Basic algebraic geometry for coding theory

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Basic algebraic geometry for coding theory

Rolf-Peter Holzapfel

To cite this version:

Rolf-Peter Holzapfel. Basic algebraic geometry for coding theory. 3rd cycle. La Havane (Cuba), 2000, pp.58. �cel-00374747�

Basic Algebraic Geometry for Coding Theory

Lectures on the CIMPA-UNSA-ICTP-UNESCO-ICIMAF School

"Algebraic Geometry and its Applications

to Error Correcting Codes and Cryptography"

Havanna, 20-th Novembre - 1-st December 2000

vspace5mm Rolf-Peter Holzapfel, Humboldt-Universitaet Berlin

December 12, 2000

Contents

1 Introduction

2

2 Algebra and Ane Geometry

3

3 Projective Geometry

12

4 Singularities

16

5 Algebraic Curves

19

6 Riemann Surfaces

25

7 Plane Curves

34

1 Introduction

According to my experience one should start after some historical interesting el-ementary codes with the original Goppa codes as in 9a] in order to understand error correcting codes living on algebraic curves over nite elds as basically described in Stichtenoth's book 7]. But this is an arithmetic-geometric jump which should be well-prepared. The aim of my lectures was to present the fun-damental background knowledges of algebraic geometry hoping that graduate students starting on this eld can understand the things better, in a more conti-neous manner. We avoid in this summery schemes, sheaves, higher cohomology groups and adeles, but the basic denitions, relations and theorems presented in my lectures are also thought as necessary preparation for deeper work in one or the other of these directions. The order of presentation is well-choosen. It corresponds to the order of proofs (in my lectures in Berlin), which are omitted here in general. We concentrate our attention to the case of algebraically closed basic elds. If it is not necessary we will announce it (quite often) and work in more generality. At some places, when derivaties play a role, we are forced to restrict ourselves to characteristic 0. Naturally, on the topological side of Riemann surface theory one has to use basically the complex numbers. Real numbers are necessary for visualisation of plane curves, their singularities and quadratic transforms. We add a MAPLE le with real curves expositing nice singularities and birational transforms. This le "CurvAlbm.mws" must be im-planted into the MAPLE-package to be readable. It plays also the role of a foto album for curves I met during the preparation of the school and and in some other lectures there. The picture "Newton's heart" discovered in Newton's knot should be taken as a symbol for all activities of the school.

I want to mention that this guideline of algebraic curve theory contains also little innovations in order to nd a most natural and eective way of presenta-tion. On the one hand we introduced the K-Dimension of K-algebras substi-tuting Krull-dimension, see 2.6. It opens some problems which could be solved by advanced students. On the other hand, I believe deeply that the more im-mediate Mittag-Leer approach to the Riemann-Roch Theorem using spaces of Laurent tails only can be used to substitute completely the adele methods in 6] for each characteristics. This should be carefully investigated and written down in detail. At least it is a good preparation for understanding the arithmetic adele theory with applications.

0

TheauthorthanksheartlytheorganizersatCIMPA,UNSA,ICTP,UNESCO,ICIMAF fortheinvitationaslectortothisinterestingschool.

2 Algebra and Ane Geometry

Denition 2.1

. LetK be a eld a commutativeK-algebra is a commutative ringR containingK as subring.

Since non-commutativeK-algebras do not play any role in this summary, we call our commutativeK-algebras shortlyK-algebras. We do the same for rings omitting the adjective "commutative", which the reader should have in mind. For instance, the ringKX1:::Xn] of polynomials with nvariablesX1:::Xn

and coecients inKis aK-algebra, which plays a fundamental role in algebraic geometry. An ideal of a ringR is a subgroup of the aditive group ofR, which is closed underR-multiplication. Besides of intersections the most important binary operations in the ordered set of ideals ofRare sums and products, dened by

a+b=fa+ba2a b2bg

ab=fa1b1+:::+akbk ai2a bi2b k2 N

+g

The denitions can be extended to sums and products of more than two ideals in obvious manner. It should be mentioned that the distributive law

(a+b)c=ac+bc

holds, which is also extendable to more than two summands. Ideals of the form

a= (a1a2:::ak) :=Ra1+Ra2+:::+Rak ai2R

are called nitely generated witha1a2:::ak as (set of) generators, sometimes

also called ideal basis ofa. Ideals (a) generated by one element only, are called principal. The trivial ideals are the principal ideals (0) and (1) =R. All other ideals are called non-trivial. The proper ideals are the ideals dierent fromR. A ring is a eld if and only if it has only trivial ideals.

Ifais a proper ideal of the aK-algebraR, then the residue class ring R=a

with addition and multiplication dened by representatives) is also aK-algebra. The idealaof a ringRis the kernel of the residue class mapR;!R=a, which is

a ring homomorphism (sendingr2Rto its residue classr moda). An maximal

ideal mis a maximal one in the set of proper ideals ofR. It is equivalent to say thatR=mis a eld. The idealpis called prime, if and only ifR=pis a domain this means a commutative ring with unit element but without zero divisors. An element 2Ris a prime element i () is a prime ideal. We say thatadivides

band write ajb, iab. Prime ideals are characterized by the implication

pjab)pjaorpjb

for all ideals a, b of R. Two ideals a, b are relatively prime, if and only if

R is a nitely generated K-algebra i there exist elements x1:::xn 2 R

(called generators) such that

R=Kx1:::xn] =ff(x1:::xn) f 2KX1:::Xn]g:

Up to isomorphy, the proper residue class rings of the polynomial ringKX] :=

KX1:::Xn] and theK-algebras generated bynelements are in bijective

cor-respondence. Namely,

Kx1:::xn]=KX]=a a=ff 2KX]f(x1:::xn) = 0g:

Not only the residue class rings of ideals ofKX] are nitely generated but also the ideals themselves. More generally, we dispose on the important

Hilbert's Basis Theorem 2.2

. Each ideal of a nitelely generatedK-algebra is nitely generated.

For a modern argument one denes noetherian ringsRby the ascending chain condition, which says that each strictly ascending chain of ideals of R must terminate (after nitely many steps). It is easy to verify the equivalence with the nite base property: each ideal ofR is nitely generated. One proves that the noetherian property ofRis preserved for the ringRT] of polynomials over

Rin one variable T. The original theorem of Hilbert follows by induction from

Kx1:::xn;1] toKx1:::xn] via factorization ofKx1:::xn;1]Xn].

We are now motivated to assume that all ringsRconsidered in this exposition are noetherian.

Now we work with the polynomial ringR =KX1:::Xn] over a eld K.

We correspond ideals with special subsets of the ane spaceAn(K) which can

be identied withKn _{forgetting the vector space structure. For a polynomial}

f =fX1:::Xn]2KX1:::Xn] we denote and dene the zero set off inK

n

by

V(f) =VK(f) =fP = (p1:::pn)2K

n_{ f(P) = 0g:}

More generally, the zero set of an ideal ais dened as

V(a) =VK(a) =fP = (p1:::pn)2K

n_{ f(P) = 0for all f 2ag:}

The zero sets V(a), a ideal in KX1:::Xn], are called algebraic subsets of

Kn₌An(K). Conversely, we correspond to each subsetM ofKnthe vanishing

ideal ofM

I(M) =IK(M) :=ff 2KX1:::Xn] f(m) = 0for all m2Mg:

It is clear that

a I(V(a)) M V(I(M)):

Moreover, for algebraic sets UWV1::Vn of K

n _{and ideals aca}

1::an of

(0) U WKn)I(U)I(W) a c)V(a)V(c) V(I(U)) V(I(W)) I(V(a)) I(V(c)) (1) I(V1::Vn) =I(V1)\::\I(Vn) (2) V(a1:::an) =V(a1\::\an) =V(a1)::V(an)

Most interesting is the precise understanding of the correspondence between algebraic sets and vanishing ideals. For this purpose we introduce the radical of an idealaof an arbitrary ring setting

Rada=Rad(a) :=ff 2R fk2afor a suitable k2N +g:

This is an ideal ofR containinga. We are motivated to look for the greatest ideal ofKX1:::Xn] containing a given idealawith the same zero locus. The

radical is obviously a natural candidate.

In general we notice the following properties:

 a Rad(a)

 a c)Rad(a) Rad(c)  Rad(a) =R,a=R  Rad(Rad(a)) =Rad(a)  ifpis prime, thenRad(p) =p

 Rad(ac) =Rad(a\c) =Rad(a)\Rad(c) .  Radam=Rada.

