an isomorphism. Thus for each i, there is a unique pi in k that maps to Ti, mod m. Thus the composition with the inverse isomorphism k[T]—>k[T]/m—>k takes Ti to pi, and this is a k algebra map. Thus it must be evaluation at p. Moreover it takes every element of m to 0. Thus p is a common zero of the ideal m, and hence of the system {fi} having coefficients in k.
Thus common solutions of the system {fi} in k[T] with coefficients in k, correspond one - one to maximal ideals m in k[T], with {fi} in m, and such that k—>k[T]/m is an isomorphism. Equivalently they correspond to maximal ideals m in k[T]/({fi}) such that k—>k[T]/m is an isomorphism. Of course there might not be any such ideals, as in the case of {T^2+1} in R[T], since ({T^2+1}) is already maximal and has quotient larger than R. This is because as we know, many systems of equations have no solutions in the coefficient field.
Note that a maximal ideal of k[T] with p = (p1,...,pn) in kn as common zero, contains the functions Ti-pi. But the ideal (T1-p1,...,Tn-pn) is already maximal since the composition k—>k[T]/(T-p) takes pi to Ti, thus surjects, so is isomorphic. So arguments like those above give one - one correspondences between the following sets:
{common solutions (a1,...,an) in kn of {fi}}
≈ {k algebra maps π:k[X1,...,Xn]—>k with π(fi) = 0, for all i}
≈ {maximal ideals {fi} in MÃk[X1,...,Xn] such that k—>k[X1,...,Xn]/M is an isomorphism}
≈ {ideals of form {fi} in (X1-a1,...,Xn-an) Ã k[X1,...,Xn]}
These correspondences are as follows: a point a in kn at which all fi = 0 yields the k algebra map π = evaluation at a, which takes all fi to 0; a k algebra map π:k[X1,...,Xn]—>k is always surjective since it already is on k, so π has a maximal ideal kernel M such that the composition
k—>k[X1,...,Xn]—>k[X1,...,Xn]/M —> k is an isomorphism, and since π(fi) = 0 for all i, thus {fi} in M; a maximal ideal M … {fi} such that the composition
k—>k[X1,...,Xn]/M is an isomorphism is always of form (X1-a1,...,Xn-an) where ai = the unique element of k such that Xi = ai mod M; finally a maximal ideal of form M = (X1-a1,...,Xn-an) … {fi} determines the point a by setting aj equal again to the unique element of k congruent mod M to Xj, and since the point a belongs to M, all fi vanish at a.
Question 2). The question is more complicated but the answer is simpler: solutions p of the system {fi}, with coordinates pi in some algebraic extension of k, correspond to all maximal ideals of k[T]/({fi}), i.e. to those maximal ideals of k[T] containing the system {fi}. This time however the correspondence is not one to one, since several solutions can correspond to the same maximal ideal. To prove it will take a little work, but one direction is elementary.
Let p be a common zero of all fi, where pj is in E, and E is algebraic over k. Consider the evaluation map π:k[T]—>E taking f(T) to f(p). The image is a domain k[p1,...,pn], since contained in the field E, and its fraction field k(p1,...,pn) is a finitely generated algebraic extension field of k, hence a finite dimensional k vector subspace of E. Multiplication by any non zero element u in the domain k[p1,...,pn] is an injective k linear map of the finite dimensional k vector space k[p1,...,pn] to itself, hence also surjective. Thus there is some v in k[p1,...,pn] such that uv = 1. Thus the image k[p1,...,pn] of the evaluation map is actually a field, i.e. k[p1,...,pn] = k(p1,...,pn). Hence the kernel, ker(π) is a maximal ideal of k[T] containing {fi}.
The converse is less elementary, but we claim it is true: i.e. if {fi} in m in k[T] is any maximal ideal, then the quotient k[T]/m is always a finite dimensional algebraic extension field of k, hence embeds in E = algebraic closure of k, via some embedding π:k[T]/m—>E. If pi = π(Ti), then the map π with kernel m, is evaluation at p in En, which is a common solution of the system {fi} with coordinates in E. Here we see that different embeddings of k[T]/m into E give rise to different solutions p corresponding to the same ideal m.
