Bibliography

[lJ Abhyankar, S. S.: Local analytic geometry. New York-London: Academic Press, 1964. [2J Aleksandrov,P.S., Efremovich,V.A.: A survey of the fundamental concepts of

topology. Moscow: ONTI 1936. [3J Algebraic surfaces. Trudy Mat. Inst. Steklov 75 (1965). English translation: Algebraic

Surfaces (by the members of the seminar of I.R.Safarevic). Amer. Math. Soc. Transla­tions 75 (1967).

[4J Ahlfors, L. V., Bers, L.: Spaces of Riemann surfaces and quasiconformal mappings. Moscow: Izdat. Inost. Lit., 1961. (Translation into Russian of two papers by Ahlfors and four by Bers, see MR 24 A229.)

[5J Artin,M.: Algebraic spaces. Mimeographed notes. New Haven, Conn.; Yale University, 1969.

[6J B6cher,M.: Introduction to higher algebra. Cambridge Mass.: Harvard Univ., 1909. [7J Borevich,Z.I., Shafarevich,I.R.: Number theory. Moscow: Nauka, 1964. Translation,

New York and London: Academic Press, 1966. [8J Bourbaki, N.: Elements of mathematics, Book III, General topology, 2 vols. Paris,

Reading, Mass.: Hermann Cie.-Addison-Wesley Publ. Co., 1966. [9J Cartan,H.: Elementary theory of analyti9 functions of one or several complex

variables. Paris-Reading, Mass.: Hermann & Cie.-Addison Wesley Publ. Co., 1963. [lOJ Cartan,H., Eilenberg,S.: Homological algebra. Princeton, N. 1.: Univ. Press, 1956. [llJ Chern,S.S.: Complex manifolds. Univ. Recife 1959. [12J Eilenberg,S., Steenrod,N.E.: Foundations of algebraic topology. Princeton, N. 1.;

Univ. Press, 1962. [13J Fam,F.: Introduction to the topological investigation of Landau singularieties.

Moscow: Mir, 1970. [14J Grothendieck,A.: Cohomologie locale des faisceaux coherents et theonemes de

Lefschetz locaux et globaux. 2 fasc., third rev. ed., Paris: Inst. Hautes Et. Sci., 1968. [15J Grothendieck,A., Dieudonne,1.: Elements de geometrie algebrique, vol. 1. Berlin­

Heidelberg-New York: Springer-Verlag, 1971. [16J Goursat,E.: A course in mathematical analysis, vol. 1. New York: Dover reprint, 1959. [17J Gunning, R. c., Rossi, H.: Analytic functions of several complex variables. Englewood

Cliffs, N. 1.: Prentice-Hall, 1965. [18J Hodge, W. V. D., Pedoe, D.: Methods of algebraic geometry, vol. II. Cambridge: Univ.

Press, 1952. [19J Husemoller, D.: Fibre bundles. New York-London: McGraw-Hill, 1966. [20J Lang, S.: Introduction to algebraic geometry. New York-London: Interscience, 1958. [21J Milnor,l.W.: Morse theory. Princeton, N. 1.: Univ. Press, 1963. [22J Moishezon, B. G.: An algebraic analogue to compact complex spaces with a sufficiently

large field of merom orphic functions. Izv. Akad. Nauk SSSR Ser. Mat. 33, 174-238, 323-367, 506-548 (1969).

[23J Mumford,D.: Geometric invariant theory. Berlin-Heidelberg-New York: Springer­Verlag, 1965.

410 Bibliography

[24J Mumford, D.: Lectures on curves on an algebraic surface. Princeton, N. J.; Univ. Press, 1966.

[25J Pontryagin,L.S.: Topologische Gruppen, second ed. 2 vols. Leipzig: Teubner, 1957. [26J de Rham,G.: Varietes differentiables. Paris: Hermann&Cie. 1955. [27J Samuel, P.: Methodes d'algebre abstraite en geometrie algebrique. Berlin-Heidelberg­

New York: Springer-Verlag, 1967. [28J Seifert, H., Threllfall, W.: Lehrbuch der Topologie. Leipzig: Teubner, 1934. [29J Serre,J.P.: Lecture Notes in Math. Vol. II, Algebre Locale multiplicites. Third ed.

Berlin-Heidelberg-New York: Springer-Verlag, 1975. [30J Siegel,C.L.: Analytic functions of several complex variables. Princeton, N. J.: Inst.

for Adv. Studies, 1950. [31J Springer, G.: Introduction to Riemann surfaces. Reading, Mass.: Addison-Wesley

Pub!. Co., 1957. [32J Vinogradov,l.M.: Elements of number theory, seventh ed. Moscow: Nauka, 1965.

Translations: New York: Dover, 1954 or Oxford: Pergamon 1952. [33J van der Waerden,B.L.: Algebra, 2 vols., fifth ed. Berlin-Gottingen-Heidelberg:

Springer-Verlag, 1960. [34J Walker,R.J.: Algebraic curves, Princeton, N. J.; Univ. Press., 1950. [35J Wallace,A.H.: Differential topology: First steps, New York-Amsterdam: W.A.Ben­

jamin, 1968. [36J Weil,A.: Introduction it l'etude des varietes Kiihleriennes. Paris: Hermann & Cie.,

1958. [37J Zariski,O., Samuel, P.: Commutative algebra, 2 vols. Princeton, N. J.; Van Nostrand

and Co. 1958/60.

Historical Sketch

This sketch does not pretend to give a systematic account of the history of algebraic geometry. Its aim is to describe in very general terms how the ideas and concepts with which the reader has become acquainted in this book were created. In explaining the research of one mathe­matician or another we often omit important work of his (sometimes even the most important) if it has no bearing on the contents of our book.

We shall try to formulate the results as closely as possible to the way the authors have done it, using only occasionally contemporary notation and terminology. In cases where this is not immediately obvious we shall explain these investigations from the point of view of the concepts and results of our book. Such places are marked by an asterisk * (at the beginning and the end).

* Naturally algebraic geometry arose first as the theory of algebraic curves. Only by going beyond the frame of rational curves do we encounter properties of algebraic curves that are characteristic for algebraic geometry. Therefore we leave aside the theory of conics, which are all rational. Next in complexity and hence the first non­trivial example are curves of genus 1, that is, elliptic curves and, in particular, non-singular curves of the third degree. And historically the first step in the development of the theory of algebraic curves consisted in a clarification of the basic concepts and ideas of this theory in the example of elliptic curves.

Thus, it would seem that these ideas developed in the same sequence in which they are now set forth (for example, in Ch. I, § 1). However, in one respect this is by no means the case. The complex of concepts and results that we now call the theory of elliptic curves arose as part of analysis, and not of geometry: as the theory of integrals of rational functions on an elliptic curve. It was precisely these integrals that originally were called by the name elliptic (they occur in connettion with the computation of the arc length of an ellipse), and later the name was transferred from them to functions and to curves. *

412 Historical Sketch

1. Elliptic Integrals. They were an object of study as early as the XVII century, as an example of integrals that cannot be expressed in terms of elementary functions and lead to new transcendental functions.

At the very end of the XVII century Jacob, and later Johann, Bernoulli came up against a new interesting property of these integrals (see J. Bernoulli [1J, Vol. 1, p. 252). In their investigations they considered integrals expressing the arc length of certain curves. They found certain transformations of one curve into another that preserve the arc length of the curve, although the corresponding arcs cannot be superposed to one another. It is clear that analytically this leads to the transformation of one integral into another. In some cases there arise transformations of an integral into itself. In the first half of the XVIII century many examples of such transformations were found by Fagnano.

In general form the problem was raised and solved by Euler. He communicated his first results in this direction in a letter to Goldbach in 1752. His investigations on elliptic integrals were published from 1756 to 1781 (see Euler [1 J).

Euler considers an arbitrary polynomial f(x) of degree 4 and asks for the relations between x and y if

dx dy Vf(x) = Vf(y) .

(1)

He regards this as a differential equation connecting x and y. The required relation is the general integral of this equation. He finds this relation: it turns out to be algebraic of degree 2 both in x and in y. Its coefficients depend on the coefficients of the polynomial f(x) and on one independent parameter c.

