What is the Genus?

Let us make a huge temporal leap, in order to reach the “Géométrie” [56] of Descartes, published in 1637, illustrating his “Discourse on the Method of Rightly Conducting One’s Reason and of Seeking Truth in the Sciences”.

During his youth, Newton had carefully studied the geometric calculus of Descartes, which served him as a source of inspiration for the development of the “calculus of fluxions”, his version of the differential calculus. This partially explains why he undertook to classify the curves of degree threeclassificationof cubicsclassificationof conics according to various species, in analogy with the classification of those of degree two, the conic sections, conic section into ellipses, parabolas, hyperbolas or pairs of lines. The following is the first paragraph of the chapter containing this classification from his work [140], published in 1711:

In the two previous chapters we dealt with curves and the polynomials defining them or the mechanisms generating them. Those curves often represented incarnations of problems involving polynomial equations, with one or more variables.

In the following extract of [17], Jakob BernoulliBernoulli, the brother of Johann, about whom we wrote in Chap. 4, analyzed various methods of construction of “mechanical” or “transcendental” curves, that is, curves which are not “algebraic” (defined by a polynomial equation).algebraiccurvecurvealgebraiccurvetranscendentalcurvemechanical Those methods created a common framework for Descartes’ algebraic curves and for the curves furnished by the differential and integral calculus: rectifying

Fagnano published his study of the lemniscate in his 1718 papers [56] and [57]. He was so proud of it that, in the 1750 edition of his collected mathematical works, he chose a lemniscate for the decoration of the front page, surmounted by the motto “Deo Veritatis Gloria” (see Fig. 6.1).

In 1751, Fagnano applied to become a member of the Berlin academy of sciences. It was on that occasion that Euler studied his works, which gave him new ideas. He arrived in 1753 at the following generalization of formula (6.1):

Starting in the late 1700s, Legendre spent several decades developing a general theory of the integrals which he called elliptic functions, and which satisfy an addition formula analogous to that established by Euler in Theorem 7.1. Here is what he wrote about his motivations in the foreword of his 1825 book [128]

At the time when Legendre published his treatise [128], Abel, a young Norwegian mathematician, was beginning to publish a huge generalization of the addition theorems for elliptic functions. This generalization dealt with all the integrals of the form: 9.1 y being an arbitrary algebraic function of the variable x. Those integrals were later called “abelian transcendents”integralabelian by JacobiJacobi in [107]. In the sequel, we use the simpler name of abelian integrals.

Confronted with his contemporaries’ difficulties in understanding his very general ideas, Abel decided to present some of them in separate articles. For instance in 1829, in the paper [3], he formulated the following version of the theorem cited in extenso in the previous chapter. This version makes no reference to the “number of algebraic relations” (and therefore to the genus)

Abel’s works, which we mentioned in the last two chapters, are the first in which the modern notion of genus appeared. Nevertheless, it was in a hidden form, a little like a secondary character remaining in the shadows, devoid of name. It is worth trying to understand better the general problems which preoccupied Abel at that time, of which [128] is only an expression. Happily for us, Abel wrote about these problems in an 1828 letter [2] to Legendre

In Chap. 9, we saw the genus timidly appearing in Abel’s investigations concerning so-called abelian integrals. Let us concentrate now on the definition of such an integral. integralabelian

As was explained by Cauchy Cauchy in the excerpt of the paper [37] discussed in the previous chapter, if one takes a path which comes back to its starting point

Shortly after Cauchy and Puiseux, Riemann came up with a radically different solution to the problem of multivaluedness of functions.CauchyPuiseux

As explained in the previous chapter, although Riemann developed a very topological vision of surfaces, he did not forget that his fundamental examples came from algebraic functions and their integrals.

Any multivalued algebraic function y(x) becomes univalued on the Riemann surface T associated with it. Consequently, if Q(x, y) is a rational function with two variables, then the algebraic function Q(x, y(x)) also becomes univalued on T. The functions of this type form the field of rational functions on the Riemann surface, which is the fundamental algebraic object associated with T.

We saw in Chap. 11 that Abel had a very ambitious program aiming to study all possible relations between abelian integrals. Even if he did not always completely prove them, he discovered many theorems concerning those relations. For instance, in [115] KleimanKleiman lists and sketches modern proofs of the theorems he encountered in Abel’s paper [1]. One of the most famous such theorems, which is explained in nearly every textbook on algebraic curves and Riemann surfaces, is the following one:

In his paper [109], which appeared in 1866, Jordan described one of the first attempts to prove a theorem on the topological classification of surfaces, that is, up to homeomorphisms.

We saw that RiemannRiemanngenusnotation for it denoted the genus by “p”, a notation which is still frequently used today, in particular for the generalizations of this notion in higher dimensions. On the other hand, Riemann did not give a name to this notion, and his definition was not the one we saw in the Introduction, in terms of holes. There is a good reason for this, namely, that the surfaces imagined by Riemann consisted of sheets which thinly cover the plane, and therefore do not admit visible holes.