The simplest statement arises if k = E is itself algebraically closed. Then we can combine the statements in 1) and 2) and see that there is a one one correspondence between solutions of the system {fi} with coordinates in k, and the set of all maximal ideals of k[T] containing {fi}. In particular, if the set {fi} does not generate the unit ideal, then there must be some maximal ideals containing the system, hence there are some common solutions. This is a several variables analog of the fundamental theorem of algebra. In its simplest form it says if k is algebraically closed, there is a one one correspondence between points of kn and maximal ideals in k[T1,...,Tn], where p = (p1,...,pn) in kn corresponds to (T1-p1,...,Tn-pn) in k[T]. This is Hilbert’s famous (“weak”) nullstellensatz, the foundation result of the whole subject of algebraic geometry, the precise dictionary between geometry and polynomial algebra.
If {fi} in k[T] where k is any field and E its algebraic closure, maximal ideals of k[T]/({fi}) correspond to (finite) Gal(E/k) orbits of the common solution set of {fi} in En. Solutions of the system {fi} lying in k^n are the fixed points of the action.
Question 3), the interpretation of points on the algebraic variety V({fi}) with coordinates in non algebraic (“transcendental”) extensions of k. For simplicity we assume k is algebraically closed, as we will henceforth always assume in this course. We know from the discussion above, such points p correspond to prime ideals Ip in k[T]/({fi}), hence the maximal ideals containing this prime ideal constitute a certain subcollection of the k valued points of the variety V({fi}) Ã kn. I.e. a point p with values in a transcendental extension E of k corresponds to a subcollection V(Ip) of k valued points.
Thus if {fi} in Ip in k[T], then the prime ideal Ip corresponds to a subvariety V(Ip) in V({fi}) in k^n, and this subvariety V(Ip) is our geometric interpretation, in kn, of the “point” p with values in the transcendental extension E of k. Moreover, if we think of a k valued point, i.e. a maximal ideal, as having dimension zero, then a “point” p with coordinates pi in E … k which generate a transcendence degree r extension k(p1,...,pn) of k in E, corresponds to a prime ideal I in ?k[T]/({fi}) of “coheight r”, so the variety V(Ip) has dimension r.
I.e. the prime ideal Ip can be joined to a maximal ideal by a chain of prime ideals of length r, but no longer, and since k(p1,...,pn) = fraction field of k[p1,...,pn] ≈ k[T]/Ip, thus as commutative algebra students may know, the transcendence degree of k(p1,...,pn) equals the Krull dimension of k[p1,...,pn], equals the coheight of Ip in [T]. (We will prove this later.) The subvariety V(I) in V({fi}) thus has dimension r. Note that “dimension” is a relative term, and here it is taken relative to the base field k. Since not all fields are algebraic over k, not all “points” are zero dimensional over k.
To sum up, if k is algebraically closed, and {fi} in I in k[T], where I is a prime ideal of coheight r, then I corresponds equivalently to an (“irreducible”) r dimensional subvariety of the variety V({fi}), and to a point with values in the extension field E = fraction field(k[T]/I) of k of transcendence degree r. (Since a prime ideal is not the proper intersection of two other ideals, here “irreducible” means the variety it defines is not the proper union of two other varieties.)
Here is a revised version of the fresh one i wrote just for you guys:
Naive introduction to algebraic geometry: the geometry of rings
I used to say algebraic geometry is the study of the geometry of polynomials. Now I sometimes call it the "geometry of rings". I also feel that algebraic geometry is defined more by the objects it studies than the tools it uses. The naivete in the title is my own.
I. BASIC TOOL: RATIONAL PARAMETRIZATION
Algebraic geometry is a generalization of analytic geometry - the familiar study of lines, planes, circles, parabolas, ellipses, hyperbolas, and their 3 dimensional versions: spheres, cones, hyperboloids, ellipsoids, and hyperbolic surfaces. The essential common property these all have is that they are defined by polynomials. This is the defining characteristic of classical algebraic sets, or varieties - they are loci of polynomial equations.
A further inessential condition in the examples above is that the defining polynomials have degree at most 2 and involve at most 3 variables. This limitation arose historically for psychological and technical reasons. Before the advent of coordinates, higher dimensions could not be envisioned or manipulated, and even afterwards it was commonly felt that space of more than 3 dimensions did not "exist" hence was irrelevant.
The dimension barrier was lifted by Riemann and Italian geometers in the 19th century such as C. Segre, who realized that higher dimensions could be useful for the study of curves and surfaces. Riemann's use of complex coordinates for plane curves simplified their study, and Segre understood that some surfaces in 3 space were projections of simpler ones embedded in 4 space.
One reason for restricting attention to equations in (X,Y) of degree at most 2 is a limitation of the basic method of "parametrization", expressing a locus by an auxiliary parameter. E.g. the curve X^2 + Y^2 = 1 can be parametrized by the variable t by setting X(t) = 2t/[1+t^2], Y = [1-t^2]/[1+t^2]. This substitutiion, along with dX = 2[1-t^2]dt/[1+t^2]^2, allows one to simplify the integral of dX/sqrt(1-X^2), to that of 2dt/[1+t^2] = 2d[arctan(t)].
The cubic Y^2=X^3 can also be parametrized, say by X = t^2, Y = t^3. But to simplify in this way the integral of dX/sqrt(1-X^3), requires us to parametrize the cubic Y^2 = 1-X^3, a problem which is actually impossible. These questions were considered first by the Bernoullis, and resolved by new ideas of Abel, Galois, and especially Riemann as follows. (Interestingly, in three variables the difficulty arises in degree 4, and 19th century geometers knew how to parametrize most cubic surfaces.)
II. NEW METHODS FOR PLANE CURVES: TOPOLOGY and COMPLEX ANALYSIS
Riemann associated to a plane curve f(X,Y)=0 its set of complex solutions, compactified and desingularized. This is its "Riemann surface", a real topological 2 manifold with a complex structure obtained by a branched projection onto the complex line. For instance the curve y^2 = 1-X^3 becomes its own Riemann surface after adding one point at infinity, making it a topological torus. Projection on the X coordinate is a 2:1 cover of the extended X line, branched over infinity and the solutions of 1-X^3 = 0.
This association is a functor, i.e. a non constant rational map of plane curves yields an associated holomorphic map of their Riemann surfaces, in particular a topological branched cover. Riemann assigns to a real 2 manifold its "genus" (the number of handles), and calculates that branched covers cannot raise genus, and the only surface of genus zero is the sphere = the Riemann surface of the complex t line. Hence if the Riemann surface of a plane curve has positive genus, it cannot be the branched image of the sphere, hence the curve cannot be parametrized by the coordinate t.
Riemann also proved a smooth plane curve of degree d has genus g = (d-1)(d-2)/2, so smooth cubics and higher degree curves all have positive genus and hence cannot be parametrized. He proved conversely that any curve whose Riemann surface has genus zero can be parametrized, e.g. hyperbolas, circles, lines, parabolas, ellipses, or any curve of degree < 3. Moreover a singularity, i.e. a point where the curve has no tangent line, like (0,0) on Y^2 = X^3, lowers the genus during the desingularization process, and this is why such a "singular" cubic can be parametrized.
One also obtains a criterion for any two irreducible plane curves to be rationally isomorphic, namely their Riemann surfaces should be not just topologically, but holomorphically isomorphic. By representing a smooth plane cubic as a quotient of the complex line C by a lattice, using the Weierstrass P function, one can prove that many complex tori are not holomorphically equivalent, by studying the induced map of lattices. It follows that there is a one parameter family of smooth plane cubics which are rationally distinct from each other.
This shows briefly the power and flexibility of topological and holomorphic methods, which Riemann largely invented for this purpose, an amazing illustration of thinking outside traditional confines.
III. RINGS and IDEALS
To go further in the direction of arithmetic questions, one would like more algebraic techniques, applicable to fields of characteristic p, algebraic numbers fields, rings of integers, power series rings,.... One can pose the question of isomorphism of plane curves algebraically, using ring theory, as follows. Since all roots of multiples of the polynomial f vanish on the zero locus of f, it is natural to associate to the curve V:{f=0} in k^2, the ideal rad(f) = {g in k[X,Y]: some power of g is in (f)}. Then the quotient ring R = k[X,Y]/rad(f) is the ring of polynomial functions on V. Moreover if p is a point of V, evaluation at p is a k algebra homomorphism R-->k with kernel a maximal ideal of R. In case k is an algebraically closed field, like C or the algebraic numbers, this is a bijection between points of V and maximal ideals of R.
In fact everything about the plane curve V is mirrored in the ring R in this case, and two irreducible polynomials f,g, in k[X,Y], define isomorphic plane curves if and only if their associated rings R and S are isomorphic k algebras. Indeed the assignment of R to V is a "fully faithful functor", with algebraic morphisms of curves corresponding precisely to k algebra maps of their rings. To recover the points from the ring one takes the maximal ideals, and to recover a map on these points from a k algebra map, one pulls back maximal ideals. (Since these rings are finitely generated k algebras and k is algebraically closed, a maximal ideal pulls back to a maximal ideal.) Any pair of generators of the k algebra R defines an embedding of V in the plane.