Euler formulates this result in another form: the sum of the integrals

SCC dx d Sfl dx. I . I· I 1~ an 1~ Isequa to a smg emtegra :

o V f(x) 0 V f(x)

CC dx fI dx Y dx

! Vf(x) + ! Vf(x) =! Vf(x) , (2)

and '}' can be expressed rationally in terms of IX and p. Euler also brings forward arguments why such a relation cannot hold if the degree of the polynomial f(x) is greater than 4. ( ) d

For arbitrary elliptic integrals of the form S V. x Euler proves a relation that generalizes (2): f(x)

SCC r(x) dx fI r(x) dx Y r(x) dx 6

o V7W + I V7W - I V7W = I V(y)dy, (3)

Historical Sketch 413

where y is the same rational function of 0( and p as in (2), and where () and V are also rational functions.

* The reason for the existence of an integral of the equation (1) and of all its special cases discovered by Fagnano and Bernoulli is the pre­sence of a group law on an elliptic curve with the equation S2 = f(t) and the invariance of the everywhere regular differential form s-ldt under translations by elements of the group. The relations found by Euler that connect x and y in (1) can be written in the form

(x, V f(x)) $ (c, V f(c)) = (y, V f(y)) ,

where $ denotes addition of points on the elliptic curve. Thus, these results contain at once the group law on an elliptic curve and the existence of an invariant differential form on this curve.

The relation (2) is also an immediate consequence of the invariance of

the form cp = I ~ • In it V f(x)

and (y, V f(y)) = (0(, V f(O())$(P, V f(l3))

'" p '" y '" y y

JCP+Jcp=Jcp+J~cp=Jcp+Jcp=Jcp, 000", 0 '" 0

where tg is the translation by g = (0(, V f(O()). Observe that we write here the equation between integrals formally, without indicating the paths of integration. Essentially this is an equation "to within a constant of integration", that is, an equation between the corresponding differential forms. This is how Euler understood them.

Finally, the meaning of the relation (3) will become clear later, in connection with Abel's theorem (see 3.). *

2. Elliptic Functions. After Euler the theory of elliptic integrals was developed mainly by Legendre. His investigations, beginning in 1786, are collected in the three-volume "Traite des fonctions elliptiques et des integrales Euleriennes (Legendre [1J).* In his preface to the first supplement published in 1828 Legendre writes: "So far the geometers have hardly taken part in investigations of this kind. But no sooner had this book seen the light of day, no sooner had it become known to scholars abroad, than I learned with astonishment as well as joy that two young geometers, Herr Jacobi in Konigsberg and Herr Abel in Christiania, have achieved in their works substantial progress in the highest branches of this theory".

* Legendre called elliptic functions what we now call elliptic integrals. The contemporary terminology became accepted after Jacobi.

414 Historical Sketch

Abel's papers on the theory of elliptic functions appeared in 1827-1829. He starts out (see Abel [1], Vol. I, No. XVI, No. XXIV) from the elliptic integral

where c and e are complex numbers; he regards it as a function of the upper limit and introduces the inverse function A(8) and the function LI(8) = V(1- c2 A2)(1- e2..12). From the properties of elliptic integrals known at that time [essentially, from Euler's relations (2) in 1.] he deduces that the functions A(8 ± 8') and LI (8 ± 8') can be simply expressed in the form of rational functions of A(8), A(8'), LI(8), and LI(8'). Abel shows that both these functions have in the complex domain two periods 2w and 2w:*

lie dx w-2 f

- 0 V(1- C2 x 2 )(1 - e2 x 2 )

He finds representations of the functions introduced by him in the form of infinite products extended over their zeros.

As an immediate generalization of the problem with which Euler had been occupied, Abel [1] (Vol. I, No. XIX) raises the question: "To list all the cases in which the differential equation

dy = +a dx (1) V(1- ciJ2)(1 - eiJ2) - V(1- c2 x 2 )(1 - e2 x 2 )

can be satisfied by taking for y an algebraic function of x, rational or irrational" .

This problem became known as the transformation problem for elliptic functions. Abel showed that if the relation (1) can be satisfied by means of an algebraic function y, then it can also be done by means of a rational function. He showed that if c I = c, e 1 = e, then a must either be rational or a number of the form Ii + -v=-Il, where fl' and fl are rational numbers and fl > O. In the general case he showed that the periods WI and WI of the integral of the left-hand side of (1), multiplied by a common factor, must be expressible in the form of an integral linear combination of the periods wand W ofthe integral of the right-hand side. * As E. 1. Slavutin has remarked, already Euler [2J drew attention to the fact that the

Y dx function S ~ has in the real domain a "modulus of multi-valued ness" similar to the

o i-x inverse trigonometric functions.

Historical Sketch 415

Somewhat later than Abel, but independently, Jacobi [1J (Vol. I, Nos. 3 and 4) also investigated the function inverse to the elliptic integral, proved that it has two independent periods, and obtained a number of results on the transformation problem. Transforming into series the expressions for elliptic functions that Abel had found in the form of products, Jacobi arrived at the concept of O-functions* and found numerous applications for them, not only in the theory of elliptic functions but also in number theory and in mechanics.

Finally, after Gauss's posthumous works were published, especially his diaries, it became clear that long before Abel and Jacobi he had mastered some of these ideas to a certain extent.

* The first part of Abel's results requires hardly any comment. The mapping x = A(O), y = LI(O) determines a uniformization of the elliptic curve y2 = (1 - c2 x2)( 1 - e2 x 2 ) by elliptic functions. Under the cor-

responding mapping f: ([:1_ X the regular differential form cp = dx y

goes over into a regular differential form on ([:1 that is invariant under translations by the vectors of the lattice 2w'll + 2m'll. This form differs

dx by a constant factor from dO, and we may assume that dO = f* -, dx y

that is, 0= J-. y The integration of the equation (1) has the following geometric mean­

ing. Let X and Xl be elliptic curves with the equations u2 = (1 - c2 x2 )

. (1 - e2 x2) and v2 = (1 - ci y2) (1 - ei y2). The point is to investigate curves C C X X Xl (which corresponds to an algebraic relation between x and y). Since an elliptic curve is its own Picard variety (see Ch. III, § 3.5), C deter­mines a morphism f : X - Xl. This makes it clear why the problem reduces to the case when y is a rational function of x. According to Theorem 4 in Ch. III, § 3, f can be regarded as a homomorphism of the algebraic groups X and Xl. Thus, Abel studied the group Hom (X, Xl) and for X = Xl the ring End X. A homomorphism f E Hom (X, Xl) deter­mines a linear transformation of the one-dimensional spaces f* :01 [X 1]_01 [X], which is given by a single number, the factor ± a in (1). See also Exercises 7, 8, and 9 to Ch. IX, § 2. *

3. Abelian Integrals. The transition to arbitrary algebraic curves proceeded entirely within the framework of analysis: Abel showed that the basic properties of elliptic integrals can be generalized to integrals of arbitrary algebraic functions. These integrals later became known as Abelian integrals.

* O-functions occured first in 1826 in a book by Fourier on the theory of heat.

416 Historical Sketch

In 1826 Abel wrote a paper (see Abel [1J, Vol. I, No. XII), which was the beginning of the general theory of algebraic curves. He considers in it an algebraic function Y determined by two equations

and X(X,y)=o,

O(x, y) =0,

(1)

(2)

where O(x, y) is a polynomial that depends, apart from x and y, linearly on some parameters a, d, ... , the number of which is denoted by cx. When these parameters are changed, some simultaneous solutions of (1) and (2) may not change. Let (Xl' Y1), ... , (XIl' YIl) be variable solutions, and I(x, y) an arbitrary rational function. Abel shows that

Xl x~

f I(x, y) dx + ... + f I(x, y) dx = f V(g) dg , (3) o 0

where V(t) and g(x,y) are rational functions depending also on the param­eters a, ai, .... Abel interpreted this result by saying that the left-hand side of (3) is an elementary function.

Using the freedom in choosing the parameters a, ai, ... Abel shows Xi

that the sum of any number of integrals f I(x, y) dx can be expressed in o

terms of jl- cx such integrals and a term of the same type as that on the right-hand side of (3). He establishes that the number jl- cx depends only on (i). For example, for y2 + p{x), where the polynomial p is of degree 2m, we have jl-cx=m-1.

Next Abel investigates for what functions I the right-hand side of (i) does not depend on the parameters a, d, .... He expresses I in the form

f{l (x'{), , and he shows that 12 = 1, and 11 satisfies a number of 2 X, Y Xy

restrictions as a consequence of which the number y of linearly independ­ent ones among the required functions I is finite. Abel shows that y ~ jl- cx and that y = jl- cx, for example, if (using a much later terminolo­gy) the curve X{x, y) = 0 has no singular points.

* The discussion of the solutions (Xl, Y1), ... , (x ll ' yJ of the system consisting of (1) and (2) leads us at once to the contemporary concept of equivalence of divisors. Namely: let X be the curve with the equation (1) and D). the divisor cut out on it by the form 0). (in homogeneous co­ordinates), where A. is the system of parameters a, d, .... By hypothesis, D). = D;. + Do, where Do does not depend on A.. Therefore all the D). = (Xl' Y1) + ... + (x ll ' YIl) are equivalent to each other. The problem with which Abel was concerned reduces to the investigation of the

PI PI" sum f cp + ... + f cp, where cp is a differential form on X, CXi and Pi are

a1 aJ.L

Historical Sketch 417

points on X, (OC1) + ... + (OCIl) "'" (Pl) + ... + (PIl). We give a sketch of a proof of Abe1's theorem that is close in spirit to the original proof. We may assume that

(OC1) + ... + (ocll) - (Pl) - ... - (PIl) = (g), g E <C(X),

(OC1) + ... + (ocJ = (g)o, (Pl) + ... + (PJ = (g)oo .

We consider a morphism g: X _]pl and the corresponding extension <C(X)/<C(g). For simplicity we assume that this is a Galois extension (the general case easily reduces to this), and we denote its Galois group by G. The automorphisms (J' E G act on the curve X and the field <C(X) and carry the points oc l , ... , ocllintoeach other, because {ocl , ... , ocll } =g-l(O). Therefore {OC1, ... ,OCIl}={(J'OC,(J'EG}, where oc is one of the points OCi. Similarly {Pl, ... ,PIl}={(J'P,(J'EG}. Representing qJ in the form udg, we see that

Il Pi t1P P

i"fll qJ= t1~GL udg= !C~G (J'u)dg . (4)

The function v = L (J'U is contained in <C(g), and Abe1's theorem follows from this. t1 E G Xi

We see that every sum of integrals ~ J f(x, y) dx can be expressed as I xj , 0

a sum of I integrals L J f(x, y) dx + J V(g) dg if the equivalence j=lO

I

L ((OCi) - 0) "'" L ((oc}) - 0) i j= 1

(5)

holds, where OCi = (Xi, Yi), OC} = (X)' Y}), and 0 is the point with X = O. From the Riemann-Roch theorem it follows at once that the equivalence (5) holds (for arbitrary OCi and certain oc) corresponding to them), with I = g (see Exercise 19 to Ch. III, § 5). Thus, the constant fl- oc introduced by Abel is the same as the genus.

If qJ E .01 [X], then also v dgE .01 [1Pl], where V= L (J'U in (4). Since t1EG

.01 [1Pl] = 0, in this case the term on the right-hand side of (3) dis­appears. Hence it follows that y ~ g. In some cases arising naturally the two numbers coincide.

We see that this paper of Abe1's contains the concept of the genus of an algebraic curve and the equivalence of divisors and gives a criterion for equivalence in terms of integrals. In the last relation it leads to the theory of Jacobian varieties of algebraic curves (see § 5). *

4. Riemann Surfaces. In his dissertation published in 1851 Riemann [1] (No.1) applied a completely new principle of investigating functions

418 Historical Sketch

of a complex variable. He assumes that the function is given not on the plane of a complex variable but on some surface that "extends in many sheets" over this surface. The real and imaginary parts of this function satisfy the Laplace equation. This function is uniquely determined if the points are known at which it becomes infinite, the curves along which cuts make it single-valued, the character of its singularities at these points, and the many-valuedness in passing through these curves. Riemann also works out a method of constructing a function from these data that is based an a variational principle which Riemann called "the Dirichlet principle".

In the first part of the paper "Theory of Abelian functions", which appeared in 1857, Riemann [1] (No. II) applied these ideas to the theory of algebraic functions and their integrals. The paper begins with the investigation of properties of the corresponding surfaces that belong, as Riemann says, to Analysis Situs. By means of an even number 2p of cuts the surface becomes a simply-connected domain. By arguments taken from Analysis Situs he shows that p = w/2 - n + 1 where n is the number of sheets and w the number of branch points of the surface over the plane of the complex variable (taken with the appropriate multiplicities).

Riemann investigates functions that, speaking generally, are many­valued on the surface, but single-valued in the domain obtained after making the cuts, and on passing through the cuts their values change by constants, the so-called moduli of periodicity of the function. The Dirichlet principle gives a method of constructing such functions. In particular, there are p linearly independent everywhere finite such functions: the "integrals of the first kind". Similarly functions are con­structed that become infinite at given points. In order to form from them functions that are single-valued on the surface one has to equate to zero their moduli of periodicity. Hence it follows that among the single­valued functions that become infinite only at m given points not fewer than m - p + 1 are linearly independent and, if m > p, this inequality becomes an equality for points in "general" position.

Riemann shows that all the functions that are single-valued on a given surface are rational functions of two of them: sand z, which are con­nected by a relation F(s, z) = O. He calls two such relations belonging to one "class" if they can be rationally transformed into one another. In that case the corresponding surfaces have one and the same number p. But the converse is not true. Studying the possible dispositions of the branch points of surfaces, Riemann shows that the set of classes depends for p > 1 on 3 p - 3 independent parameters, which he calls "moduli".

* The surfaces introduced by Riemann closely correspond to the con­temporary concept of a one-dimensional complex analytic manifold; these are the sets on which analytic functions are defined. Riemann

Historical Sketch 419

raises and solves the problem of the connection of this concept with that of an algebraic curve. (The appropriate result is called Riemann's existence theorem.)

This circle of Riemann's ideas did not by any means become clear at once. An important role in their clarification is played by Klein's lectures [2], in which he emphasizes that a Riemann surface a priori is not connected with an algebraic curve or an algebraic function. A definition of a Riemann surface that differs only terminologically from the presently accepted definition of a one-dimensional analytic manifold was given by H. Weyl [1].

Riemann's paper marks the beginning of the topology of algebraic curves. The topological meaning of the dimension p of the space Q1 [X] is explained in it: it is half the dimension of the first homology group of the space X(CC).

Analytically Riemann proves the inequality leD) ~ degD - p + 1. The Riemann-Roch equality was then proved by his pupil Roch.

Finally, in this paper the field k(X) emerges for the first time as an original object connected with a curve X, and the concept of a birational isomorphism appears. *

5. The Inversion Problem. Already Abel had raised the question of the inversion of integrals of arbitrary algebraic functions. He observed, in particular, that the function inverse to the hyperelliptic integral con­nected with V 1p(x) has periods equal to half the value of this integral taken between two roots of the polynomial 1p (see Abel [1], Vol. II, No. VII).

Jacobi drew attention to the fact that we are concerned here with a function of a single complex variable having more that two periods if the integral is not elliptic, and that this is impossible for a reasonable function. If X is a polynomial of degree 5 or 6, Jacobi proposes to consider the pair of functions

x dx Y dx u=J-+J-, ov'X ov'X

x xdx Y xdx v= J lI'v + J lI'v'

o vX 0 vX

He suggests expressing x - y and xy as analytic functions of the two variables u and v and conjectures that this expression is possible by means of a generalization of O-functions (see Jacobi [1], Vol. II, Nos. 2 and 4). This conjecture was verified in a paper by Gopel [1] published in 1847.

The second part of Riemann's paper [1] (No. II) on Abelian functions is concerned with the connection between O-functions and the inversion

420 Historical Sketch

problem in the general case. He considers a series in p variables

O(v) = L eF (m)+2(m,v) ,

m ( 1)

where m = (m1' ... , mp) ranges over all integral p-dimensional vectors, v = (V1' ... , vp), (m, v) = ~miVi> F(m) = ~Ct.j/mjmz, Ct.jl = a/j. This series converges for all values of v if the real part of the quadratic form F is negative definite. The main property of the function 0 is the equality

O(v + nir) = O(v) , O(v + Ct. j) = eLj(v)O(v) , (2)

where r is an integral vector, Ct. j a column of the matrix (Ct.jl), and Lj(v) a linear form.

Riemann shows that one can choose cuts a1' ... , ap ' b1, ... , bp which make his surface simply-connected, and a basis U1, .•. , up of everywhere finite integrals on this surface such that the integrals Uj over al are equal to 0 for j+ I and to ni for j= I, and the same Uj over bl form a symmetric matrix (Ct.j/), satisfying conditions under which the series (1) converges. He considers the function 0 corresponding these coefficients Ct.j / and the function O(u - e), where U = (U1' ... , up) (the Ui are everywhere finite integrals) and e is an arbitrary vector.

Riemann shows that O(u - e) has on the surface p zeros 171' ... , 17p or vanishes identically. For a suitable choice of the lower limits in the integrals Ui in the first case

(3)

where the congruence is taken modulo integral linear combinations of the periods of the integrals Ui. In this way the points 171' ... , 1'Jp are uniquely determined. In the second case there also exist points 171' ... , 17 p -2 such that

(4)

Riemann knew that the periods of any 2n-periodic function of n variables satisfy relations similar to those that are necessary for the convergence of the series defining the O-function. These relations between the periods were described explicitly by Frobenius [1J, who showed that they are necessary and sufficient for the existence of non-trivial functions satisfying the functional equation (2). Hence it follows that these relations are necessary and sufficient for the existence of a mero­morphic function with given periods that cannot be reduced by a linear change of variables to a function of a smaller number of variables. One only has to apply the theorem that every 2n-periodic analytic function can be represented as a quotient of entire functions satisfying the functional equation of the O-function. This theorem, stated by Weierstrass, was proved by Poincare [2]. In 1921 Lefschetz [1J proved

Historical Sketch 421

that when the Frobenius relations hold, the O-functions determine an embedding of the manifold (f;"/o' (where 0, is the lattice corresponding to the given period matrix) into a projective space.

* The inversion problem is connected with questions which in this book we only touched upon incidentally, often without proofs. The matter concerns the construction of the J aco bian variety of an analytic curve and properties of arbitrary Abelian varieties (see Ch. III, § 3.5, and Ch. VIII, § 1.3).

If 0 is a fixed point on a curve X, then f(x) = x - 0 is evidently an algebraic family of divisors of degree zero on X. The basis of this family coincides with X. By the definition of the Jacobian variety Jx of X (we recall that this is the name for the Picard variety if X is a curve) there exists a morphism cp: X -Jx , which is an embedding if the genus p of X is different from O. It can be shown that cp*: 0,1 [Jx] _ 0,1 [X] is an isomorphism. Therefore in the representation

(5)

the 2p-dimensionallattice 0, C CP consists ofthe periods of p independent differential forms OJ E 0,1 [X]. Riemann also starts out from this analytic specification of the Jacobian variety and then develops an algebraic method of investigating it.

If Do is an arbitrary effective divisor of degree p, then g(Xl' ... , xp) = Xl + ... + xp - Do determines a family of divisors of degree 0 on X. A basis of this family can be taken to be the factor space XP/Sp of the product of p copies of X with respect to the symmetric group acting by permutations of the factors. By the definition of the Jacobian variety there exists a morphism 1p: XP /S p - Jx . It follows easily from the Riemann­Roch theorem that it is an epimorphism and one-to-one on an open set in Jx . Therefore it is a birational isomorphism. In the analytical represen­tation 1p takes the form (3), and by definition it is not one-to-one at those points (Xl, ... , Xp) for which l(xl + ... + xp) > 1. It follows from the Riemann-Roch theorem that this is equivalent to the condition l(K - Xl - ... - xp) > 0, that is (because degK = 2g - 2), to

Xl + ... +xP"'~ - Yl_ - ... _ - Yp -2

for certain points Yl' ... ' Yp-2. The latter relation is the same as (4), to within the additional term K, hence to a shift by a point in Jx.

The Frobenius relations are the condition for the analytic manifold eel/O, to be projective. They are written down in Ch. VIII, § 1.4 [formulae (3) and (4)]. *

6. Geometry of Algebraic Curves. So far we have seen how the concepts and results that nowadays form the foundation of the theory of

422 Historical Sketch

algebraic curves have been created under the influence and within the framework of the analytic theory of algebraic functions and their integrals. A purely geometric theory of algebraic curves developed independently of this trend of research. For example, in a book published in 1834 Plucker found formulae connecting the class, the degree of a curve, and the number of its double points (see Exercise 13 to Ch. IV, § 3). There he also proved the existence of nine points of inflexion on a plane cubic curve (see Exercise 4 to Ch. VIII, § 1). But research of a similar kind took second place in the mathematics of the time no deeper ideas were linked with it.

Only in the period following the era of Riemann did the geometry of algebraic curves occupy a central place in the contemporary mathematics, alongside the theory of Abelian integrals and Abelian functions. Basically this change of view point was connected with the name of Clebsch. Whereas for Riemann the foundation is the function, Clebsch takes as the fundamental object the algebraic curve. One can say that Riemann considered a finite morphism f: X -4 IP 1, and Clebsch the algebraic curve X itself. In the book by Clebsch and Gordan [1] a formula is deduced for the number p of linearly independent integrals of the first kind (that is, for the genus of the curve X), which expresses it in terms of the degree ofthe curve and the number of singular points (see Exercise 12 to Ch. IV, § 3). There it is also shown that for p = 0 the curve has a rational parameterization, and for p = 1 becomes a plane cubic curve.

An error Riemann had made turned out to be exceptionally useful for the development of the algebraic-geometrical aspect of the theory of algebraic curves. In the proof of his existence theorems he had regarded as obvious the solubility of a certain variational problem: the "Dirichlet principle". Before long Weierstrass showed that not every variational problem has a solution. Therefore Riemann's results remained unfounded for some time. One of the ways out was an algebraic proof of these theorems: they were stated essentially in algebraic form. These investi­gations, which were undertaken by Clebsch (see Clebsch and Gordan [1]), furthered considerably the clarification of the essentially algebraic­geometrical character of the results of Abel and Riemann, hidden under an analytic cloak.

The trend of research begun by Clebsch achieved its bloom in the work of his pupil M. Noether. Noether's ideas are particularly clearly out­lined in his joint paper with Brill [1]. In it the problem is raised of developing the geometry on an algebraic curve lying in a projective plane, as the collection of results that are invariant under biunique (that is, birational) transformations. The foundation is the concept of the group of (coincident or distinct) points of the curve. They consider systems of groups of points that cut out on the original curve linear systems of

Historical Sketch 423

curves (that is, systems whose equations form a linear space). It can happen that all groups of such a system contain a common group G, that is, consist of G and another group G'. The system of groups G' obtained in this way is called linear. If the dimension of the linear (pro­jective) space of equations of the cutting curves is equal to q and the groups G' consists of Q points, then the system is denoted by g~). Two groups of one and the same system are called corresidual. Clearly this corresponds to the modern concept of equivalence of effective divisors, and if G is contained in a linear system g~), then in the modern notation deg G = Q, l( G) > q + 1 (we recall that l( G) is the dimension of the vector space and q that of the corresponding projective space).

Every group of points determines a largest linear system g~) con­taining all the groups corresidual to the given one. The numbers q and Q are connected by the Riemann-Roch theorem, which is proved purely algebraically.

Of course, the Riemann-Roch theorem presupposes a definition analogous to the canonical class. It is given without appeal to the concept of a differential form, but the connection with this concept is very easily established. For if a curve of degree n has the equation F = 0 and is smooth, then the differential forms W E Ql [X] can be written in the form

W = ;, dx, where <p is a homogeneous polynomial of degree n - 3 v

(Ch. III, § 5.4). It can be shown that if the curve has only the simplest singularities, then this expression remains valid if it is required that <p vanishes at all singular points. These polynomials are said to be associ­ated. Associated polynomials of degree n - 3 determine the linear system that is an analogue to the canonical class.

In their paper Brill and N oether consider a mapping of a curve into the (p - i)-dimensional projective space defined by the associated polynomials of degree n - 3. Its image is called a normal curve. They show that a single-valued (in present-day terminology, birational) correspondence of curves reduces to a projective transformation of normal curves (provided that the curves are not hyperelliptic).

Noether [1] applied these ideas to the investigation of space curves. In modern language we can say that his paper is concerned with the study of the irreducible components of the Chow variety of curves in three­dimensional space.

7. Many-Dimensional Geometry. At the beginning of the second half of the nineteenth century many special properties of algebraic varieties of dimension greater than 1, mainly surfaces, had been found. For example, cubic surfaces had been investigated in detail, in particular, Salmon and Cayley had proved in 1849 that on any cubic surface without

424 Historical Sketch

singular points there are 27 distinct lines. However, for a long time these results were not combined by any general principles and were not connected with the deep ideas that had been worked out at the time in the theory of algebraic curves.

The decisive step in this direction was taken, apparently, by Clebsch. In 1868 he published a small note [1] in which he considers algebraic surfaces from the point of view (using modern terminology) of birational isomorphism. He considers everywhere finite double integrals on the surface and mentions that the maximal number of linearly independent among them is invariant under birational isomorphism.