As we explained before, Riemann did not study algebraic curves embedded in the projective plane by themselves, but only up to birational transforms. Nevertheless, his techniques, passing through the associated Riemann surface or through the abelian integrals of the first kind attached to them, allow one to prove properties of such curves. For instance, one has

One year after Clebsch’s paper [45], Cayley published an article which cites it, but without retaining the term “genus” suggested by Clebsch. Cayley introduced a rival term, the “deficiency”deficiency. This term was used for more or less half a century, mainly by British mathematicians, before being abandoned in favor of “genus”. Here is the way in which Cayley explained the reason behind his choice (see [38, pp. 1–2])

As explained in the previous chapter, Cayley used the curves passing through the singular points of C in order to study a given plane projective curve C. Max Noether called such curves the adjointsadjointcurvecurveadjoint of C. One has the following theorem, going back to RiemannRiemann [161], then refined by ClebschClebsch, GordanGordan and Noether:

Riemannsurfacesurfaceabstract Riemann In 1882, Klein published his book [116] in which he tried to explain the physical intuitions lying behind Riemann’s theory. In the introduction, he argued that Riemann’s surfaces do not necessarily arise from functions, but that one may reverse this process

Riemann surfaces were introduced to build new domains of definition on which multivalued algebraic functions become univalued. One of the advantages of algebraic functions which was lost in this procedure was the ability to use a single parameter in order to describe the function in its full domain of definition (which becomes the whole Riemann surface). In fact, this advantage is still available if one lifts the function to the universal covering of the Riemann surface, as shown by the following uniformization theoremtheoremuniformization

In the introduction of his 1928 thesis [183], WeilWeil explained the usefulness of the notion of genus in arithmetic:genusin arithmetic

In order to have an overview of the path we have traveled so far, let us discover the viewpoint described by Weil [190] in 1981

Summarizing what we have seen so far, the genus is a measure of the complexity of abstract objects: algebraic curves (defined over an arbitrary field, etc.) and topological surfaces (orientable, etc.) This is in contrast with the degree, which partially measures the embedding into an ambient space.

The works of Riemann had a great impact on his contemporaries, so much that during the rest of the nineteenth century, with few exceptions, algebraic geometers directed their efforts toward the development of complex algebraic geometry, that is, the study of the geometry of the sets of complex solutions to systems of polynomial equations in several variables.

We saw in the last chapter’s quotations that the first definitions of notions of genus for algebraic surfaces were formulated for those contained in the three-dimensional complex projective space

The Italian school of algebraic geometry of surfaces, whose most famous representatives were CastelnuovoCastelnuovo, EnriquesEnriques and SeveriSeveri, privileged geometric methods when studying a surface, via the algebraic families of curves it contained. The simplest such families are those which depend linearly on the parameters: the linear systems of curves, which generalized the linear series of groups of points on algebraic curves, considered by BrillBrill and NoetherNoetherMax (see the beginning of Chap. 28).

It was the classification of algebraic curves which served as a model for that of surfaces, developed by Enriques with the help of CastelnuovoCastelnuovo between 1890 and 1914. It was explained in detail in [33] and [71]. Let us begin with the way EnriquesEnriques presented the principle of the classification of curves in [70]

The various notions of genus which we have discussed up to this point do not tell us how to effectively compute the genera of an algebraic surface if we start from a defining equation f(x, y, z) = 0 in ℂ3 (which determines its birational equivalence class).

In his 1933 paper [65], Du ValDu Val said that an isolated singular point O of an algebraic surface S in ℙ3does not affect the conditions of adjunctionconditions of adjunction if it does not impose conditions on the surfaces adjointadjointsurfacesurfaceadjoint to S. That is, if those surfaces are not obliged to pass through the point O.

One may notice that the definitions of the various notions of genera of algebraic surfaces described in the previous chapters are all algebraic, that none is purely topological. We saw that the Italian geometers preferred to give such definitions starting from models of a birationally equivalence class situated in three-dimensional projective space, in which the curves of singularities were in general unavoidable. The question of the topological meaning of the genus deals instead with a smooth model, situated in a higher-dimensional space.

The question of the topological invariance of the plurigenera resisted much longer. In fact, they are invariants of the differentiable structure, but not of the topological structure.

We saw in Chap. 28 that, in order to define a birationally invariant notion of genus for algebraic surfaces, ClebschClebsch, CayleyCayley and NoetherNoetherMax considered certain surfaces passing through their singular locus. Those surfaces are analogs of the adjoint curves of a plane algebraic curve, passing in a controlled way through their singular points (see Chap. 22).

In his 1909 paper [167], