Genus, differential forms, and algebraic geometry

[Hurkyl]

I decided earlier this week that I was going to compute by hand the genus of an elliptic curve. I've had a miserable (but enlightening!) time!

I eventually stumbled upon the trick in Shafaravich: I should be looking at the rational differential forms, and counting zeroes & poles of things.


But I still feel like there should be a way to do it without resorting to the holomorphic stuff. (Though, I suppose I don't have enough intuition for algebraic geometry to have any right to think so. )


This is what I had been doing:

I decided to consider the complex projective elliptic curve E : y² z = x³ - x z².

Let U be the open affine subset consisting of all of the points of the form (u : v : 1). This is the affine curve defined by v² = u³ - u.
Let V be the open affine subset consisting of all of the points of the form (s : 1 : t). This is the affine curve defined by t = s³ - s t².

Then {U, V} is an open cover of E. Let W be their intersection. On W, the change of variable relations are ut = s and vt = 1.

(Does omega -- Ω -- show up right?)

Ω, the differential forms on U, is simply the C-module generated by {du, dv}, satisfying the relation 2v dv = (3u² - 1) du.

Ω[V] is the C[V]-module generated by {ds, dt}, satisfying the relation (1 + 2st) dt = (3s² - t²) ds

And we have maps from both of these into the C[W]-module Ω[W]. The collection of all of the relevant relations is:

v² = u³ - u
t = s³ - s t²
ut = s
vt = 1
2v dv = (3u² - 1) du
(1 + 2st) dt = (3s² - t²) ds
u dt + t du = ds
v dt + t dv = 0


and I was trying to find the intersection of the images of the two maps. (As C-vector spaces, I suppose) But I just couldn't figure out how to do it. After much work, I eventually stumbled across the global differential form... but I have absolutely no idea how I would go about proving that was the only one (up to a constant).

So, I suppose my problem is that I just don't know how I would compute the intersection of the images of these two maps -- is this a tractable problem at all, in this case or in general?

 

[mathwonk]

what is your definition of the genus? if it is the topological one, then the compoutation is trivial, since genus is an integr it is unchanged under continuous defirmation, so deform your cubic to three lines in a triangle, then notice there is one hole in a triangle! so the genus is one.

another approach is to project onto the x axis, sending (x,y) to x. and count branch points. i.e. you get a 2 to 1 cover which has only one preimage over certain points, in your example, over 0,1,-1, and infinity. the hurwitz formula says the genus of the double cover must be then one, since the genus of the x-axis (a sphere) is zero. i.e. hurwitz just says that if you triangulate the double cover you get 2 edges over every edge and 2 faces over every face, but only one vertex over those 4 vertices, so v'-f'+e' = 2(v-f+e) - 4 = 2(2) - 4 = 0 = 2g-2,, so the genus is 1.

if your definition is the vector dimension of the space of global holomorphic differentials, then you can easily construct one ala shafarevich, by considering dx/y. to prove there cannot be more than one, note that a holomorphic one form and its complex conjugate, give harmonic forms on the manifold, and if there are mnokre of these than the rank of the first homology group, then one would get a linear combination which is exact, in the sense of havng zero integral over every homology mcycle, hence defines a global harmonic function.

but such a function is constant by the maximum principle.

thus the topology always tells you there can never be more than g inmdependent differentials and the real problem is to construct them. constructing them via the rational forms approach in shafarevich is due to riemann, in his original paper.if your definition is the dimension of the cohomology group h^1(O), please consult my notes on the RRT on my webpage, p.41, where the formula is proved:

1-g = 1 - (d-1)(d-2)/2, for a smooth curve of degree d. when d = 3, you get

1-g = 0. or just notice this number g = (d-1)(d-2)/2 is the number of holes in a polygon with d sides, and again for a triangle we get one.

or notice that you differential form has no zeroes, hence the "hopf theorem", due again to riemann in this case, says the euler characteristic of your surface is zero, = 2g-2, so g = 1.

 

[Hurkyl]

The definition I have at my disposal that I actually understand is the vector space dimension of the global regular differential forms.

I also have the arithmetic genus defined in Hartshorne, but I don't understand the Hilbert polynomial at all, so that one's right out.


I guess I should point out that I'm not specifically interested in complex elliptic curves -- I just figured that amongst all the possible choices of fields and varieties, that would be the simplest nontrivial case to look at.


After seeing the idea, I think I can reproduce Shafarevich's approach to computing the genus of any hyperelliptic curve.

I was sort of hoping there was a dumb brute force approach -- even if you would never do it that way in practice, sometimes it's instructive to see! I was trying to cobble something together with Gröbner bases and linear algebra, but I'm still relatively new to actually computing things with Gröbner bases and modules.


But seeing different approaches is always good. How much of what you said can be formulated in the general case of an algebraic curve? Or an algebraic variety?

 

[matt grime]

All of it. And it works in almost any characteristic. The Riemann-Hurwitz formula breaks down if the ramification indices are divisible by the characteristic of the field.

I have come to the conclusion that geometry is actually relatively straight forward if we could just get the geometers to use plain English and stop blinding us with fancy terms.

 

[mathwonk]

if you just want to construct a differential, just differentiate the equation,

i.e. if your equation is y^2 = x^3 + x, then you get 2ydy = (3x^2+1)dx.

then dx/y = 2dy/(3x^2+1). these two expressions show the differential is well defined. you can check holomorphicity by noting that the map defined by x ramifies exactly where y vanishes, so dx and y have the same zeroes.

 

[mathwonk]

and note this differential has no zeroes or poles. since any two differentials can be divided to yield a function, if there were two of these then their quotient would yield a function with no zeroes or, poles hence constant.

so the space of them is one dimensional.

is that explicit enough?

 

[mathwonk]

let me try to make this as detailed and explicit as possible.

1) on a smooth projective curve, every non constant function has both zeroes and poles, and the same number of each, counted properly.

(for a rational function f, p is a pole of f iff p is a zero of 1/f.)

2) the quotient of two differentials (the one in the bottom being non zero) is a rational function.

3) hence any smooth plane curve having a differential with no zeroes and no poles, has genus one, where genus is defiend as the vector dimension of the space of differentials with no poles.

proof: if Z is a differential with neither zeroes nor poles, and W is any other differential with no poles, then W/Z is a rational function with no poles, hence no zeroes, hence constant. so W is a constant mkultiple of Z. thus Z is a basis of the space of differentials with no poles. mhence the genus is one.

corollary: the plane cubic with equation y^2 z = x^3 - x z^2, has genus one.
proof: we need only exhibit a differential with neither zeroes nor poles.
we go affine by setting z = 1, getting equation y^2 = x^3 - x. now we take d of both sides, getting, 2y dy = [3x^2 - 1]dx.
now we divide by, well you know, getting dx/y = 2dy/[3x^2-1]. we claim this is our desired differential.

"clearly" the expression on the left shows there are poles at most where y = 0. but at such points the function x^3 - x, has a simple zero, hence its derivative 3x^2 - 1 has no zero. so the expression on the right shows there is no pole at these points either.

so this is a differential with no poles. as to zeroes, the only zeroes of the expression on the elft must be where dy = 0, but the onlyt zeores of the expression on the right are where dx = 0, and since our curve is smooth, dx and dy do not have any common zeroes.

thus this differential has no zeroes or poles, except possibly at infinity, and I leave it to you to check there.

If you consult shafarevich, in the chapter III on differential forms, either sections III.5.4-5 in the 1974 edition or sections III.6.4-5 of the 1994 edition, you will find the genus formula I gave above for all hypersurfaces. I.e. he explicitly computes the number of regular differentials on any smooth hypersurface.

then he does it again on all plane hyperelliptic curves, which includes cubics, but by a method which works on all smooth plane curves.

hows that? (the first three results I gave as facts are also proved in shafarevich.)