These ideas were developed in Noether's paper [2], which consists of two parts. As is clear from the very title, in it he considers varieties of an arbitrary number of dimensions. However, the major part of the results refers to surfaces. This is typical for the whole subsequent period of algebraic geometry: although very many results were in fact true for varieties of arbitrary dimension, they were stated and proved only for surfaces.

In the first part Noether considers "differential expressions" on a variety of arbitrary dimension, and it is interesting that he writes down an integral sign only once. Thus, here the algebraic character of the concept of a differential form already becomes formally obvious. N oether considers only forms of maximal degree. He shows that they make up a finite-dimensional space whose dimension is invariant under single­valued (that is, birational) transformations.

In the second part he considers curves on surfaces. (Only the last section contains some interesting remarks on three-dimensional varieties.) Noether gives a description of the canonical class (in modern terminology) by means of associated surfaces, analogous to the way in which this was done earlier for curves. He raises the question of the surfaces V that cut out on a curve C lying on V its canonical class (again in modern terminology). He calls the curves cutting them out on V associated with C and gives an explicit description for them which leads him to a formula for the genus of a curve on a surface. This formula essentially is the same as (1) in Ch. IV, § 2.3; however, an understanding of the fact that an associated curve is of the form K + C was achieved only 20 years later in the work of Enriques.

In the same paper Noether investigates the concept of an exceptional curve, which contracts to a point under a birational isomorphism.

The most brilliant development of the ideas of Clebsch and Noether came not in Germany, but in Italy. The Italian school of algebraic geometry exerted an immense influence on the development of this branch of mathematics. Undoubtedly many ideas created by this school have so far not been fully understood and developed. The founders ofthe Italian geometrical school are Cremona, C. Segre, Bertini. Its most im­portant representatives are Castelnuovo, Enriques, and Severi. Castel-

Historical Sketch 425

nuovo's papers began to appear at the end of the 1880's. Enriques was a pupil (and relative) ofCastelnuovo. His papers appeared at the beginning of the 1890's. Severi began to work about ten years after Castelnuovo and Enriques.

One of the main achievements of the Italian school is the classifi­cation of algebraic surfaces. As a first result we can quote here a paper of Bertini [1] in which a classification of involutory transformations of the plane is given. The matter concerns (in present-day terminology) the classification, to within conjugacy in the group of birational aut om or­phisms of the plane, of all group elements of order 2. The classification turns out to be very simple, in particular, it is easy to derive from it that the factor space of the plane with respect to a group of order 2 is a rational surface. In other words, if a surface X is unirational and a morphism f: IP2 ~ X is of degree 2, then X is rational.

The general case of Liiroth's problem for algebraic surfaces was solved (affirmatively) by Castelnuovo [1]. After this he raised the problem of characterizing rational surfaces by numerical invariants and solved it in [2]. The classification of surfaces that we have explained in Ch. III, § 5.7, was obtained by Enriques in a series of papers, which was completed in the first decade of our century (see Enriques [2]).

In the context of Liiroth's problem for three-dimensional varieties Fano investigated certain types of these varieties, suggesting a proof of the fact that they are not rational. Enriques had shown that many ofthem are unirational. This would give a negative solution of Luroth's problem, but Fano pointed out many obscure places in the proof. Some intermediate propositions turned out to be not true. The problem was solved finally when the last pages of this book were already written. V.A. Iskovskii and Yu. I. Manin have shown that Fano's basic idea can be made to work. They have shown that smooth hypersurfaces of degree 4 in IP4 are non­rational (the fact that some of them are unirational had been proved by B. Segre). Simultaneously Griffiths and Clemens have found a new analytical method of proving that certain varieties are non-rational, for example, smooth hyper surfaces of degree 3 in IP4 (see Exercise 18 to Ch. III, § 5). Of course, these results are only the first steps on the way to a classification of unirational varieties. *

The main tool of the Italian school was the investigation of families of curves on surfaces -linear and algebraic (the latter were called continuous). This led to the concept of linear and algebraic equivalence (in our book linear equivalence is simply called equivalence). The con­nection between these two concepts was first investigated by Castelnuovo [3]. He discovered a link of this problem with an important invariant of

* A third method to construct non-rational but unirational threefolds was discovered by Artin and Mumford. See v.A.lskovskii and Yu.I.Manin, Mat. Sborn. 86 (1971),140-166, C.H.Clemens and P.Griffiths, Ann. of Math. 95 (1972) 281-358; M.Artin and D.Mumford Proc. of Lond. Math. Soc. XXV (1972) 75-95. (Footnote to corrected printing, 1977).

426 Historical Sketch

the surface, the so-called irregularity. We do not give here the definition of irregularity, which was used by Castelnuovo; it is connected with ideas close to the cohomology theory of sheaves. Formula (1) below gives another interpretation of this concept.

Castelnuovo [3J proved that if not every continuous system of curves is contained in a linear system (that is, if algebraic and linear equivalence are not one and the same thing), then the irregularity of the surface is different from zero. Enriques [1J proves the converse proposition. Furthermore, he shows that every sufficiently general curve (in an exactly defined sense) lying on a surface of irregularity q is contained in an algebraically complete (that is, maximal) continuous family, which is stratified into linear families of the same dimension, and the basis of the stratification is a variety of dimension q. Castelnuovo [1J showed that the fibering of linear systems (that is, classes of divisors) determines on a q-dimensional basis of the fibering constructed by Enriques a group law by virtue of which this basis is an Abelian variety and is therefore uniformized by Abelian (2q-periodic) functions. This Abelian variety does not depend on the curve from which we have started out and is determined by the surface itself. It is called the Picard variety of this surface.

The irregularity turned out to be connected with the theory of one­dimensional differential forms on the surface, the beginning of which goes back to Picard [1 J; in this paper it is proved that the space of every­where regular forms is finite-dimensional. In 1905 Severi and Castelnuovo proved that this dimension is the same as the irregularity; in our notation

(1)

Severi [1J investigated the group of classes relative to algebraic equivalence and proved that it is finitely generated. His proof is based on a connection of the concept of algebraic equivalence with the theory of one-dimensional differential forms. Namely, an algebraic equivalence nlCl + ... +nrCr~Oisequivalenttothefactthatforsomeone-dimensional differential form the set of its "logarithmic singularities" coincides exactly with the curves C1, ... , Cr, taken with the multiplicities nl' ... , nr •

(A curve C is a logarithmic singularity of multiplicity n for a form OJ

iflocally OJ = n f- 1 d f, where f is a local equation of C.) Picard had already proved earlies that the so-defined relation of equivalence by means of differential forms gives rise to a finitely generated group of classes (see Picard and Simart [1J).

8. The Analytic Theory of Manifolds. Although a considerable part of the concepts of algebraic geometry arose in analytic form, their algebraic meaning cleared up in time. Now we pass on to concepts and

Historical Sketch 427

results which are essentially connected with analysis (at least from the present point of view).

At the beginning of the 1880's there appeared papers by Klein and Poincare devoted to the problem of uniformization of algebraic curves by automorphic functions. The aim was to uniformize arbitrary curves by functions that are now called automorphic, just as elliptic functions uniformize curves of genus 1. (The term "automorphic" was proposed by Klein, previously these functions were called by various names.) Klein [1] (No. 84) started out from the theory of modular functions. The field of modular functions is isomorphic to the field of rational functions, but one can consider functions that are invariant under various subgroups of the modular group and so obtain more complicated fields. In particular, Klein considered functions that are automorphic relative to the group

consisting of all transformations z ~ az + db in which a, b, e, and dare ez+

integers, ad - be = 1, and

(: :) == (~ ~) (mod 7) .

He proved that these functions uniformize the curve of genus 3 with the equation xgx l + XIX 2 + x~xo = o. The fundamental polygon of this group can be deformed so as to obtain new groups uniformizing new curves of genus 3.

A similar train of ideas lay at the basis of the papers by Klein [1] (No. 101-103) and Poincare [1], (p.92, 108, 169), but Poincare used for the construction of automorphic functions the series that now bear his name. They both conjectured correctly that every algebraic curve admits a uniformization by the corresponding group and made substantial progress towards a proof of this result. However, a complete proof was not achieved at the time, but only in 1907 by Poincare (and independently by Koebe). An important role was played by the fact that by this time Poincare had investigated the concept offundamental group and universal covering.

The topology of algebraic curves is very simple and was completely studied by Riemann. In the investigation of the topology of algebraic surfaces Picard developed a method that is based on a study of the fibres of a morphism f: X ~ lPl. The point is to find out how the topology of the fibre f-l(a) changes when the point a E lPl changes, in particular, when this fibre acquires a singular point. He proved, for example, that smooth surfaces in lP3 are simply-connected (see Picard and Simart [1], Vol. I). By this method Lefschetz [2], [3] obtained many deep results on the topology of algebraic surfaces, and also on varieties of arbitrary dimension.

428 Historical Sketch

The study of global properties of analytic manifolds began fairly recently (Hopf [1J, A. Weil [2J). This domain developed vigorously in the 1950's, in connection with the creation and application by Cartan and Serre ofthe theory of analytic coherent sheaves (see Cartan [1J and Serre [1J). We do not give a definition of this concept-it is an exact analogue of the concept of an algebraic coherent sheaf (but we must emphasize that the analytic concept was introduced before the algebraic one). One of the basic results of this theory was the proof that the cohomology groups (and, in particular, the groups of sections) of an analytic coherent sheaf over a compact manifold are finite-dimensional. In this context Cartan gave a definition of an analytic manifold that is based on the concept of a sheaf and expressed the idea that the definition of various types of manifolds is linked with the specification of sheaves of rings on them.

9. Algebraic Varieties over an Arbitrary Field. Schemes. Formally the study of varieties over an arbitrary field began only in the twentieth century, but the foundations for this were laid earlier. An important role was played here by two papers printed in one and the same issue ofCrelle's Journal in 1882. Kronecker [1J investigates problems that would now­adays be referred to the theory of rings of finite type without divisors of zero and of characteristic O. In particular, for integrally closed rings he constructs a theory of divisors.

The paper by Dedekind and Weber [1J is devoted to the theory of algebraic curves. Its aim is to give a purely algebraic account of a considerable part of this theory. The authors emphasize that they do not use the concept of continuity anywhere, and their results remain true if the field of complex numbers is replaced by the field of all algebraic numbers.

The principal significance of the paper by Dedekind and Weber lies in the fact that in it the basic object of study is the field of rational functions on an algebraic curve. Concrete (affine) models are employed only as a technical tool, and the authors use the term "invariance" to denote concepts and results that do not depend on the choice of model. In this paper the whole account becomes to a significant degree parallel to the theory of fields of algebraic numbers. In particular, the analogy between prime ideals of a field of algebraic numbers and points of the Riemann surface of a field of algebraic functions is emphasized (we could say that in both cases we are concerned with the maximal spectrum of a one-dimensional scheme).

Interest in algebraic geometry over "non-classical" fields arose first in connection with the theory of congruences, which can be interpreted as equations over a finite field. In his lecture at the International Congress

Historical Sketch 429

of Mathematicians in 1908 Poincare says that the methods of the theory of algebraic curves can be applied to the study of congruences in two unknowns.

The ground for a systematic construction of algebraic geometry was prepared by the general development of the theory of fields and rings in the 1910's and 20's.

In 1924 Artin published a paper (see Artin [1J, No.1) in which he studied quadratic extensions of the field of rational functions of one variable over a finite field of constants, based on their analogy to quadratic extensions of the field of rational numbers. Of particular importance for the subsequent development of algebraic geometry was his introduction of the concept of the '-function of this field and the formulation of an analogue to the Riemann hypothesis for the' -function. We introduce (which Artin did not do) a hyperelliptic curve defined over a finite field k for which the field in question is of the form k(X). Then the Riemann hypothesis gives a best possible estimate for the number N of points x E X that are defined over a given finite extension K/k, that is, for which k(x) c K (just like the Riemann hypothesis for the field of rational numbers gives a best possible estimate for the asymptotic distribution of prime numbers). More accurately, the Riemann hypothesis is equivalent to the inequality IN - (q + 1)1 < 2gvq, where q is the number of elements of the field K and g the genus of the curve X.

Attempts to prove the Riemann hypothesis (which, as becomes clear at once, can be formulated for any algebraic curve over a finite field) led in the 1930's to work by Hasse and his pupils on the theory of algebraic curves over an arbitrary field. Here the hypothesis itself was proved by Hasse [1J for elliptic curves.

Strictly speaking, this theory concerns not curves but the correspond­ing fields offunctions, and the authors nowhere use geometric terminology. With this style one can become acquainted in the book by Hasse [2J (see the sections devoted to function fields). The possibility of this bi­rationally invariant theory of algebraic curves is connected with the uniqueness of a smooth projective model of an algebraic curve. Therefore great difficulties arise in applying this approach to the many-dimensional case.

On the other hand, in a sequence of papers published under the general title "Zur algebraischen Geometrie" in the Mathematische Annalen between the end of the 1920's and the beginning of the 1930's, van der Waerden made progress in the construction of algebraic geometry over an arbitrary field. In particular, he set up a theory of intersections (as we would say nowadays, he defined the ring of classes of cycles) over a smooth projective variety.

430 Historical Sketch

In 1940 A. Weil succeeded in proving the Riemann hypothesis for an arbitrary algebraic curve over a finite field. He found two ways of proving it. One of them is based on the theory of correspondences of the curve X (that is, divisors on the surface X x X), and the other on an analysis of its Jacobian variety. Thus, in both cases many-dimensional varieties are brought into play. The book by Weil [1] contains the construction of algebraic geometry over an arbitrary field: the theory of divisors, cycles, intersections. Here "abstract" (not necessarily quasiprojective) varieties are defined for the first time by the process of pasting together affine pieces (similar to Ch. V, § 3.2).

A definition of a variety based on the concept of a sheaf is contained in the paper by Serre [2], where the theory of coherent algebraic sheaves is constructed, for which the recently created theory of coherent analytic sheaves served as a prototype (see § 8).

Generalizations of the concept of an algebraic variety, close in spirit to the later concept of a scheme, were proposed at the beginning of the 1950's. Apparently the first and for the time very systematic working out of these ideas is due to Kahler [1], [2]. The concept of a scheme, as well as the majority of results in the general theory of schemes, is due to Grothendieck. The first systematic account of these ideas is contained in a lecture by Grothendieck [1].

Bibliography for the Historical Sketch

Abel,N.H.: [1] Oeuvres completes. Christiania 1881. Artin,E.: [1] Collected papers. New York-London: Addison-Wesley, 1965. Bernoulli,J.: [1] Opera omnia, vol. I-IV. Lausannae et Genevae: Bosquet 1742. Bertini,E.: [1] Ricerche sulle transformazioni univoche involutorie nel piano. Ann. Mat.

Pura Appl. (2) 8 (1877). Brill,A., Noether,M.: [1] Uber die algebraischen Funktionen und ihre Anwendung in

der Geometrie. Math. Ann. 7 (1873). Cartan,H.: [1] Varietes analytiques complexes et cohomologie, Coli. sur les fonctions de

plusieurs variables. Bruxelles, March 1953. Castelnuovo,G.: [1] Sulla razionalita delle involuzioni piane. Rend. Accad. Lincei 2

(1893). [2] Sulle superficie di genere zero. Mem. Soc. Ital. Sci. 10 (1896). [3] Alcuni proprieta fondamentali dei sistemi lineari di curve tracciati sopra una superficie, ibid. [4] Sugli integrali semplici appartenenti and una superficie irregolars. Rend. Accad. Lincei 14 (1905).

Clebsch,A.: [1] Sur les surfaces algebriques. C. R. Acad. Sci. Paris 67, 1238-1239 (1868). Clebsch,A., Gordan, P.: [1] Theorie der Abelschen Funktionen. Leipzig: Teubner, 1866. Dedekind,R., Weber, H.: [1] Theorie der algebraischen Funktionen einer Veranderlichen.

J. Reine Angew. Math. 92 (1882). Enriques, F.: [1] Sulla proprieta caratheristica delle superficie irregolary. Rend. Accad.

Bologna 9 (1904). [2] Superficie algebriche. Bologna 1949.

Euler,L.: [1] Integral calculus, Vol. I, Ch. VI. [2] Opera omnia, Ser. I, Vol. XXI, 91-118.

Frobenius, G.: [1] Uber die Grundlagen der Theorie der Jakobischen Funktionen. J. Reine Angew. Math. 97, 16-48, 188-223 (1884).

Gopel: [1] Theoriae transcendentium Abelianarum primi ordinis adumbrato levis. J. Reine Angew. Math. 3S (1847).

Grothendieck,A.: [1] The cohomology theory of abstract algebraic varieties. Internat. Congr. Math. Edinburgh 1958.

Hasse,H.: [1] Zur Theorie der abstrakten elliptischen Funktionenkorper. J. Reine Angew. Math. 17S (1936). [2] Zahlentheorie. Berlin: Akademie-Verlag, 1950.

Hopf,H.: [1] Zur Topologie der komplexen Mannigfaltigkeiten. Studies and essays presented to R. Courant. New York 1958, 167-187.

Jacobi,C.G.J.: [1] Gesammelte Werke. Berlin 1881. Kahler, E.: [1] Algebra und Differentialrechnung. Berichte Math. Tagung. Berlin 1953.

[2] Geometria arithmetica. Ann. Mat. Pura Appl. (4), 4S (1958).

432 Bibliography for the Historical Sketch

Klein, F.: [1] Gesammelte Mathematische Abhandlungen, Vol. III. Berlin: Springer-Verlag, 1923. [2] Riemannsche Fliichen. Berlin 1891-1892.

Kronecker,L.: [1] Grundziige einer arithmetischen Theorie der algebraischen GraBen. J. Reine Angew. Math. 92 (1882).

Lefschetz,S.: [1] Numerical invariants of algebraic varieties. Trans. Amer. Math. Soc. 22 (1921). [2] L'Analysis situs et la geometrie algebrique. Paris 1924. [3] Geometrie sur les surfaces et les varietes algebriques. Mem. Sci. Math. 40 (1929).

Legendre, A. M.: [1] Traite des fonctions elliptiques et des integrales Euleriennes, 3 vols. Paris 1825-1828.

Noether,M.: [1] Zur Grundlegung der Theorie der algebraischen Raumkurven. J. Reine Angew. Math. 93 (1882). [2] Zur Theorie des eindeutigen Entsprechens algebraischer Gebilde von belie big vielen Dimensionen. Math. Ann. 2 (1870), 8 (1875).

Picard,E.: [1] Sur les integrales des differentielles totales algebriques de premiere espece. C. R. Acad. Sci. Paris 99 (1884).

Picard, E., Simart, G.: [1] Theorie des fonctions algebriques de deux variables independentes. Paris 1897-1906.

Poincare, H.: [1] Oeuvres, Vol. II. Paris 1916. [2] Sur les proprietes du potentiel et sur les fonctions Abeliennes. Acta Math. 22, 89-178 (1899).

Riemann,B.: [1] Gesammelte Werke. Serre,J.P.: [1] Quelques problemes globaux relatifs aux varietes de Stein. ColI. sur les

fonctions de plusieurs variables. Bruxelles, March 1953. [2] Faisceaux algebriques coherents. Ann. of Math. (2) 61 (1955).

Severi,F.: [1] La base minima pour la totalite des courbes algebriques tracees sur une surface algebrique. Ann. Ecole Norm. Sup. 25 (1908).

Weil,A.: [1] Foundations of algebraic geometry. New York 1946. [2] Sur la theorie des formes differentielles attachees a une variete analytique complexe. Comm. Math. Helv. 24 (1947).

Weyl,H.: [1] Die Idee der Riemannschen Flache. Berlin 1923.

Subject Index

Addition of points on plane cubic curve 148

Affine variety 36 Algebraic curve, branch 123 - -,genus 171 - -, irreducible 5 - -, moduli 178 - -, plane 3 - -, rational 5 - -, real 336 - family of divisors 154 - group 150 Analytic space 357 Automorphic form 393 - function 399

Bezout's theorem in projective space 199 - - in product of projective spaces 199 - - on curve 145 Bundle, corresponding to divisor 278 -,cotangent 274 -, linear 277 -, normal 275

Canonical class 170 Centre of u-process 104 Characteristic class of a linear bundle 279 Chow coordinates of a variety 66 Chow's lemma 282 Class, canonical 170 -, characteristic of a linear bundle 279 - of divisors 131 - - - of degree zero on an algebraic

curve 145 Closed embedding of schemes 248 - subscheme 248 - subvariety 264 Closure of an ideal 96 Codimension of subvariety 53 Combinatorial surface 334

Complete intersection 401 - variety 265 Completion of module 96 Complex dimension 345 - torus 347 Convergence in ring of formal power series

94 Coordinate ring of a closed subset of an

affine space 16 Cotangent bundle 274 Criterion of Chevalley-Kleiman 292 - of Grauert 293 - ofNakai-Moishezon 293 - for being ruled 180 - for rationality 180 Curve, hyperelliptic 175 Cycle on algebraic variety 206

Decomposition, incontractible, of closed set into irreducibles 23

Degree of divisor on curve 140 - of mapping 116 Dense set 15 Differential form, invariant 168 - of a function 156 - - - -- at a point 75 Dimension of divisor 137 -, complex 345 - of complex space 357 - of q uasiprojective variety 53 - of ring 232 - of topological space 232 Direct sum of sheaves 272 - - of vector bundles 273 Divisor 127 -, effective 127, 360 -, locally principal 133 - ofa form 133 - of a function 129 - of differential form 170

434

Divisor of hyperplane section 133 - of meromorphic function 363 - of poles of a function 129 - of zeros of a function 129 - on analytic manifold 360 - class 131

Elliptic integral 391 Embedding of schemes, closed 248 Equivalence of divisors 131 -, algebraic 154 - -, of cycles 207 -, numerical 288 Exceptional subvariety 105

Factor bundle 275 - sheaf 296 Factorial variety 112 Family, algebraic, of divisors 154 -, -, of varieties 68 - of vector spaces 268 Field of rational functions on quasipro­

jective variety 38 - - - - as irreducible closed subset

of affine space 24 - - - - of plane irreducible algebraic

curve 8 - - - - of variety 263 Finiteness theorem 304 Flat ring 96 Form, automorphic 393 -, differential, invariant 168 -, -, domain of regularity 164 -, -, rational 164 -, -, regular one-dimensional 156 -, -, -, r-dimensional 161 Fraction, meromorphic 360 Frobenius relations 356 Function, analytic on complex variety 345 -, automorphic 399 -, elliptic 390 -, meromorphic 362 -, rational, domain of definition 25 -, -, regular at point 24 -, regular at point 33 -, -, on closed subset of affine space 16 Fundamental polygon 400

General position of divisors 182 Generic point 230 Genus of algebraic curve 171 Group, algebraic 150 -, modular 392

Group of divisor classes 131 -, Picard 133

Harnack's theorem 337

Subject Index

Hilbert's basis theorem XIII - Nullstellensatz XIII Holomorphically convex space 407 - - -, complete 407 Homomorphism of sheaves of modules

271 Hopf manifold 348 Hyperelliptic curve 175 Hypersurface in an affine space 16 - as closed subset 18 -, projective 40

Ideal of closed subset in the ring of regular functions on large closed subset of an affine space 18

- - - - of an affine space 16 - - - - of a projective space 30 Image of closed set under rational mapping

26 - of homomorphism of sheaves 295 Incontractible decomposition of closed

set into irreducibles 23 Integral, elliptic 391 Intersection index of divisors 195 - - - - in general position 183 - - of a curve and a divisor 287 - - of cycles on a differentiable mani-

fold 317 - - of effective divisors at a point 182 In variant differential form 168 Inverse image, proper, under cr-process

209 - -, -, of subvarieties 287 - - of scheme 250 Isomorphism of birational quasiprojective

varieties 39 -, birational of closed subsets of an affine

space 26 -, - of plane irreducible algebraic curves

11 - of closed subsets of an affine space

20 - of quasiprojective varieties 36

Jacobian variety 155

Kernel of homomorphism of sheaves 295 -, local, of cr-process 101

Subject Index

Kernel of 6-function 387 - with centre in smooth subvariety 284 - - - at point of quasiprojective

variety 103 - - - - of projective space 99

Lattice 347 Linear bundle 277 - divisors, system of 138 Local equations of subvariety 90 - parameters at a point 81 - ring of an irreducible subvariety 188 - - of point 72 - - - - of a spectrum 227 - - of prime ideal 71 Liiroth's problem 174 - theorem 9

Manifold, analytic 344 -,-,complex 344 -, -, of elliptic type 344 -, -, of hyperbolic type 382 -, -, of parabolic type 344 -, Hopf 348 -, -, generalized 378 -, orienting class of 313 Mapping, finite of affine varieties 48 -, -, of quasiprojective varieties 49 -, holomorphic 345 - of spectra associated with homomor-

phism of rings 224 -, rational, of irreducible subset of affine

space 25 - -, of plane irreducible algebraic curves

11 -, -, regular at a point 26 -, regular, of closed subjects of affine

space 18 -, -, of quasiprojective varieties 34 -, restriction, of sheaf 235 -, unramified regular at a point 117 -, Veronese 40 Meromorphic fraction 360 Modeloffield 106 -, relatively minimal 107 Module, completion 96 -, length 189 -, of differentials of ring 160 Morphism of families of vector spaces 268 - of ringed spaces 242 - of varieties 263 -, rational, of schemes 246

435

Multiplicity of singular point of a plane curve 80, 185

- of intersection of divisors in a sub­variety 190

- - - of effective divisors at a point 182

Nakayama's lemma 83 Neighbourhood of point 15 - - -, infinitely small 251 Nil-radical 227 Normal bundle 275 Normalization of quasiprojective variety

113 - of varieties in finite extension of func­

tion field 266 - theorem 52 Noetherian scheme 252

Order of ramification of mapping of curves 327

Ordinary singular point 123 Orientation of differentiable manifold 312 - of canonical manifold X (CC) 312 - of triangulation 335 Ovals of real curve 340

Parameters, local, at a point 81 Pasting of schemes 246 Picard group 133 Poincare series 393 Poincare's theorem on complete irre-

ducibility 351 Point, generic 230 Polygon, fundamental 400 Presheaf 234 - of regular functions 235 -, structure 235 Principal open set 37 - - - of spectrum 229 Product of closed subsets of affine space

16 - of cycles 206 - of schemes 256 Projection with centre in subspace 40 Projective space over a ring 247 - variety 36 Projectization of vector bundle 285

Quadratic transformation 215 Quasiprojective variety 33

Ramification point 117 Rational differential form 164

436

Rational variety 174 Reducibility of topological space 230 Reducible Space 230 Regular vector field 305 Restriction mapping of sheaf 235 - of divisor 134 Riemann-Roch theorem 176 Riemann's existence theorem 351 Ring of formal power series 84 Ringed space 242 Ruled surface 108

Scheme 244 -, affine 245 -, defined over a field 310 -, diagonal 258 -, group 258 -, Noetherian 252 - of finite type 252 - over ring 244 -, reduced 250 -, separated 258 Section of vector bundle 270 Set, dense 15 Sheaf 238 -, associated with a presheaf 242 -,coherent 299 -, corresponding to vector bundle 272 -, - to divisor 278 -, dual 272 -, exterior power 272 -, invertible 279 -, locally free 272 - of analytic functions 345 - of differential forms 273 - of fibres 241 - of modules over a sheaf of rings 272 -, structure, of ringed space 243 Simple sequence 186 Singular point with distinct tangents 123 Space associated with divisor 137 -, projective over a ring 247 -, reducible 230 Spectrum of ring 224 - - -, maximal 223 -, regular point 227 -, simple point 227 -, specialization of point 230 Structure presheaf 235 - sheaf of ringed space 243 Subbundle 274 Subscheme, closed 248

Subject Index

Subset, affine open of closed subset of projective space 32

-, - - of projective space 14 -, closed, of affme space 14 -, -, irreducible 22 -, -, of projective space 31 -, -, of quasiprojective variety 33 - -, reducible 22 -, open, of affine space 15 -, -, of closed subset of projective space

32 -, -, of quasiprojective variety 33 Subsheaf 295 Subspace of complex space 358 Subvariety, closed 264 -, codimension of 53 -, expectional 105 -, local equations 90 - of quasiprojective variety 33 Support of divisor 127 - - - on analytic manifold 361 - of locally principal divisor 133 - of sheaf 297 Surface, combinatorial 334 -, elliptic 179 -, Euler, characteristics 336 - of type K3 179 - - - -, analytic 378 -, ruled 108

Tangency of a line and a variety 73 Tangent bundle of quasiprojective variety

77 - - of variety 274 - cone 79 - space at point of spectrum 227 - - of variety 73 Taylor series of a function 85 Tensor product of vector bundles 273 - - of sheaves 272 Theorem of Siegel 365 - on closure of projective image of variety

44 - - dimension of intersection with

hypersurface 57 - - - of fibres of mapping 60 Topology complex, of quasiprojective

varieties 89 - real, of quasiprojective varieties 89 -, spectral 228 Transition matrix of vector bundle 270

Subject Index

Transversality of subvarieties 82 Triangulation 333

Uniformization of curve 390 U nirational variety 174 Unramified covering of topological space

346

Variety 263 -, Abelian 152 -, affine 36 -, associated form 66 -, complete 265 -, factorial 112 -, Jacobian 155 -, non-singular of codimension 1 90

Variety, normal 109 -, Picard 155 -, projective 36 -, quasiprojective 33 -, rational 174 -, simple point 79 -, singular point 78 -, smooth 77 -, unrational 174 -, Veronese 40 Vector bundle 268 - -,dual 272 - -, exterior power 272 Veronese mapping 40 - variety 40

Weierstrass' preparation theorem 95

437

List of Notation

IAn XxY

n-dimensional affne space 14 product of affine varieties X and Y 16 product of quasiprojective varieties X and Y 42 product of schemes X and Y 255

k[XJ coordinate ring of closed subject X of affine space 16 ring of regular functions on quasiprojective variety X 34

Illx ideal of closed subset X of affine space 16 ideal of closed subset X of projective space 31

cry ideal of closed subset Y in ring of functions on closed set X) Y 18 V(u) hypersurface with equation u(x) = 0 in closed subset of affine space 18

subset of spectrum determined by the element u 229 f* mapping of function corresponding to regular mapping f 19

mapping of function corresponding to rational mapping f 27 mapping of divisors corresponding to regular mapping f 134

Ll diagonal of varieties 21 diagonal of schemes 258

k(X)

!pn

D(f)

Vm

dimX codimxY

A. (!Jx

e x

dx(f) k[[TJJ X'

v

field of rational functions on closed subset X of affine space 24 field of rational functions on quasi projective variety X 38 field of rational functions on variety X 263 n-dimensional projective space 30 principal open set in quasiprojective variety 37 principal open set in scheme 229 Veronese mapping 40 dimension of variety X 53 codimension of subvariety Y in X 53 local ring of prime ideal p 71 local ring of point x of variety 72 local ring of point x of spectrum 227 tangent space at point x of variety 73 tangent space at point x of spectrum 227 differential of function f at point x 75 ring of formal power series 85 normalization of quasiprojective variety X 113 normalization of variety X 226 normalization mapping of quasiprojective variety 113 normalization mapping of variety 266

List of Notation

deg(f) (f)

Div(X) CI(X) (F) Pic (X) £,(D) /(D) degD df Q"[X]

Q"(X)

hr

degree of mapping f 116 divisor of function f 129 equivalence of divisors 131 group of divisors 130 group of divisor classes 130 divisor of form F 133 Picard group 133 space associated with divisor D 137 dimension of divisor D 137 degree of divisor D on curve 140 differential of function f 156 module of r-dimensional regular differential forms on X 161 space of r-dimensional rational differential forms on X 164 = dimQ" [X] 168

(w) divisor of differential form w 170 Kx canonical class of X 170 g(X) genus of curve X 171 (D1 , ..• ,Dn)x (D1 , ..• , Dn)e (Dl,···,Dn) Spec A "cp As Af

eb,eb.F (!J

Fx (!Jx

£'(E)

£IE detE e Nx/y

£'D ED X(CC) WM

(!J.n

fan .R(X)

O(z)

intersection index of divisors D1 , ••. , Dn at point x 183 intersection index of divisors D1 , ..• , Dn in variety C 190 intersection index of divisors D1 , ..• , Dn 183, 195 spectrum of ring A 224 mapping of spectra associated with homomorphism cp 224 localization of ring A with respect to mUltiplicative system S 225 localization of ring A with respect to multiplicative system {fn} 226 mappings defining a sheaf 234 structure presheaf (shea!) on spectrum of ring 235 fibre of sheaf F at point x 241 structure sheaf of ringed space 243 space of sections of vector bundle E 271 sheaf corresponding to bundle E 272 determinant of bundle E 273 tangent bundle 274 normal bundle 275 sheaf corresponding to divisor D 277 bundle corresponding to divisor D 277 space of closed points in complex topology 309 orienting class of manifold M 313 sheaf of analytic functions 344 holomorphic mapping corresponding to morphism f 345 field of meromorphic functions on analytic manifold X 363 theta function 387

439