 Rad(a+c) =Rad(Rad(a) +Rad(c))

 a,care relatively prime, iRad(a) andRad(c) are relatively prime.  Rada=T

fprime ideals of R containingag

We present now the basic versions of Hilbert's Nullstellensatz (HN). For three of them one has to assume thatK= K is an algebraically closed eld ( Kdenotes the algebraic closure ofK).

Hilbert's Nullstellensatz 2.3

(ane versions) The correspondencesI,V be-tween ideals of KX] = KX1:::Xn] and algebraic sets in K

n _{restrict to the}

following bijective correspondences:

(HN.0) fpoints ofKng , fmaximal ideals of KX]g

(HN.1) firreducible algebraic sets ofKng , fprime ideals of KX]g

By denition, an irreducible algebraic set cannot be composed by joining any two smaller algebraic sets. Each algebraic setV is a join of nitely many irreducible algebraic sets. They are uniquely determined by V and called (irreducible) components ofV.

The original version of Hilbert's Nullstellensatz is the following:

(HN) LetK be an arbitrary leld. Each systemf1:::frofK-polynomials inn

variables, which is not relatively prime, has at least one common zero in 

Kn_.

A ring is called factorial or a unique factorization domain (UFD), if each el-ement, which is neither 0 nor a unit, has a unique (up to unit factors and numeration) multiplicative decomposition into prime factors. Our third basic result is the following:

Theorem 2.4

. For each eldK the polynomial ringKX1:::Xn] is factorial.

The key of proof is the following implication: ifRis factorial, then also the ring

RT] of polynomials in one variable with coecients inRis. This can be proved by means of the

Gau~-Lemma 2.5

. If the domauinR is factorial, then for each pair of poly-nomialsf g2RT] it holds that

g:c:d:(coefficients of fg) =g:c:d:(coeff: of f)g:c:d:(coeff: of g)

holds.

Thereby g.c.d. denotes the greatest common divisor, which is uniquely dened for nitely many elements of the UFD-domainRup to a unit factor.

We want to dene the dimension of algebraic manifolds in the most easy and natural manner. For this purpose associate "rings of functions". The coordinate ring ofaKX1:::Xn] is the residue class ring

Kx1:::xn] =KX1:::Xn]=a xi=Ximoda:

The elements are understood as polynomial functions onV(a). Namely,f(P)2

Kforf 2Kx1:::xn] andP2V(a) is dened asF(P), whereF 2KX1:::Xn]

is an arbitrary representative off. Belonging to athe dierence of two repre-sentatives vanishes on V(a). Therefore the denition of f(P) is correct. The pair

Va= (V(a) Kx1:::xn])

is called ane algebraic manifold attached toa. The residue class ring

Kx1:::xn] is also denoted byKV] (withV=Va) and is called the coordinate

ring ofV. For prime idealspthe coordinate ring is a domain. Its quotient eld - denoted byK(V) - is called the (algebraic) function eld of K(Vp), andVp

of misunderstandings of notations because p-adisations will not occur in this paper. Now we have also a more precise denition of the ane spacesAnK as

"ringed space" (V(0) = Kn_{ KX}

1:::Xn]). We write V W i V = Va, W=Vb andabinterpreting the naturalK-algebra homomorphism

KW] =KX1:::Xn]=b;!KV] =KX1:::Xn]=a

as restriction of functions.

In the case of principal ideals we write alsoVf instead ofV(f). Sometimes

we use also the notationVf1:::fr instead of V (f

1:::f

r). Now we are exible. If

Lis an extension eld ofK we denote and dene the zero set of theK-ideala

inLn _by

Va(L) :=fP = (p1:::pn)2L

n_{ f(P) = 0for all f 2ag}

Denition 2.6

. The (polynomial,K-algebra or shortly K;) Dimension

DimKR=DimK(R) of a K-algebra R is the maximal natural number n such

thatRcontains the polynomial ringKX1:::Xn] up to isomorphy ofK-algebras

- supposed it exists. If it does not exist, then we set the Dimension equal to

1. We use the capital "D" in Dimension in order to distinguish it from the

K-dimension of vector spaces. For each nitely generated K-algebras the Di-mension is nite.

If R is a eld, then we call the K-Dimension also the transcendence degree of R over K. The K;Dimension of an ane algebraic manifold V = Va is

denoted and dened by

DimKV=DimK(V) :=DimKKV]:

With help of some commutative algebra one can prove that

V W)Dim_K(V)Dim_K(W)

DimKKX1:::Xn] =DimKK(X1:::Xn) =n=DimK AnK DimK(R) =DimK(QuotR) = transcendence degree ofQuot R over K

If R is a K-domain with quotient eld QuotR :=fr=s 0 6=s r 2Rg, L an

algebraic extension eld of K contained in R and S is a K-algebra, algebraic overR, then

DimK(R) =DimL(R) =DimL(S) =DimK(S):

IfLis an arbitrary algebraic extension eld ofK, then

DimKR=DimL(R_K L):

If there is no doubt about the basic eldK(or K) we work with, then we write shortlyDimR orDim Vinstead ofDimKR orDimKV, respectively.

Example 2.7

. The Fermat polynomialf =X3

;Y3

;1 has coecients in each

the algebraic set VQ(f) =f(10)(01)g consits of two points only, which has

naive dimension zero. But the algebraic setsVR(f), VC(f) have real dimension

1 or 2, respectively. The latter set is a punctured torus (elliptic curve without

1) and has complex dimension 1. In the case of rational numbers we come

also to dimension1, if we take the coordinate ringQxy] =QXY]=(f) or the

function eld Q(xy) into consideration. They have Q-Dimension 1. We get

the same dimension forRxy] andCxy] over R in both cases and overC for

the latter one.

After the denition of dimension it is quite natural to ask for calculations. Most cases are not so simple as the above example. Especially, one wants to know when a set of algebraic equations denes an ane algebraic curve, which has dimension 1 by denition. Calculation programs for dimensions exist (e.g. CASA, SINGULAR, MACAULEY). We will present now the basic theorems, which one needs for such procedures. First we introduce another "codimension", which looks not so natural at rst glance but is o.k. and near to calculations. We give also an idea why both notions of dimension must coincide.

We denote the set of all prime ideals of a ringRby PrimR(the usual nota-tion in higher algebraic geometry isSpec R, the spectrum ofR, Grothendieck).

PrimRdenotes the subset of all prime ideals ofRconsisting not only of zero

divisors. Neglecting zero divisor ideals we denote the set of minimal elements ofPrimRbyPrim

minR. So, minimal prime ideals of a domain are the

min-imal ones among all prime ideals excluding (0). For an idealaof a ringR we introduce also

Primmina=Primmin(a) :=fminimal elements of Prim R containingag:

Principal Ideal Theorem 2.8

. LetRbe a noetherian domain, The following properties are equivalent:

(i) pis a minimal prime ideal ofR

(ii) p2Prim_min(f) for a suitable 06=f 2RnR.

For the factorial ring R =KX1:::Xn], K algebraically closed, the minimal

prime ideals correspond to irreducible hypersurfaces ofAnK. SinceKX

1:::Xn]

is factorial, the biunivoque correspondence extends - up toK-factors - to the

set of irreducible polynomials.

Ascending chains of prime ideals have to stagnate by denition of noetherian rings. A deeper result is the following

Proposition-Denition 2.9

. Each strictly descending chain of prime ideals of a noetherian ringR

php_h;1ph;2 :::::

terminates after nitely many steps. For each p2Prim R there is a maximal

number h(p) of possible descend steps starting from p. It depends only on p. This maximal numberh(p) is called the hight ofp. More generally, we call

the hight ofa.

The background for the descending property of prime ideals is the

Relative Principal Ideal Theorem 2.10

. For each ideala= (f1:::fr) of

the (noetherian) ringR it holds that h(a)r.

Theorem 2.11

. For each proper idealaofR the setPrimmin(a) is nite and

Rada=\

Primmin(a):

With the Relative Principal Theoerem one feels and really can prove that any prime ideal p of KX1:::Xn] contains h(p) K-algebraically independent

ele-ments and not more. Changing to the coordinate ring

KV] =KX1:::Xn]=p V=Vp

the elements ofpare mapped to the zero class. One can prove that the polyno-mialK-Dimension ofKV] is complementary to hight:

Proposition 2.12

. With the above notations it holds that

h(p) +DimKKVp] =n:

This motivates already to dene the Codimension of idealsaof nitely generated

K-algebrasRand corresponding ane algebraic manifoldsVa in the caseR=

KX1:::Xn] as

CodimKa :=h(a) =:CodimKVa:

The above proposition can be generalized to

Theorem 2.13

. For each idealaof a noetherianK-algebra it holds that

CodimKb+DimKR=b=DimKR:

(1)

In the standard literature one denes the Krull dimension ofR=pas maximal length of ascending prime ideal chains connectingpwith a maximal ideal. But the relation (1) is not true for all noetherian rings R if one substitutes there ourK-Dimension by Krull dimension. When it is true for all (prime) ideals, then R is called a Cohen-Macauley ring. For instance, the polynomial rings

KX] =KX1:::Xn] are Cohen-Macauley. Here is no dierence between Krull

and polynomialK-Dimension. Cohen-Macauley rings can be also characterized by the property that all maximal prime ideal chains joining two given prime idealspqhave the same length.

The hights of prime ideals p of KX], hence the Dimension of Vp can be

calculated by means of Hilbert's theory of syzygies. Knowing an ideal basis

f1::frofpone starts with the exact sequence

of noetherian KX]-modules, where the middle homomorphism sends the canon-ical basis ofKX]rto the elementsf1:::fr. S1is called the rst syzygy module

of the ideal basis. This procedure can be extended in the same manner using (nitely many) generators ofS1. So one gets withr=r0longer exact sequences

0!S_d!KX]rd!KX]rd;1:::::!KX]r0 !KX]!KX]=p

We stop, if thed-th syzygy moduleSdis a freeKX]-module. Hilbert proved that

the numberdexists, is smaller thannand independent of the choices of bases in the modules. Moreover this number coincides with the heighth(p). The heights can be eectively calculated via Groebner bases of ideals and modules. For instance, the computer packages SINGULAR, CASA and MACAULEY work on this line.

Important for us are function elds of algebraic curves. They have a simple structure.

Theorem 2.14

. The function eld of an irreducible curve C in AnK has two

generators, ifK is either a nite eld, an algebraically closed eld or a eld of characteristic0:

K(C) =K(xy) =K(x)y] x transcendental overK,y algebraic overK(x).

So each such curve is "almost" determined by one equation with two variables only: consider the minimal polynomial ofy overK(x). This polynomial denes a plane curve. So the theorem has the following

Theorem 2.15

(Geometric Version). Each irreducible algebraic curveC over a nite eld, an algebraically closed eld or a eld with characteristic 0 has a plane model. This means that there exists a plane irreducible curveC0 with the

same function eldK(C) =K(C0).

The proof of the theorem joins two basic results of eld theory.

Denition 2.16

. LetLbe an algebraic eld extension of K. An element2L

is called separable over K, i the minimal polynomial p(T)2KT] of has

only simple zeros. Lis separable overK, i each element ofLis. A eldK of characteristic p(inclusively p= 0) is called perfect, i each element of K has ap-th root inK.

It is clear that all elds of characteristic 0, all nite and algebraically closed elds are perfect.

Proposition-Denition 2.17

(of primitive elements). Each nitely generated separable algebraic eld extensionLofK can be generated by one element only:

L=K(). Such an elementis called aprimitive element of the eld extension

Proposition-Denition 2.18

(F.K.Schmidt). Each eld extensionLof a per-fect eld K of K-Dimension (transcendence degree) r is separably generated. This means that there existr K-algebraically independent elementsx1:::xr2L

such thatLis a separable extension ofK(x1:::xr). Each set of such elements

is called separating transcendental basis ofL.

Taking into account that the function eld C(K) of an ane algebraic irre-ducible curve is nitely generated we have only to summerize the above propo-sitions (and denitions) for proving Theorem 2.14.

Plane irreducible curves (inA 2

K) correspond bijectively (up to aK-factor)

to irreducible polynomials in two variables with coecients inK. It is an old problem to ask whether an irreducible curve in AnK can be described by n;1

equations. By rather recent research one knows

Theorem 2.19

(Cowsik-Nori). Assume that the charactristic of K is a prime number. Then each ane algebraic curveCover Kis a (settheoretical) complete intersection. This means thatC( K) is the intersection of (the algebraic sets of)

n;1 hyperplanes.

Remark 2.20

. For elds of characteristic0 a similar result is only known for connected smooth curves (Ferrand-Szpiro-Mohan Kumar). In general it remains to be an open problem.

3 Projective Geometry

The n-dimensional projective set Pn(K) consists of points (z

0 : z1 : ::: : zn),

zi2K,i= 0:::n, not all zero simultaneously. Precisely, it is dened as factor

set

Pn(K) = (Kn +1

n fOg)=K

whereK operates multiplicatively on Kn+1 considered asK-vector space. In

the case of complex numbersPn(C) has the structure of a connected and

com-pact topological space descending the topology ofCn

+1 along the quotient map Cn

+1

n fOg ;!Pn(C) (factor topology). It has a holomorphic aneCn-atlas

consisting ofn+ 1 cards isomorphic toCn, namely

Ui=f(z0:z1:::::zn)2 Cn

+1 z

i6= 0g i= 0:::n:

Especially, one identiesAn(C) withU

0 sending (z1:::zn) to (1 :z1 :::::zn)

and consideresPn(C) as (algebraic / analytic) compactication ofCn. The same

can be done withUi instead ofU0. The complements are the hyperplanes Pn(C)nUi=: Hi : Zi = 0 i= 0:::n:

The setH0is also denoted byH1called the innite hyperplane. A polynomial

F(Z0:::Zn) 2 KZ0:::Zn] is called homogeneous of degree d, i it is aK

-linear combination of monomials

Zi :=Zi 0 0 :::Z in 0  d=i 0+:::+in

of degree d. The topological closures of ane complex manifolds in Cn are

also compactications. They appear as projective complex manifolds inPn(C),

which will be dened now. A projective algebraic set inPn(K) is the zero set

of an ideal A of KZ0:::Zn] generated by homogeneous polynomials. Such

ideals are called homogenoeus ideals. For algebraically closed elds K there is a bijection

falgebraic sets inKng m

fproj: alg: sets inPn( K)without irred: components inH 1g

where irreducible sets and compononents are dened in obvious analogy to the ane case. The map from below to above is the intersection with U0( K) = Pn( K)nH

1. We have to explain the map from the upper to the lower set. For

this purpose we have to homogenize polynomials in n variables. This is done for arbitrary eldsK by the homogenization map

h : KZ

1:::Zn];!KZ0Z1:::Zn]

f 7! F =fh :=Zdeg f 0 f(Z

1=Z0:::Zn=Z0):

It is linear, injective, multiplicative (compatible with multiplication of polyno-mials), degree preserving but not additive and not surjective. The image con-sists of all homogeneous polynomials ofKZ1Z0:::Zn], which are not divisible

byZ0. The inverse map sends a homogeneous polynomialF(Z0Z1:::Zn) to

a_{F =f(Z}

1:::Zn) :=F(1Z1:::Zn).

The homogenization map extends to

h:

fideals of KZ1:::Zn)]g ;! fhomogeneous ideals of KZ1:::Zn)]g

corresponding

a7!A=ah := (fh f

2a)

(theKZ0Z1:::Zn)]-ideal generated by thef

h's). Conversely, we dene

a_{A :=f}a_{F =F(1Z}

1:::Zn)F 2Ag

getting ideals in the polynomial ring of n variables. One checks easily the following properties and compatibilites (with obvious notations).

(a+b)h=ah+bh (ab)h=ahbh (a\b)h=ah \bh (Rada)h=Rad( ah) a_{(A+B) =} a_A+ a_B a_{(AB) =} a_Aa_B a_{(A\B) =} a_A\ a_B a_{(RadA) =Rad}a_A a_(ah) = a pis prime iffphis:

A basic motivation for the introduction of projective spaces is the following theorem, which can be proved by means of the Relative Principal Ideal Theorem.

Theorem 3.1

. Each system of homogeneous polynomials

F1:::Fn2KZ0Z1:::Zn]

has at least one zero inPn( K).

This means geometrically that the intersection of nhypersurfaces in Pn( K) is

not void. For instance, the intersection of two dierent plane projectiveK-lines consists of a point inP

2(K). This was a historical starting point.

Essential homogeneous ideals of KZ0Z1:::Zn] are precisely those whose

radicals are dierent from the maximal homogeneous idealM0= (Z0Z1:::Zn).

In analogy to the ane geometry the zero set correspondence

fessential homogeneous radical ideals ofKZ0Z1:::Zn]g

m

sendingAto

V(A) =f(z0:z1:::::zn)2

Pn( K)F(z

0z1:::zn) = 0for all F 2Ag

is bijective.

Now we intersect the a projective algebraic set V =V(A) with Kn_{. Then}

we get an ane algebraic set denoted bya_{V, namely}

V(A)\Kn= aV(A) =V(A)nH1=V(

a_A):

The ane algebraic set aV(A) coincides with aV(B) if and only if V(A) and

V(B) coincide up to components lying in the innite hyperplane. Conversely, we send ane algebraic setsV0=I(a) to the projective ones (V0)

h :=V(ah)

Pn(K). With obvious notations the following relations are immediate:

a_{I(V) =I(}a_V) (I(V0)) h =I((V 0) h) V(aA) = aV(A) V(ah) =V(a)h:

In the case of algebraic closed elds we get a bijective correspondence between ane algebraic sets in Kn_{and projective algebraic sets in}Pn( K) without

com-ponents inH1.

In both spacesKn_andPn(K) there is a Zariski topology dened by

comple-ments of all algebraic sets as a topology basis of open sets. In this topology the projective algebraic setsV(ah) is the closure ofV

0(a) :=V(a) K

n _inPn(K).

Thanks to Hilbert's Nullstellensatz the correspondenceV0 7! V = (V0) h does

not depend on the choice of the vanishing ideala of V0, if K is algebraically

closed. If K is the eld of complex numbers, then V is also the topological closure ofV0 in

Pn(C), which is also regarded as a complex compactication.

For essential homogeneous idealsAKZ0:::Zn] we dene the attached

projective manifold in analogy to ane geometry as pair

VA= (V(A) KV(A)])

of a projective algebraic set and a coordinate ring. The coordinate ring is dened asKVA] =KZ0Z1:::Zn]=A. A projective (algebraic) variety (dened) over

K is a projective manifold VP, which belongs to an essential homogeneous

prime idealPofKZ0:::Zn].

We writeV W, ifV =VA, W=VB and AB. The projective space PnKis dened to beV

0=V(0), (0) the zero ideal ofKZ0:::Zn]. The projective

algebraicL-sets ofV=VA, La eld extension ofK, is written and dened as V(L) =fP = (p0:::::pn)2

Pn(L) F(P) = 0forallhomogeneousF 2Ag:

The projective dimension ofVis dened to be equal to

Dimh

The reason for the;1-shift is the following: For proper idealsaofKZ1:::Zn] it holds that Dimh KKah] =Dim KKa] =DimVa=DimV a h:

In geometric words, the projective closure applied to ane manifolds is dimen-sion preserving as it should be. For algebraically closed elds we notice the following

Hilbert's Nullstellensatz 3.2

(projective version). The zero set correspon-dence between ideals of KZ0Z1:::Zn] and projective algebraic sets in

Pn( K)

restricts to bijective correspondences (HN.0)h fpointsofPn( K)g m fmaximalessentialhomogenousidealsofKZ0Z1:::Zn]g (HN.1)h firreduciblealgebraicsetsinPn( K)g m fessentialhomogenousprimeidealsofKZ0Z1:::Zn]g (HN.2)h fprojectivealgebraicsetsinPn( K)g m fessentialhomogeneousradicalideals 2KZ0Z1:::Zn]g

The most classical version is

(HN)h Each essential homogeneous ideal of KZ

0Z1:::Zn], has at least one

4 Singularities

Now we are able to dene and describe the singularities of a projective man-ifolds. We start with hypersurfaces. Let F be a homogeneous polynomial in

KZ0:::Zn] of degreed >0. We say that P 2

Pn(K) is a singularity of VF,

i the homogeneous gradient (or Jacobian) ofF

(@F=@Z) = (@F=@(Z0:::Zn)) := (@F=@Z0::::@F=@Zn)

consisting ofn+ 1 polynomials, vanishes atP. At least in characteristic 0 the pointP really belongs toV(F), if the singularity condition

(@F=@(Z0:::Zn))(P) = 0:

is satised. This follows from the Euler Identity

Z0@F=@Z0+Z1@F=@Z1+:::::+Zn@F=@Zn=dF

which can be easily proved starting with monomials. IfF =fh is the

homoge-nization off 2KZ1:::Zn] andP = (1 :p1:::: :pn) = (p1:::pn)2Vf(K),

then it holds that P is a singularity ofVf if and only if the (ane) gradient

(or Jacobian) off

gradf := (@f=@(Z1:::Zn)) = (@F=@Z1::::@F=@Zn)

vanishes together withf at P. So the ane singularities are described by the

n+ 1 equations

@f=@Z1=::::=@f=@Zn= 0 =f(Z1:::Zn):

The gradient and homogenizing operations are almost commuting (up to Z0

-power factors at monomials). With this knowledge it is easy to check that the ane set ofK-singularities ofVf coincides with ane part of the projective set

ofK-singularities of VF. With the notations

SingVF =V@F=@(Z 0:::Zn ) PnK SingVf =Vf\V_@f=@(Z 1:::Zn ) AnK

this means that the singular loci are related by

(SingVf)(L) =An(L)\(SingVF)(L)

whereLis any extension eld ofK. For P = (p1:::pn)2

An(K) we dene the cotangent space of An atP as

space of linear functions vanishing atP, precisely:

T

P(AnK) : :=K(Z

1;p1) +:::+K(Zn;pn)

We have a natural coordinate map : T

P ;!Kn, (Z_i;p_i)7!(0:::1:::0).

The tangent space TP(AnK) is the dual of the K-vector space T 

composition of the gradient map, substitution of point P and inverse of the coordinate map

dP : KZ];!KZ]n;!Kn ;!T

P

sendsf(Z1:::Zn) to the linear polynomial

dPf :=X

(Zi;p_i)@f=@Z_i(P) =<;!PZgrad_P(f)>

called the dierential off at P. The following properties are easy to check:

dP(f+g) =dPf+dPg dP(fg) =g(P)dPf+f(P)dPg:

The equationdPf = 0 describes the tangent hyperplane TP(Vf) ofVf at P, if

P is not a singular point (remember to gradient properties in analysis). These considerations can be extended from hypersurfaces to complete inter-sections of Dimensionn;r Vf=V (f 1:::f r): f 1=:::=fr= 0 f= (f1:::fr)2KZ1:::Zn] r_:

P 2Vf(K) is a singular K-point of this manifold, i the rank of the Jacobian

matrix (@f=@Z) = (@fi=@Zj) is smaller thanr. Non-singular points are called

regular points. Vfis calledK-smooth i it has no singularK-points. It is called

(absolutely) smooth i there is no singular K-point on it. Since the singular points are dened by nitely many algebraic equations depending only on the dening equations of a manifold we receive

Theorem 4.1

. The singular locus SingV is a closed algebraic subvariety of

V=Vf.

IfP is a regular point ofVf(K), then the tangent space overK ofVf at P

is dened by a linear system of equations:

TP(Vf) : (@(f1:::fr)=@(Z1:::Zn))(P)

t_(Z

1;p1:::Zn;pn) = 0:

LetVbe an ane K-variety,P aK-point on it. The local ringO_P of Vat P

is the subring of the function eld

O_P =O_PV=ff=gfg2KV] g(P)6= 0g K(V):

A ringR is called local, i it has a unique maximal ideal. The unique maximal ideal ofO_P is

mP =mPV=ff=gfg2KV] g(P)6= 0 f(P) = 0g

which is the complement inOP of the groupO

P of units ofOP. The dierential

mapdP can be extended fromKZ] =KZ1:::Zn] toDP : OP ;! OP setting

Obviously,DP restricts to the powers

DP : mkP ;!mkP

ofmP for eachk2N. The residue eld kP :=OP=mP containingKacts on the

residue modulesmk=mk+1in obvious manner. As noetheriank

P- andK-modules

they are are nite-dimensional K-vector spaces. If K = K is algebraically closed, the residue eld kP coincides with K because it is nitely generated,

hence algebraic over K.

Theorem 4.2

. If P is a regular point of the K-algebraic (complete intersec-tion) varietyV, thenDP induces an isomorphism

P : mP=m2

P 

;!T

P(V):

That's the reason why the cotangent space can be identied with mP=m2

P at

regular K-points. So one denes for all algebraic manifolds at regular points

T

P(V) =mP=m2

P TP(V) = (mP=m2

P):

Regular points of arbitrary algebraic varietyV over K are dened now by the conditioin dimK(mP=m

2

P) = DimV. Moreover one has the following general

regularity criterion.

Theorem 4.3

. A point P of the algebraic K-variety Vis regular, if and only if the maximal idealmP of the local ringO_P has an ideal basist1:::td of length

d=DimV. Such an ideal basis is called a regular system of parameters (of V

atP).

More generally, for a local domain (Rm) with residue class eld k = R=m a regular system of parametersis dened as ideal baisis ofmof lengthdimk(m=m

2).

A local ring which has a regular system of parameters is called a regular local ring.

Theorem 4.4

. Each regular local domain is a unique factorization domain. Especially, it is a normal domain.

Thereby we call a domainR normal, i it coincides with its normale closure in its quotient eld. The normal or integral closure ofRin a eldLcontainingR

is dened by

L\ fzeros of normalized polynomials of RT]g

5 Algebraic Curves

For local rings at points of algebraic curves there is no dierence between nor-mality and regularity. Namely, it is not dicult to prove the following useful

Theorem 5.1

. LetR be a localK-domain with maximal ideal m6= (0). Then

the following properties are equivalent: (i) mis a principal ideal

(ii) R is a principal domain

(iii) R is factorial of K-Dimension1 (iv) R is normal of K-Dimension1

(v) R is a discrete valuation ring (vi) R is regular ofK-Dimension 1.

Property (v) means that the quotient eldQofRhas a (surjective) discrete valuation v: Q;!Z 1, which is dened by the following properties:

(0) v(r) =1 ,r= 0,

(1) v(rs) =v(r) +v(s),

(2) v(r+s)minfv(r)v(s)g,

for all rs2Q. The valuation comes from the multiplicative extension of the

correspondencet 7!1, wheret is an arbitrary local parameter (generating m).

From the valuation v one gets back R, R and m as v;1(

N 1), v

;1(0) or

v;1( N

+ 1), respectively.

Proposition 5.2

. The point P of an irreducible algebraic curve C is regular if and only ifO_P is normal. Almost all local rings O_P,P 2C( K) are normal.

"Almost all" means (for curves): up to nitele many points.

Theorem 5.3

. For (absolutely) smooth projective curves C over an alge-braically closed eld K there is a bijective correspondence

C( K),discrete valuation rings of the function eld K(C).

It correspondsPto the local ringO_P. The corresponding valuations are denoted

byvP. Forf 2K(C) the numberv_P(f)2Z 1is called the zero order off

atP and

is the pole order off at P.

Assume that K = K is algebraically closed. For singular points P of hy-persurfacesV =Vf AnK we look for a measure of deviation from regularity.

Without loss of generality we assume thatP =O 2V(K) is the zero point. We

dene rst the contact order atOofV and the lineL=La=Ka,o6=a2K

n_,

as the zero order vO(f(ta))> 0 of the polynomialf(ta) 2Kt] at O. If this

order is equal to 1 thenO must be a regular point ofV andLcrosses V atO. So it is natural to dene the multiplicity ofV atO as

O(V) :=minfv_O(f(ta))L=Kaline through Og:

A lineT =KawithvO(f(ta))> O(V) is called a tangent line ofV at O.

Plane Curve Examples 5.4

: The curvesY2

;X2(X+1) = 0 andY2

;X3=

0 have multiplicity 2 respectively 3 atO. In the rst case the singularityO is a double point. Singularities of multiplicity 3 are triple points. In the rst case there are two curve branches at O, each of them has a tangent line, crossing each other at O. In the latter case there is only one tangent. Triple points of this kind are called cusps. There is also the possibility of triple points with three dierent crossing branch tangents. Take for example the curve with equation

Y4

;Y3+X2Y +X4= 0.

Above we dened multiplicities of hypersurface singularities because Vf is a

hypersurface inAnK. Especially forn= 2, we are able to calculate multiplicities

of plane curve singularities as demonstrated in the above examples. We would like to determine also singualirity multiplicities of curves in An  Pn. This

can be done more generally for varieties Vf, f = (f1:::fr) 2 KZ1:::Zn]

r_.

Knowing the complications for the general denition with acceptable qualities, I make only a step in this direction:

Denition 5.5

. The contact order of Vf and La at O 2 Vf(K) is the zero

orderv0(P) at 0 of the greatest common divisor polynomial

P(t) = g:c:d:Kt](f

1(ta):::fr(ta):

We would like to resolve curve singularities. The purely algebraic way is the normalization. LetC be an irreducible curve. The ane coordinate ring is not normal in general. But there are only nitely many local rings O_P which fail

to be normal, see Proposition 5.2. These are the singularities. A local global principle of algebra applied to our situation yields

Proposition 5.6

. The coordinate ring KC] is normal if and only if all local ringsO_P in K(C) are normal.

So , the normal closureK]C] ofKC] in the function eldK(C) is a ring which

has only regular local rings. Moreover, it is a nitely generated KC]-module and therefore a nitely generated K-algebra. Looking at the ideal dened by

relations of generators we see thatK]C] is the coordinate ring of the curve ~C

dened by this ideal. The function elds ofCand ~Care the same. So we get a smooth model ~C ofC together with a map

~

C;!C P~7!P

(2)

sending discrete valuation ringsOP~ ofK ~C] = ]

KC] to its intersectionO_P :=

KC]\OP~. This map (2) is called the singularity resolution ofC. We presented

this algebraic procedure of singularity resolution only for ane curves. But it extends to projective curves via nitely many ane cards. We remember to Theorem 5.3 to see that in the case of algebraically closed eldK = K we re-solved all singularities of a given projective irreducible curve. The normalization method leads to a proof of

Theorem 5.7

. Each irreducible projective curve over an algebraically closed eld has a unique smooth projective model together with a unique singularity resolution map (2). Each two projective curve models of the same function eld have, up to isomorphy, the same smooth model.

We want to resolve singularities in a more constructive and more visible geometric manner in the case of plane curves. Since each irreducible curve has a plane model by Theorem 2.15 this is most important. We will explain now the -process at the point O of the ane plane A

2. Geometrically,O will be

substituted by the space of all tangent lines throughO, which is a (projective) line itself. This procedure has an algebraic realization as surfaceS in

A 2 P 1= ( A 2 U0)( A 2 U1) Ui: ti6= 0 i= 12

dened by the equationT0Y =T1X. Substitutet1bytandt0bys. Then

S=f(xy(s:t))2A 2 P 1 sy=tx g: Outside ofOP

1the points ofSare uniquely determined by the A

2-coordinates

(xy)6= (00). This is understood as an (algebraic) isomorphism

SnOP 1  ;!A 2 nO

restricting the natural surjective algebraic projection

: S ;!A

2 (xy(t

0:t1))7!(xy)

with the exceptional line ;1(O) =: L = P

1. It is clear that the tangents

throughO represented by (x : y) correspond to the point on L  S with the

coordinate (s:t) = (x:y) because ofsy=tx.

The procedure is local. It can be easily dened at any regular point P of a projective algebraic surfaceX. The corresponding -process P : S ;! X

is also called the blowing up of the point P of X or ofX at P, and S is the blown up surface. It is a projective surface again. Examples. The plane curve

C : Y2

;X2(X + 1) = 0 has a double point O. The two curve branches

throughO are separated after blowing up this point. Namely, look at the pair of crossing tangent lines of the branches. They represent and appear as two dierent points on the exceptional line L = ;1(O). This means that the

branches cross the exceptional line at these points. The singularity is resolved. For cusps as represented by the pointO of the curveC: Y2

;X3= 0 one needs

also one -processes to resolve the singularity. In this case the exceptional line appears as tangent of the transformed curve.

Theorem 5.8

. LetC be a plane projective curve over an algebraically closed eld K of characteristic 0. There exist nitely many -processes at curve sin-gularities such that the properly transformed curve ~C is smooth.

The proof needs divisors on smooth projective surfacesX and intersection in-dices of them. Divisors D =P

kiCi onX areZ- linear combinations of

irre-ducible (projective) curvesCi on X called (prime) components of D (assume

ki 6= 0). They form an additive group denoted byDiv X. Eective divisors are

those with only positive coecients. The additive subsemigroup of all of them is denoted byDiv+X. Prime divisors are the irreducible curves endowed with

co-ecient 1. The support of the divisorD is the algebraic setS

Ci. On a suitable

neighbourhood of any pointP 2X( K) each eective divisor is described by a

local equation f = 0,f =fP 2 O_PX. In this equationf is uniquely determined

up to a unit of the local ringO_PX because it is factorial. The relationf(P) = 0

means thatf belongs to the maximal idealmPX of the local ring. Now letD,E

be two eective divisors intersecting each other properly atP, that means their supports do so. The local equations ofD,E atP aref = 0 respectivelyg= 0. Then the ideal (fg) inO_P is contained in m=m_PX. One can prove that the

residue class modulem=(fg) is a nite dimensional K=O_P=m_P-vector space.

The intersection index ofD,Eat P is denoted and dened by (DE)_P :=dimK OPX=(fPgP):

The denition extends to eective divisors D (and E) not going through P

taking a unitfP ofOP instead ofmP. Then the local intersection index at P

is 0. It is easy to see that the intersection index is symmetric and biadditive whenever dened. We globalize the intersection index setting

(DE) = X

P2C

(DE)P:

This denition is correct ifE,D have no common prime component because in this case there exists only a nite number of intersection points. For a gener-alization to all divisors we rstly introduce principal divisors. Let 06=f be an

element of the quotient eld K(X). It has a unique factorization

f =pk1 1 :::p kr r ql1 1 :::q ls s

in prime factors ofO_P for each point P of our smooth surface X. The prime

factors dene prime divisorsCi : pi= 0 and C0

j : qj = 0. AroundP, say on a

Zariski-open setU, this decomposition denes a local principal divisor (fU) :=k1C1+:::+krCr;(l1C

0

1+:::+lsC 0

s):

Outside ofU there are only nitely many closed irreducible curves. We have to add these, along whichf vanishes and to subtract those along whichf has a pole, each with the correct multiplicity coming from prime factorizations as above. At the end we get with obvious notations a (global) principal divisor

(f) =X

C vC(f)C:

It holds that (f g) = (f) + (g) for any two principal divisors. The principal

divisors form a subgroupPX ofDiv X. The residue class group

PicX :=Div X=PX

is called the divisor class or Picard group ofX. We writeD E i D andE

belong to the same divisor class and callD andE linearly equivalent.

Proposition 5.9

. The intersection index((f)D) with a principal divisor (f) vanishes, whenever dened.

Moving Lemma 5.10

. For two prime divisors C, D there exists a divisor

ED avoiding the given curvesC as component.

On this way one can dene correctly (CD) as (CE). Especially

selnter-sections (C2) are dened now. By

Z-linear extension the intersection pairing

(  ) : Div(X)Div(X);!Pic(X)Pic(X);!Z:

is well-dened.

We come back now to the -processes = P : S ;!X with exceptional

line L = ;1(P), wishing to understand why singularities can be resolved by

them. ForD2Div+Xwe denote by (D) the divisor onSwhich has the same

local equations on ;1(U) asD has onU, for any set of small Zariski-open sets

U covering X. The inverse image map  restricts to P

X ;!P_S. Together

we get group homomorphisms

: Div X

;!Div S PicX ;!PicS:

For the intersection indices it holds that ( (D)

 (E)) = (D

E) D E2Div X:

The proper (pre-) image C0 = 0(C) on S of an irreducible curve C on X is

dened to be the closure of ;1

P (CnfPg) on S. The denition extendsZ-linearly

to

0 : Div X

For a curveCwith multiplicity k=P(C) at P it holds that

(C) = 0(C) +k

L ( 0(C)

L) =k:

The singularityP ofC splits into intersection points ofC0 withLand the sum

of multiplicities of these points is smaller thanP(C). For instance, ifP is an

ordinary singularity, then all intersection points ofC0 andLare regular points

ofC0. One can prove also

( 0(C 1) 0(C 2)) = (C1C2);k1k2 ki=P(Ci) especially (C0 2) = ( 0(C) 0(C)) = (C2) ;k2:

6 Riemann Surfaces

A Riemann surface X is a reell two-dimensional connected smooth manifold with a holomorphic complex structure. This means: there is an atlas consisting of (at most countable many open) cards (homeomorphisms)X U 

;!V C

such that any pair of cardsU1, U2 is connected by biholomorphic

transforma-tions over the intersectionU1\U2. Most interesting for us are compact

Rie-mann surfaces. An example is the complex torus dened as coset spaceC=!,

! =Z! 1+

Z!

2a lattice in

C, inheriting the complex structure of C in

obvi-ous manner. The following theorem presents us all smooth complex projective curves as compact Riemann surfaces.

Connectedness Theorem 6.1

. Each complex ane or projective algebraic variety is connected in the complex topology.

Notice that the analogeous statement for real algebraic varieties is wrong even in the case of plane curves. For example the set E(R) A

2(

R) of real points

of the elliptic curveE : Y2 =X3

;X splits into two connected components,

one of them is an oval. Things are complicated for real plane curves in general. Hilbert's 16-th problem is dedecated to them.

Holomorphic mapsX ;!Y of Riemann surfacesX,Y are dened locally via

cards of suitable atlasses in obvious manner. The holomorphic mapsX ;!P 1

form a eld, the eldM(X) of meromorphic functions ofX. For an irreducible smooth complex projective curveCthe function eldC(C) coincides withM(C).

For instance, M(P 1) = C(P 1) =C(Z) =QuotCZ]

is the eld of complex rational functions. The ring of holomorphic functions on

U X is denoted by O(U). Meromorphic functions onU are the elements of

the quotient eldM(U) =QuotO(U). In contrast to the compact (projective)

case there are holomorphic functions which are not rational, e.g. the exponential function onC. From elementary function theory one knows

Discreteness Theorem 6.2

. The set of zeros of a holomorphic function on a Riemann surface is discrete.

Maximum Theorem 6.3

. If a holomorphic mapf :U ;!C has an absolute

maximum inside of the Riemann surfaceU, thenf is constant.

Corollary 6.4

. Each global holomorphic function on a compact Riemann sur-faceX has to be a constant: O(X) =C.

Namely, the absolut value function jfj : X ;! R is contineous. Since X is

compact, there is anm2X with maximal valuejf(m)j.

Corollary 6.5

(Liouville). Absolutely bounded holomorphic functions onC are

For each point P 2 X one has a discrete valuation v_P : M(X) ;! Z 1

corresponding to each function its zero order atP. Especially, for P=1 2P 1

one nds forf g2Cz] using the local parameter 1=zthere:

v1(f=g) =;degf(z) +deg g(z) g6= 0:

Covering Theorem 6.6

. Let': X;!Y be a holomorphic map of Riemann

surfaces,Q2Y,X compact. Then the preimage';1(Q) is nite,is surjective

andY is compact.

Denition 6.7

. The multiplicity of 'atP is dened asmultP(') =vP(f),

wheref ='jU : U ;!C sendingP to0,U a card around P.

Denition 6.8

. Ramication points P 2 X of ': mult_P(') > 1 branch

points: all'-images of ramication points.

Both are discrete, hence nite sets of the compact Riemann surfaces X or Y, respectively: Use the local criterion: f0(z) = 0 for ramication points together

with Discreteness and Covering Theorem. The multiplicity of ' at P is also called ramication order or ramication indexof 'at P. It is also frequently denoted byeP(').

The local degree of'atQis dened as

dQ(') := X

P7!Q

multP('):

Proposition-Denition 6.9

. dQ(') is constant, that means not depending

onQ. It denes the degree of': deg ' :=dQ(').

Corollary 6.10

. ': X ;!Y is an isomorphism i deg '= 1.

Corollary 6.11

. The following properties are equivalent: (i) X ₌P

1

(ii) there existsf 2M(X) with precisely one zeroP andv_P(f) = 1

(iii) there existsf 2M(X) with precisely one poleP andv_P(f) =;1.

Proposition 6.12

. Let X be compact Riemann surface, f 2 M(X) then

P

P2XvP(f) = 0.

Denition 6.13

. Let " be a triangulation, of the compact Riemann surface

X andv,k,s the number vertices, edges respectively faces. The number

e(X) =e (X) :=v;k+f

The denition is correct, that means it does not depend on the choice of trian-gulation. For instance, the Euler number ofP

1 (Riemann sphere) is equal to 2

by Euler's Polyhedron Theorem. With same methods as already Euler used one can prove that the Euler number of a torus is equal to 0. The genus, dened below, of the Riemann sphere or of a torus is 0 or 1, respectively.

Denition 6.14

. The number g = gtop _{= g(X) := 2;e(X) is called the}

(topological) genus ofX.

There is a theorem of dierential topolology, which says that each Riemann surface can be embedded smoothly asR-manifold into R

3. The compact

Rie-mann surface appear on this way as compact real surfaces in R

3 with some

holes. We connect two such topological surfacesX,Y by opening both, each at a point, and join the arising open holes by an open cylinder. The new compact topological surface we denote byX Y. Triangulations on X and Y can be

joined in obvious manner to a triangulation onXY. It is elementary to prove

the following additive property of the topological genus and its consequences:

Lemma 6.15

. g(XY) =g(X) +g(Y) g(X) = #fholes of Xg 0.

By means of lifting triangulations including the setB =B(') of branch points as (some) vertices it is again a matter of elementary combinatorics to prove for coverings': X0

;!X of compact Riemann surfaces the

Hurwitz' Genus Formula 6.16

. (i) e0 ;#0=d(e ;#), (ii) 2g0 ;2 =d(g;2) +P P0 2X 0(multP 0(');1), whered=deg ', # = #B('), #0 = #';1(B),g=g(X),g0=g(X0).

Corollary 6.17

. With the above notations it holds that

(i) g= 0 anddeg ' >1)B(')6=

(ii) if g= 1 then: g0= 1

,'is unramied (that means B(') =),

(iii) g0

g0

(iv) if g2 and 'is not isomorphic theng0> g.

Remember to the notions of divisors and principal ones on curves and the degree map. It is easy to transfer them to Riemann surfaces. For a complex smooth complete intersection curveC =V(F1:::Fn;1) 

Pn, and hypersurfacesH :

G= 0,Gnot in (F1:::Fn;1), we introduce the intersection index atP with the

help of an (arbitrary) auxiliary homogenous polynomialG1 of the same degree

as that ofGsatisfyingG1(P)6= 0 via restriction of functions

The intersection divisor is dened as

HC := X

P2C

(HC)_PP 2Div+C:

It holds thatH1CH2C for two hypersurfacesH1,H2 of the same degree.

Therefore the global intersection index (HC) :=deg HCis well-dened. For

hyperplanes H it is called the degree of the embedding C ,!Pn.

Lemma 6.18

. For plane smooth curvesP

2

C: F = 0 it holds that

deg(HC) =deg F.

You are not far away now to prove

Bezout's Theorem 6.19

. For two smooth plane projective curves Ci: Fi =

0 it holds that

(C1C2) =deg C1C2= (deg C1)(deg C2) = (deg F1)(deg F2):

There are two important consequences, namely the genus formulas

g(C) = (d;1)(d;2)=2

for smooth projective plane curvesC of degreedand

deg K= 2g;2 = 2gtop;2

for each canonical divisor onC (dened below). One has to combine Hurwitz' genus formula and Bezout's theorem for a proof of the rst formula and to apply the Hurwitz formula to any covering (non-constant meromorphic function)

f : C;!P

1knowing the canonical divisors on P

1.

Dierential formsare dened locally around a pointOofXasf(z)dz, where

zis a local parameter atOandfa meromorphic function.. The local dierential form g(t)dt at P is identied with the former i f(z)dz = g(t)dt. So each local dierential form denes a global one. The vector space of meromorphic dierential forms onX is denoted by $X. It is clear that for each meromorphic

dierential form!6= 0 it holds that

$X=M(X)!:

We dispose also on valuations of meromorphic dierential forms at points P

settingvP(fdt) :=vP(f),ta local parameter at P. We get surjective maps

vP : $X;!Z 1

on this way. They dene zero and pole orders of dierential forms at points. The space of dierential forms on X without poles on an open set U X is

denoted by $X(U). Its elements are called holomorphic or regular dierential

The residue of a meromorphic dierential form!at P is the number

resP! = 1_2i

I

!

where one has to integrate along a small circle around P such that inside of the circle there is no pole of ! except, perhaps, at P. Knowing the Residue Theorem for meromorphic functions on onC it is via triangulation not dicult

to prove

Residue Theorem 6.20

. For meromorphic dierential forms ! on compact Riemann surfaces it holds that

res ! := X

P2X

resP!= 0:

Keep in mind that $X is a one-dimensionalM(X)-vector space. Since

(f!) = (f) + (!)

for f 2 M(X), ! 2 $_X, the dierential forms sit in precisely one class in

PicX, called the canonical class. The elements of this class are called canonical divisors. Furthermore we dene for divisorsDonXthe function and dierential form spaces

L(D) :=ff 2M(X) (f) ;Dg $(D) :=f!2$_X (!)Dg:

The corresponding dimensions are nite and are denoted by l(D) or l(D),

respectively. We have

 L(O) =C

 $(O) = regular dierential forms

 $(K) =C for canonical divisorsK= (!)

 ()(!))();(!) = O  L(D) =O forD < O  DE)L(D) L(E) and $(D)$(E)  DD0 )L(D)=L(D0), hence l(D) =l(D0), and $(D)_{= $(}D0), hencel(D) =l(D0).

We have also for divisorsD,E a bilinear map

L(E;D)$(E);!$(D) (f!)7!f!

especially, for any canonical divisorE=K= (!) one gets an isomorphy

L(K;D)$(K) =L(K;D) 

Any xed canonical divisorK denes an involution on the divisor group

: Div X

;!Div X D7!D :=K

;D:

It is convenient to keep in mind the dual style of writing

L(D)= $(D) l(D) =l(D) =l(K

;D) =dim$(D):

Riemann-Roch Theorem 6.21

(topological or geometric version). For each divisor D on a compact Riemann surfaceX it holds that

l(D);l(D) =deg D+ 1

;g g=gtopthe topological genus of X:

Especially, forD=O one gets (RR)0 g

an _{:=l(K) =dim$(O) =g=g}t_op.

We dened the analytic genus gan _{and change to}

Riemann-Roch Theorem 6.22

(analytic version). For each divisor D on a compact Riemann surfaceX it holds that

l(D);dim$(D) =deg D+ 1;gan:

We attack the proof of the analytic version by Mittag-Leer theory. For this purpose we need (nite) tails of Laurant-series. ForP 2X,ta local parameter

at P, f 2 M(X), ! 2 $_X, we assume that v_P(f) =m =v_P(!). Then there

exist unique Laurent series

f =amt;m+:::+a ;1t ;1+ Taylor series in t a m6= 0 !=fdt= (amt;m+:::+a ;1t ;1+ Taylor series)dt:

Principal tails look like

amt;m+:::+a ;1t ;1 (a mt;m+:::+a ;1t ;1)dt

One can prove that

Proposition 6.23

. With the above notations the local residues of dierential forms can be algebraically expresed as

resP!=a;1

independently of the local parameter choice.

Corollary 6.24

. If! is regular at P, thenresP!= 0.

Mittag-Leer Theorem 6.25

. For given principal tails!1:::!r at

dier-ent points P1:::Pr 2X there exist dierential forms ! with principal tails !i

atPi,i= 1::r, which are regular elswhere, if and only if

res(!1+:::+!r) =a (1)

;1+:::+a (r) ;1= 0:

One direction comes immediately from the Residue Theorem.

Corollary 6.26

. For any negative divisor D = m1P1+:::+mrPr < O it

holds that

dim$(D) =;deg D+dim$(O);1

It is not dicult to prove that the analytic version of the Riemann-Roch The-orem is equivalent to Mittag-Leer TheThe-orem. The proof direction ()) is an

easy exercise, deduce it from (RR);below. We give some steps for proving the

other direction.

1-st step: Substitute l(D) = 0 in the 6.22 to get (RR); l(D);dim$(D) =deg D+ 1;g

an _{for allD < O.}

This follows easily from 6.25 by counting constants (coecients) in the given tails there.

2-nd step: Observe thatDis negative forD > K. The proof of

(RR)>K l(D);dim$(D) =deg D+ 1;gan forD > K

can be easily reduced to the rst step. 3-rd step: (D) :=l(D);l(D)

;deg D is an increasing function:

For this step is sucient to prove that(D+P)(D) forP 2X,D2Div X.

Lett be a local parameter atP and

D=dP +:::other points::::

The mapf 7!ftd+1yields an exact sequence

0;!L(D);!L(D+P);! O_P=m_P =C: It follows that l(D+P) =l(D) + l(D) =l(D ;P) +  201 hence l(D) =l(D) ++ ;1:

We show that== 1 is impossible. Assume the contrary.