Similarly, if f (irreducible) in k[X,Y,Z] defines a surface V:{f=0} in k^3, (k still an algebraically closed field), then not only do points of V correspond to maximal ideals of R = k[X,Y,Z]/(f), but irreducible algebraic curves lying on V correspond to non zero non maximal prime ideals in R. Again this is a fully faithful functor, with polynomial maps corresponding to k algebra maps. In particular the pullback of maximal ideals is maximal, but now the pullback of some non maximal ideals can also be maximal, i.e. some curves can collapse to points under a polynomial map.
To give the algebraic notion full flexibility, in particular to embrace non Jacobson rings with too few maximal ideals to carry all the desired structure, Grothendieck understood one should discard the restriction to rings without radical and expand the concept of a "point", to include irreducible subvarieties, i.e. consider all prime ideals as points, as follows.
IV. AFFINE SCHEMES
If R is any commutative ring with 1, let X (= "specR") be the set of all prime ideals of R, with a topological closure operator where the closure of a set of prime ideals is the set of all prime ideals containing the intersection of the given set of primes. (Intuitively, each prime ideal contains the functions vanishing at the corresponding point, so their intersection is all functions vanishing at all the points of the set, and the prime ideals containing this intersection hence are all points on which that same set of functions vanishes. So the closure of a set is the smallest algebraically defined locus containing the set.) This closure operator defines the "Zariski topology" on X.
Now any ring map defines a morphism of their spectra by puling back prime ideals, and in particular a morphism is continuous, although this alone says little since the Zariski topology is so coarse. Notice now maximal ideals may pull back to non maximal ones, e.g. under the inclusion map Z-->Q of integers to the rationals, taking the unique point of specQ to a dense point of specZ. Maximal ideals now correspond to closed points, and in particular there are usually plenty of non closed points. Intuitively, every irreducible subvariety has a dense point, and together these "points", one for each irreducible subvariety, give all the points of specX.
If K is a ring, a "K valued point" of X is given by a ring homomorphism R-->K, not necessarily surjective. E.g. if K is a field, the pullback of the unique maximal ideal of K is a not necessarily maximal, prime ideal P of R, the K valued point. Even if the point is closed, i.e. if P is maximal, we get information on which maximal ideals correspond to points with coefficients in different fields. If say k = the real field, and f is a polynomial over k, then a k algebra map g:k[X,Y]/(f)-->k has as kernel a maximal ideal corresponding to a point of {f=0} in k^2, i.e. a point of {f=0} in the usual sense, with real coefficients. The coordinates of this point are given by the pair of images (g(X),g(Y)) in k^2 of the variables X,Y, under the algebra map g, which after all is evaluation of functions at our point. But if say f = Y-X^2, the map from k[X,Y]/(f) -->C taking X to i, and Y to -1, corresponds to the C (complex) - valued point (i,-1), in C^2 rather than k^2.
More generally, if I is any ideal in Z[X1,...,Xn] generated by integral polynomials f1,...fr, and A is a ring, a ring homomorphism Z[X1,...,Xn]/I -->A takes the variables Xj to elements aj of A such that all the polynomials fi vanish at the point of A^n with cordinates (a1,...an). I.e. the map defines an "A valued point " of the locus defined by I. E.g. if M is a maximal ideal of R,we can always view the coordinates of the corresponding point in the residue field R/M, i.e. the point M of specR is "R/M valued".
This approach lets us recover tangent vectors too, in case say of a variety V with ring R = k[X1,...,Xn]/I, where radI = I, and k is an algebraically closed field. Consider the ring S = k[T]/(T^2), with unique maximal ideal (t) generated by the nilpotent element t. Then we claim tangent vectors to V correspond to S valued points (over spec(k)), i.e. to k algebra maps R-->S. E.g. if R = k[X], and we map R-->S by sending X to a+bt, then the inverse image of the maximal ideal (t) is the maximal ideal (X-a), and two elements of (X-a) have the same image in S if and only if they have the same derivative at X=a. Thus S valued points of V are points of the "tangent bundle" of V.