 

[mathwonk]

remark: these results are not trivial, as genus >0 implkies non rational, and just proving non rational is quite hard. just try it by hand on your example without the theory of the genus, which came from geometry via riemann.

the interesting and hard problem then is to decide rationality for example where the genus does not do it for you. e.g. a cubic threefold (hypersurface in P^4), has no global regualr differentials, but can still be shown to be non rational.

this was a big spectacular result by clemens - griffiths in 1972, also by artin -mumford, and 2 russian mathematicians. one must find another, subtler invariant that vanishes on rational avrieties but not on these objects.

the one used by clemens griffiths agin came from geometry, the intermediate jacobian, also computable as a cycle class group.

these methods from geometry decide the purely algebraic question of whetehr every subfield of a purely transcendental field of rational functions, like k(X,Y,Z), must itself be pure transcendental, the "luroth problem".

mumford also gave another proof (unpublished) using Prym varieties.

the details were given by Beauville.

A. Beauville, Les singularites du diviseur

 

[mathwonk]

references on cubic threefolds

mumford also gave another proof (unpublished) using Prym varieties.

the details were given by Beauville.

A. Beauville, Les singularites du diviseur theta de la Jacobienne intermediare de l'hypersurface cubique dans P^4, Algebraic threefolds (Proc. Varenna 1981), 190-208: Lecture Notes 947, Springer-Verlag, Berlin- Heidelberg- New York (1982).

or in English:

Smith, Varley, A Riemann singularities theorem for Prym theta
divisors, with applications, Pacific Journal of Math, vol. 201, no.
2, Dec 2001, 479-509.

 

[mathwonk]

ok here is another calculation: suppose C is a smooth curve with a differential having exactly 2 zeroes and no poles. i claim C has genus 2.
proof: i guess you have to construct another regular differential, hence also having 2 zeores. then in the 2 dimensional space spanned by these, you can find a non zero one with a zero at an given point. so if you are given another differential with no poles, it must have 2 zeroes, and you can find a linear combination of the first two with a zero at one of those zeroes. then the quotient of your differential by that linear combination has only one zero and one pole, hence defines a map from C to P^1 of degree one or less. but a curve with a non zero differential cannot have a map of degree one to P^1 since that would be an isomorphism, and P^1 has no non zero differentials.

so our map is constant and hence our given differential must be a linear combination of the first two. so any smooth curve with a differential having 2 zeroes and no poles, has genus 2. (e.g. a plane quartic with one ordinary double point is the image of such a curve under a map identifying two points to the double point.)

 

[mathwonk]

i am doing my best here to make this concrete (no cohomology!). is it helping?

it was riemann by the way who showed how to write down differentials on plane curves using rational functions.

 

[mathwonk]

by the way, the idea of the genus works to show non rationality for plane curves of degree 3 or mroe, surfaces in 3 space of degree 4 or more, threefolds of degree 5 or more, etc...

but it is believed that actually smooth hypersurfaces of degree 4 or mroe are never rational, but no argument is known in high dimensions.

 

[mathwonk]

helooo? helooo?

 

[Hurkyl]

I'm still here. Just haven't worked out what to say yet.

 

[mathwonk]

well the post i stand by is #7. that is what I think may be what you want. and it has nothing fancy in it.

i.e. to compute dimension of space of differentials as one, we need to produce one and then prove there are no others. the trick of dividing two differentials to get a rational function is the most direct way, and was also classical, going at least back to weyl and maybe riemann.

i.e. looking at the differewntials as a vector space over the rational functions as well as over the complex numbers is a basic trick. in fact that is the underlying trick in the serre weil proof of "serre duality" (the roch part of riemann roch) via adeles.

sorry i keep going off on a tangent. but i have spent 30 years contemplating these objects and I like them.

 

[Hurkyl]

Oh bleh, my response got killed.


I'm not opposed to seeing fancy techniques -- it's just nice to see the "simple" techniques applied too. It was easy to turn the problem into a "simple" one, but I was mildly irritated I couldn't solve the problem with "simple" techniques!


Once I saw to use rational differentials, it was easy enough to compute the genus of the elliptic curve -- and would now be even easier using the tricks you've mentioned.


Post #10 was very useful -- based on the elliptic curve case, I had been wondering if differential forms had the property that {number of zeroes} - {number of poles} = 0.

Of course, I already knew that wasn't true, but forgot: I had worked out that if x is the coordinate on the line, that dx has a double pole at infinity, and no zeroes.


And now that I realize that in the three cases: g=0,1,2, that {number of zeroes} - {number of poles} = 2g - 2, the Riemann-Roch theorem (at least a special case of it) finally makes sense! I had never quite fathomed the canonical class before -- it's amazing how quickly a definition can make sense when one starts wondering about the idea it defines!

 

[mathwonk]

i am very glad you asked this question as it motivated me to give a simple proof of the result.

i have more difficulty showing that a smooth plane quartic has genus 3, but again shafarevich's proof of the adjunction formula for hypersurfaces will do iit. i just do not remember how he proceeded and don't like to peek.

the key point seems to be that the 4 zeroes of a regular differential are all collinear. but why?

i.e. I recall his proof hinges on showing all regular differentials are given by multiplying dx/[

 

[mathwonk]

post#10 hides another useful fact: i.e. it is not entirely trivial that a degree one map to a smooth curve is an isomorphism.

e.g. the degree one map t goes to (t^2,t^3) is not an isomorphism from P^1 to the cuspidal curve x^3=y^2.

 

[Hurkyl]

Oh, something I forgot to repost -- I (think I) understand the topological definition of genus for a dimension 2 manifold, but not in a way that carries over to the case of an algebraic variety!

a smooth plane quartic has genus 3

What, then, is a "smooth plane quartic"? I know elliptic curves can be written as quartic equations, and they certainly don't have genus 3!

 

[mathwonk]

the degree of a (reduced)( plane curve of degree d equals (d-1)(d-2)/2 - a contribution coming from any singular points.

a smooth plane quartic has genus 3 but there do exist plane quartics of form y^2 = x^4 + 1 say, which have genus 1 because they have a singularity at infinity that subtracts 2 from the genus.

for more general algebraic varieties i was taking your preferred definition of genus as the number of differential n - forms where n is the dimension of the variety, so for surfaces it would be 2forms and for threefolds it would be 3 forms. etc...there are homology groups of course of such varieties, let's see, the first homology group has dimension twice that of the space of 1 forms, but for 3 folds say, the hodge decomposition of the 3 cycles yields forms of type (0,3), (1,2), (2,1) and (3,0) of which only the forms of type (3,0) give the genus.

for special 3 folds with no (0,3) forms, such as smooth 3 folds in P^4, we have the space of (1,2) forms and the space of (2,1) forms reflect the topology of the 3 fold but there are no holomorphic 3 forms so the "genus" is zero.i have some notes on plane curves i will try to put on my webpage, with pictures of curves with singular points, showing how the singularity contributes to the genus.

i have course notes on plane curves (a la walker), basic algebraic geometry ( al la shafarevich), and geometry of surfaces ( a la beauville).

 

[mathwonk]

i also have notes on sheaf cohomology (a la kempf and serre and mumford).

 

[mathwonk]

a singularity is a point where both dx and dy = 0, i.e. where both ?f/?x and ?f/?y = 0.

 

[mathwonk]

lets stick to surfaces, i.e. complex curves. then the euler characteristic equals the altyernating sum V-E+F of vertices minus edges plus faces, for any triangulation of the surface, and this is also equal to the number of zeores of any vector field, with finitely many zeroes, or dually to minus the number of zeores of a differwntial form, i.e. a covector field.

so the eulet characteristic is 2-2g and the number of zeroes of a differwential is 2g-2.for a higher dimensional manifold, the euler characteristic is the number of zeroes of a vector field. it also equals the alternating sum of the number of vertices faces edges etc of any triangulation, and also the alternating sum of the ranks of the topological homology or cohomology groups.

the other key invariant for a complex surface, i.r. algebraic real 4 maifold, is the number of zeroes of a 2 form i.e. c1. but a 2 form has a curve of zeores so we must take the self intersection of this curve to get an integer so this is called c2^2, so the chern characters o0f a surface are c1 and c2^2.

the noiether formual says that (1/12)[c1+c2^2] = chi(O) = 1-h^1(O) +h^2(O) = 1 - h^1(O) + number of regular 2 forms.

and for simply connected surfaces, h^1(O) = 0, so for those, 1 - the number of regular 2 forms = (1/12)[number of zeroes of a vector field + self intersection number of the curve of zeroes of a 2 - form].

this is the so called noether foprmula for algebraic surfaces.

i suggest you see my notes on RRT on my webpage. i wrote them for some one bright, naive, and curious like you.

 

[Hurkyl]

I thought I had bookmarked your page, but alas I seem not to have the link! What is it?

By the way, maybe it would be a good idea to put the link in your profile here on PF?

 

[mathwonk]

http://www.math.uga.edu/~roy/

class notes are at the bottom, research (p)reprints are slightly higher up.

the photo is me as a child.

 

[mathwonk]

basically differential forms reflect the topology to some extent, esssentiaklly the cohomology, or dually topological cycles modulo boundaries, in different dimensions.e.g. for a comp[lex curve, i.e. real 2 manifold, there is homology only in dimensions zero one and two. the zero and two dimensional homology are both rank one, so the one dimensional case is the interesting one.

there H1 is an abelian group of even rank 2g, where also V-E+F = 2-2g, for Vertices, Edges, Faces, of any triangulation of the (compact connected) real surface.

there are also exactly a g dimensional space of holomorphic one forms, and conjugate to these a g dimensional spacxe of antiholomorphic one forms, that together make up a 2g dimensional space dual to the one cycles/

these two types of forms, holomorphic and antiholomorphic, are called (1,0) and (0,1) forms, and together span the space of harmonic one forms.for higher dimensional smooth complex varieties, always even dimensional real manifolds, there is homology in every dimension again, and by poincare duality the dimensions are symmetrical, with top and zeroth ranks beiing one, and the others less predictable.

for smooth sufaces in projective 3 space, H^1 is always zero, so the only interesting dimesnion is that of H^2, and among these one wants to know which ones are represented by algebraic curves on the surface.

for threefolds in P^4 again H^1 and H^5 are zero so one wants to know H^2 and H^3. for some reason the mifddle dimension is always the most interesting, i guess by lefschetz theorem that the others are mostly predictable from taking a hyperplane section, so really come from lower dimensions.

in dimension 3, there are harmonic forms of types (3,0), (2,1), (1,2) and (0,3), with symmetrical ranks. for a cubic threefold, there are no (0,3) or (3,0) forms and the harmonic forms of type (1,2) are 5 dimensional and in a sense take the place of the holomorphic differential forms on a curve.

thus although the geometric genus is zero, there is an intermediate jacobian variety, an algebraic group, of dimension 5 associated to a cubic threefold that contains much information about the threefold.

so the link between the topology and the differential forms is fairly clear, but the hard problem in higher dimensions is to determine which homology cycles are algebraic.

for surfaces, it is the 2 cycles that are of type (1,1) that come from algebraic curves, but which 2 cycles on a threefold come from algebraic curves is harder to be sure of.

so I guess

 

[mathwonk]

if you will tell me how toa dd information to my profile i will do so. i just tried without success to put my web address there.

 

[mathwonk]

computing the genus

lets get back to the question: how do you compute the genus of a smooth plane curve? There are several definitions available each with its own difficulties but we take this to mean computing the vector dimension of the space of regular differential forms.

in general riemann studied the problem of computing the number of rational functions with a given set of points as poles, or more precisely with a given divisor (points with multiplicities) dominating the pole divisor.

computing the genus is a special case, the most important one, of this problem. i.e. let w be a given regular differential. then since the quotient of two differentials is a rational function, every other regular differential is defiend by multiplying w by a rational function whose pole divisor is dominated by the zero divisor of w.

i.e. to get another differebntial, we must multiply the given one w by a rational function. to get a differential which is regular, the multiplying function must have its poles canceled by the zeroes of w. thus we are solving a riemann roch problem, how many rational functions exist with pole divisor dominated by the zero divisor of w?

the difficulty in this problem is reversed for algebraic curves as opposed to compact riemann surfaces: for riemann surfaces, homology and the maximum principle for harmonic functions limits the number of meromorphic functions with given poles, and the difficulty is to produce some.

for algebraic functions we can write them down, and without homology, the difficulty is to limit the number of them from above.

the basic result then replacing the maximum principle, is that a rational function on a projective curve with no poles at all is a constant. this is a corollary of the hilbert nullstellensatz and proved in shafarevich near the beginning. this is a non trivial result.

then a corollary of that is the fact that, on an affine plane curve, i.e., a closed hypersurface in A^2, the only locally rational functions with no poles are global polynomials.

now with that, we can compute the genus of a smooth projective plane curve as follows. write down one global holomorphic differential, given on the affine plane by w = dx/[?f/?y] where ? means curly d. note that since the curve is defined by f = 0, the differential df = 0, so ?f/?y dy + ?f/?x dx = 0, so dx/[?f/?y] = -dy/[?f/?x].

and since f is smooth, either ?f/?x or ?f/?y is non zero at each point, hence these expressions show the differential has no poles in the affine part of the plane at least.
i hope it also has none at infinity.

now to get another regular differential we have to multiply w by a rational function with poles dominated by the zeroes of w. computation shows that there are no zeroes of w in the affine plane. hence we must multiply by a rational function with no poles in the affine plane, hence by the nullstellensatz corollary above, it must be a polynomial there.

then all the zeroes of our fixed differential are at infinity, and so our polynomial must not only be regular in the affine plane, but also must have at infinity a pole divisor dominated by the zeroes of our differential w at infinity.

computation shows w has zero divisor at infinity cut out on the curve by (d-3) times the line at infinity where d is the degree of the original curve.

hence our polynomial in the plane can only have a pole at infinity bounded by the divisor (m-3) times the divisor cut by the line at infinity. this means our polynomial can only have degree at most d-3. thus the genus of a smooth plane curve of degree d, equals the dimension of the space of polynomials of degree d-3 or less in the affine plane. this dimension is again a special case of the riemann roch theorem, but one we know how to compute.

namely the dimension of that space of polynomials is the binomial coeficient "d-1 choose 2" = (1/2)(d-1)(d-2), the genus of a smooth plane curve of degree d.

to recap, to begin the computation of how many regular differentials there are, we begin by writing down an explicit one, and computing its zeroes. then we are reduced to computing how many rational functions have poles canceled by those zeroes.

by clever choice of the explicit differential, all zeroes are at infinity, so this reduces ultimately to the fact that a rational function with no poles in the affine plane is a polynomial.

 

[mathwonk]

note in the computation above, the zero diviusor of a differential on a smooth curve of degree d, always has degree d(d-3).

i guess the general riemann roch theorem is harder to prove since the dominating divisor is not always equivalent to a multiple of a line.

presumably for those that are, one could imitate this proof.

in fancy language, not all line numdles on a plane curve are restrictions of line bundsles on the plane. in fact there are only a countable number of line bundles on the plane, and a curve has a whole jacobian variety of line bundles.

put another way, multiples of line bundles on the plane would have degree dn for some n, but a line bundle on a curve can have any degree, and there are a huge number of each degree.

so in the g dimensional jacobian variety of line bundles of degree d(d-3) there is in fact only one line bundle which is a restriction of one from the plane, and fortunately its the one we want to calculate here.

i.e. the plane only has one line bundle of each degree, while a curve of genus g, has a g dimensional family of them!

so this method of solving the riemann roch problem works precisely on that one special line bundle of that degree that comes from the plane.

how interesting. maybe i can teach this stuff better next time, since maybe i am beginning to understand some of it.

 

[Haelfix]

Thats a pretty broad and impressive arsenal you have there Mathwonk, I must admit some of it went over my head (different language for the same concepts)

Im much more used to Riemann Roch with theta functions and abel's theorem (eg in the context of Riemann surfaces). Also in terms of sheaf cohomology and the index theorem.

But the algebraic curve aspect goes swoosh.

 

[mathwonk]

what do you refer to as rrt with theta functions? rrt on an abelian variety? i.e. the dimension of h^0(M) where M is a multiple of the theta divisor on an abelian variety?

or the riemann singularity theorem that for line bundles L of degree g-1 on a curve of geneus g, that the dimension of h^0(L) equals the order of vanishing of the theta function on the jacobian at L?

or something else? (i also have a set of notes on the theta geometry of jacobian varieties. in the book, lectures on riemann surfaces, edited by gomez - mont, from world scientific.)

. Lectures on Riemann Surfaces: Proceedings of the College on Riemann Surfaces, International Centre for Theoretical Physics, Trieste, Italy, 9 Nov.-18 DEC., 1987
M. Cornalba, X Gomez-Mont, A Verjovsky
Format: Paperback
Pub. Date: January*1989

 

[mathwonk]

riemann surfaces are almost the same as algebraic curves. i.e. a riemann surface is a complex one manifold. as an example, take any equation f(z,w) in C^2 with no point where both partials vanish.

the implicit function theorem says the zero locus is a complex manifold. then close up or compactify the complex plane C^2 as the projective plane, adding in a copy of C, and one more point at infinity.

there is a way to view this compact complex 2 manifold as a union of three copies of C^2, so the points at infinity look the same as other points. then one can use the equations for the curve in the other two copies of C^2 to check whether the curve is still a manifold at infinity as well.

if so, one gets the "riemann surface" of the original algebraic curve. if one considers the variable w as a multivalued function of z, defined implicitly by the equation f(z,w) = 0, this is the riemann surface of that function. i.e. the projection of (z,w)-->z, from the curve to the z axis, is a multi sheeted cover of the z line, with as many sheets as there are choices of values for w, over each point z.


now a (reduced) algebraic equation always defines a multivalued implicit function somewhere locally, even if the curve has some singular points, i.e. both partials may vanish somewhere, but the riemann surface should be a manifold.

so riemann "desingularized" the curve by lifting it out of the plane and unkinking those points where the partials vanish, and adding in new points where pairs of points had been identified, so the algebraic plane curve always differs at most at a finite numebr of points from a plane curve.

then riemann proved that every compact one manifold can be mapped into the plane as an algebraic plane curve, usually with a few singular points. and he estimated the maximum degree of the curve from the genus of the manifold.

 

[mathwonk]

the rrt notes on my webpage cover some more of these aspects, not theta functions, but index theorems, sheaf cohomology and riemann surfaces and rrt.

i also discuss there hirzebruch rrt, and using sheaf cohomology to compute rrt on plane curves.

 

[mathwonk]

there is a functor from irreducible algebraic curves over C, to riemann surfaces. each rational map of irreducible algebraic curves extends uniquely to a holomorphic map of riemann surfaces.

given a pair of embedded smooth algebraic curves, a holomorphic map of their riemann surfaces defines a unique algebraic map of the curves.

two irreducible algebraic curves are birationally isomorphic,a nd if non singular they are isomorphic, if and only if their associated riemann surfaces are holomorphically isomorphic.

in fact if their jacobian varieties are isomorphic by an isomorphism that takes the theta divisor of one isomorphically, or even homologically, to the theta divisor of the other, the riemann surfaces and curves are isomorphic.

 

[mathwonk]

try this on for size:

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 dF/dw (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 not 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 dF/dw (a,b) != 0 or
dF/dz (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 P^2, by copies of the plane C^2, we eventually get the surface S, which is in fact compact, and comes equipped with a holomorphic map S-->X in P^2, 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.

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 not 0), and these ƒi determine back the map ƒ. Indeed the ƒi determine the holomorphic map S0-->C^n = {z0 not 0}in P^n.

Analyzing ƒ by the poles of the ƒi
Note that since the ƒi are holomorphic in ƒ-1(z0 not 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” <sum> 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 topological 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: <sum>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 equiv 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 equiv 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) = <sum> 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 Picd(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 Cg.
(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.

Next we discuss how to clasify all Riemann surfaces of genus g, using the idea of a moduli space. (to be continued?)