Take f 2 L(D+P)nL(D), ! 2 $(D)n$(D+P). It follows f! 2 $(;P)

because (f!) = (f) + (!) ;(D+P) +D=;P and (f!)_P =;P because

We get a contradiction to the Residue Theorem: 0 =res !=resP!6= 0:

3-rd step: For negative divisors N < O and D0 > K we know already that

the Riemann-Roch formula holds, that means(D0) =(N) = 1

;gan. For an

arbitrary divisor D chooseN , D0 as above such that N < D < D0. Since 

is increasing we get also(D) = 1;gan, which is the analytic Riemann-Roch

formula.

Now we are a littlebit prepared for the use of Laurent tails in order to prove the analytic version of Riemann-Roch Theorem. Consider the local Laurent tail space space TP consisting of all Laurent tails at P. The subspace TP(kP),

k2Z, consists, by denition, of all tails 2TP withvp() ;k. For a divisor

D=k1P1+:::+krPr we dene the subspace T(D) =Mr i=1 TPi(kiPi) M Q6=P i TQ(0Q)

of the global Laurent tail spaceT :=L

P2XTP. For instance,T(O) is the space

of nite sums of regular (local) tails. Now we introduce the truncation map

D : M(X);!T(D)

which cuts away (to set 0) from the local Laurent series of a meromorphic func-tion all summands ofvP-value ;v_p(D), wherev_P(D) is dened as coecient

ofD atP (0 ifP doesn't occur inD). For instance, the image ofOconsists of

nite sums of principal tails. We get exact sequences 0;!L(D);!M(X);!T(D);!H1(D)

;!0

Exactness on the laeft side comes from the denitions. The right part is nothing else but a denition of the rst cohomology group H1(D) as cokernel of the

truncation mapD. Stepwise one can to prove the following facts:

Proposition 6.27

. h1(D) :=dimH1(D)<

1.

Riemann-Roch Theorem 6.28

(cohomological version).

l(D);h1(D) =deg D+ 1

;h1(O).

Theorem 6.29

(Serre duality). There is a canonical isomorphism H1(D) =

$(D)_ onto the dual space of $(D).

The proof is based on the residue map

T(D)$_X(D);!C (tail sum!)7! X

P2X

resP(tailP!)

It factorizes throughH1(D)

Corollary 6.30

. h1(D) =dimH1(D) =dim$(D) =l(D).

Now we are able to nnish the analytic Riemann-Roch Theorem. We have only to substitute in the cohomological version h1(D) by l(D) = dim$(D) and

h1(O) byl(O) =l(O) =l(K) =gan to get 6.22.

ForD=Kwe get

l(K);l(K) =deg K+ 1

;l(K)

We know thatl(K) =l(K) =l(O) = 1 anddeg K= 2gtop

;2. We substitute

both and receive 2l(K) = 2gtop, hence gan = l(K) = gtop. It clear that

the most classical geometric version follows now immediately from the analytic version.

We mention without detailed proofs some applications of the Riemann-Roch Theorem.

Theorem 6.31

. Each compact Riemann surface X can be embedded into a projective space PN.

So - up to isomorphy - there is no dierence between Riemann surfaces and smooth projective complex algebraic curves. For embeddings one uses comple-tions of maps

(f0:f1:::::fn) : X ,!

Pn P 7!(f

0(P) :::::fn(P))

dened outside of the locus of common zeros of any basisf0f1:::fn ofL(D)

of suitable divisorsD onX.

Most important for curves of genus>2 are canonical embeddings working -by denition - with a canonical divisorD=K. Sometimes the canonical divisor does not embed the Riemann surface. This happens if and only if the curve is hyperelliptic. But the canonical map is a covering map onto its image curve for each curve of genus2.

7 Plane Curves

We write a short dictionary for plane projective curves over algebraically closed elds. In the le "RealCurves.mws" we present a list of nice real pictures of plane curves visualizing singularities and resolution (steps) with the ;process. To

open the le one has to implant it into MAPLE.

Flex point (also inection point): Regular pointP of a projective curveCwith tangent line contactingCof order3, or equivalently: the tangent hyperplane

ofCatP meetsCatP with multiplicity3. There are only nitely many ex

points on C.

Flex tangent: Tangent at a ex point.

Tangent hyperplane of a projective curveCat a regular pointP: A hyperplane meetingC atP with multiplicity2.

Double tangent: Projective lines of a plane projective curve C, which is a tangent at two dierent points ofC.

Ordinary singularity: Plane curve singularity with multiplicitym >1 having

mdierent tangent lines.

Node: Ordinary double point of a plane curve singularity.

Cusp (sometimes hypercusp): Curve singularityP with only one tangent line of the curve atP.

Simple cusp: Hypercusp of plane curve with multiplicity 3.

Dual curve Cof a projective plane curveC(not a line): The algebraic closure

of the curve in the dual space %P 2 of

P

2-lines consisting of all tangents on C.

The correspondenceC7!Cis an involution on the set of projective curves not

being lines. This involution restricts to the subset of all irreducible curves and to the set of Pl&ucker curves.

Plucker curve: Plane irreducible projective curve with at most nodes and simple cusps as singularities.

Class of a projective curve C: maximal number n = n(C) = n

Q(C) of

tangents at regular points ofC through a xed point Q2P

the same for almost all pointsQ2 P

2. It coincides with the degree deg C of

the dual curveC.

Plucker formulas (for Pl&ucker curves). LetC be a projective curve of degree

n >1. Set

d=d(C) := #fdouble points ofCg d :=d(C)

s=s(C) := #fcusps of Cg s=s(C):

It holds that a) d=

fdouble tangents of Cg,

b) s= #

finflection points ofCg,

a*) d= #fdouble tangents of C

g, b*) s= #finflection points of C g 1) n=n(n ;1);2d;3s(class formula), 2) s= 3n(n

;2);6d;8s(inection point formula),

1*) n=n(n ;1);2d ;3s, 2*) s= 3n(n ;2);6d ;8s.

Clebsch's genus formulafor Pl&ucker curvesC: The genusg=g(C) of (a smooth model of)C is equal to 1=2(n;1)(n;2);(d+s).

Canonical map of smooth projective curves C (Riemann surfaces) of genus

g2 is the regular (holomorphic) map

(!0:!1:::::!g;1) : C;! Pg

;1

extending

(1 :f1:::::fg;1) : P7!(1 :f1(P) :::::fg;1(P))

where ! = !0, !1 =f1! ,..., !g;1 =fg;1! is a basis of the space of regular

dierential forms onC.

Canonical model: The image curve of a canonical map.

H(F) is the determinant of (@2F=@ z

i@zj), i j = 012. The intersection of

H(C) andCcoincides with the set of inection points ofC.

n-gonal curve (also superelliptic curve): n-cyclic covering ofP

1. In

character-istic 0 such a curve has a plane model of ane equation type

Yk_{=pn(X) pn(X)2KX]of degree n:}

Picard curve (smooth): 3-gonal curve of genus 3. In characteristic 0 such a curve has a plane model of ane equation type

Y3=X4+aX3+bX2+cX+d=p

4(X) p4(X)without multiple zero:

Hyperelliptic curve: smooth 2-gonal curve. In characteristic 0 such a curve C

has a plane model of ane equation type

Y2=p

n(X) pn(X)without multiplezero n= 2m+ 1odd:

The genusg of a smooth model C is equal to m. The point 1 := (0 : 1 : 0)

is regular only in the elliptic case g = m = 1. The projection (xy) 7! x

onto the x-axes extends projectively to the canonical map C ;! P

1, which

is a 2-sheeted Galois covering. Among the smooth curves of genusg 2 the

hyperelliptic curves of are characterized by each of the following properties: (i) The canonical map C ;! Pg

;1 factorizes through a 2-sheeted Galois

covering onto a smooth rational curve.

(ii) There is a 2-sheeeted Galois covering ofC ontoP 1.

(iii) The canonical map C;!Pg

;1 is not an embedding.

Rational curve: Curve of genus 0. A smooth rational curve is isomorphic toP 1.

Smooth projective plane rational curves are the quadrics (degree 2 curves) and the projective lines.

Elliptic curve: Smooth curve of genus 1. In characteristic 0 such a curve has a plane model of ane equation type

Y2=X3+aX2+bX+c=p

3(X) p3(X)without multiple zero:

All smooth projective plane cubic curves are elliptic.

Genus 2 curve: All genus 2 curves are hyperelliptic. In characteristic 0 such a curve has a plane model of ane equation type

Y2=X5+aX4+bX3+cX2+dX+e=p

5(X) p5(X)without multiplezero:

genus 3 curve: The non-hyperelliptic smooth genus 3 curves are characterized by the property to have a smooth plane quartic model. These are the canonical