V. SCHEMES
One next defines a scheme as a space with an open cover by affine schemes, by analogy with topological manifolds, which have an open cover by affine spaces. For this we need to be able to glue affine schemes along open subsets, so we need to understand the induced structure on an open subset of V = specR. A basis for the Zariski topology on specR is given by the open sets of form V(f) = {primes P in specR with f not in P}. Intuitively this is the set of points where f does not vanish. (The analogy is with a "completely regular" topological space whose closed sets are all cut out by continuous real valued functions.)
On the set V(f), the most natural ring is R(f) = {g/f^n: g in R, n a non negative integer}/{identification of two fractions if their cross product is annihilated by a non neg. power of f}. I.e. since powers of f are now units, anything annihilated by a unit must become zero, so g/f^n = h/f^m if for some s, f^s[gf^m - hf^n] = 0 in R. Intuitively these are rational functions on V which are regular in V(f). This construction defines an assignment of a ring to each basic open set V(f) in V, i.e. it defines a sheaf of rings on a basis for V, and hence on all of V, by a standard extension device. This sheaf is called O, perhaps in honor of the great Japanese mathematician Oka, who proved much of the foundational theory for analytic sheaves.
Then one develops a number of technical analogues of properties of manifolds, in particular of compactness, and Hausdorffness, now called properness and separation conditions. Since the Zariski topology is very coarse, the usual version of Hausdorffness almost always fails but there is a better analogue of separation which usually holds. The point is that Hausdorffnes has a descriptiion in terms of products, and algebraic or scheme theoretic products also differ from their topological versions.
In making these constructions, mapping properties come to the fore, and are crucial even for finding the right definitions, so categorical thinking is essential. It is also useful to keep in mind, that some technically valuable varieties are not separated even in the generalized sense. I.e. sometimes one can prove a theorem by relaxing the requirement of algebraic separation.
VI. COHOMOLOGY
To really take advantage of methods of topology one wants to define invariants which help distinguish between different varieties, i.e. to measure when they are isomorphic, or when they embed in projective space, and if so then with what degree and in what dimension. One wants to recover within algebra all the rich structure that Riemann gave to plane curves using classical topology and complex analysis. Since the Zariski topology is so coarse, again one must use fresh imagination, applied to the information in the structure sheaf, to extract useful definitions of basic concepts like the genus, the cotangent bundle, differential forms, vector bundles, all in a purely algebraic sense. This means one looks at "sheaf cohomology", i.e. cohomology theories in which more of the information is contained in the rings of coefficients than in the topology. This is only natural since here the topology is coarse, but the rings are richly structured. Computing the genus of a smooth plane curve V over any algebraically closed field for instance, is equivalent to calculating H^1(V,O), where O is the structure sheaf.
The first theory of sheaf cohomology for algebraic varieties was given by Serre in the great paper Faiseaux Algebriques Coherent, where he used Cech cohomology with coefficients in "coherent" sheaves, a slight generalization of vector bundles. (They include also cokernels of vector bundle maps, which are not always locally free where the bundle map drops rank. This is needed to have short exact sequences, a crucial aspect of cohomology.) Cech cohomology is analogous to simplicial or cellular homology, in that it is calculable in an elementary sense using the Cech simplices in the nerve of a suitable cover, but can also become cumbersome for complicated varieties. Worse, for non coherent sheaves which also arise, the Cech cohomology sequence is no longer exact.
Other constructions of cohomology theories by resolutions ("derived functors"), e.g. by flabby sheaves or injective ones, have been given by Grothendieck and Godement, which always have exact cohomology sequences, but they necessarily differ from the Cech groups, hence computing them poses new challenges. (Just as one computes the topological homology of a manifold from a cover by cells which are themselves contractible, hence are "acyclic" or have no homology, one also computes sheaf cohomology from a resolution by any acyclic sheaves - sheaves which themselves have trivial cohomology. This is the key property of flabby and injective sheaves.)
As in classical algebraic topology, no matter how abstract the definition of cohomology, it becomes somewhat computable, at least for experts, once a few basic exactness and vanishing properties are derived. A fundamental result is that affine schemes have trivial cohomology for all coherent sheaves. This makes it possible to calculate coherent Cech cohomology on any affine cover, without passing to the limit, e.g. to calculate the cohomology H*(O(d)) of all line bundles on projective space. But once the affine vanishing property is proved for derived functor cohomology, it too allows computation of the groups H*(O(d)).
VII. SPECIAL TOPICS
It is hard to prove many deep theorems in great generality. So having introduced the most general and flexible language, one often returns to the realm of more familiar varieties and tries to study them with the new tools. E.g. one may ask to classify all smooth irreducible curves over the complex numbers, or all surfaces. Or one can study the interplay between topology and algebra as Riemann did with curves, and ask in higher dimensions what restrictions exist on the topology of an algebraic variety. Hodge theory, i.e. the study of harmonic forms, plays a role here.
Instead of global questions, one can focus instead on singularities, the special collapsing behavior of varieties near points where they do not look like manifolds. Brieskorn says there are three key topics here: resolution, deformation, and monodromy. Resolution means removing singularities by a sort of surgery while staying in the same rational isomorphism class. Deformation means changing the complex structure by a different sort of topological surgery which allows the singular object to be the central fiber in a family of varieties whose union has a nice structure itself. This leaves the algebraic invariants more nearly constant than does resolution. Monodromy means studying what happens to topological or other subvarieties of a smooth fiber in a family, as we "go around" a singular fiber and return to the same smooth fiber.
E.g. if a given homology cycle on a smooth fiber is deformed onto other nearby smooth fibers, when it goes around the singular fiber and comes back to the original smooth fiber, it may have become a different cycle! I.e. if we view the homology groups on the smooth fibers as a vector bundle on the base space, sections of this bundle are multivalued and change values when we go around a singularity, just as a logarithm changes its value when we go around its singularity at the origin.
People who like to study particular algebraic varieties may look for ones that are somewhat more amenable to computation that very general ones, e.g. curves, special surfaces, group varieties like abelian varieties. The latter is my area of specialization, especially abelian varieties arising from curves either as jacobians, or as components of a splitting of jacobians induced by an involution of a curve (Prym varieties).
Others study curves, surfaces and threefolds which occur in low degree in projective space such as curves in projective 3 space, or as double covers of the projective plane or of projective 3 space branched over hypersurfaces of low degree such as quadrics. Dual to varieties of low dimension are those of low codimension, e.g. the study of general projective hypersurfaces, varieties defined by one homogeneous polynomial. Some study vector bundles on curves, or on projective space.
Some examine how varieties can vary in families. One beautiful and favorite object of study are called "moduli" varieties, which are a candidate for base spaces of "universal" families of varieties of a particular kind, the guiding case always being curves. A very active area is the computation of the fundamental invariants of the moduli spaces M(g) of curves of genus g, and of their enhanced versions M(g,n), moduli of genus g curves with n marked points.
Another very rich source of accessible varieties is the class of "toric" varieties, ones constructed from combinatorial data linked to the exponents of monomials in the defining ideal.
VIII. PRERECQUISITES
To do algebraic geometry it obviously helps to know algebraic topology, complex analysis, number theory, commutative algebra, categories and functors, sheaf cohomology, harmonic analysis, group representations, differential manifolds,... even graphs, combinatorics, and coding theory! But one can start on the most special example that one finds attractive, and use its study to motivate learning some tools. This is a commonly recommended way to begin.
If you look here now at the talks in my birthday conference, you will see you already recognize some of the words:
can anybody follow posts 14,15 usefully at all? It is hard to summarize a 150 year old subject in 3 pages. Of course it is made easier by the fact that I don't know that much about it.
if you really want to learn the foundations of modern algebraic geometry, I recommend reading Mumford's "Redbook on algebraic geometry" for everything but the cohomology, and Serre's Annals paper FAC (or Hirzebruch's book on Topological Methods..) for the cohomology.
If you want to also understand what all the abstraction has to do with curves and classical algebraic geometry, I recommend reading first Griffiths' notes on Algebraic Curves, lectures from China, and Shafarevich's Basic Algebraic Geometry, vol 1.
If you read these 4 works, you will know an awful lot of algebraic geometry. After that, you can also read Hartshorne's book and benefit from it. A useful brief reference for this will also be Atiyah - MacDonald on Commutative Algebra, or their graddaddy Zariski - Samuel.
After learning cohomology from anywhere, you can read some wonderful specialized works such as A-C-G-H: Geometry of Algebraic Curves; and Beauville: Complex Algebraic Surfaces; also Barth- Hulek - Van de Ven's book on Surfaces; and Lange - Birkenhake: Complex Abelian Varieties. A great paper on a single 3 dimensional example is Clemens-Griiffiths' monumental Annals paper: The intermediate Jacobian of the cubic threefold.
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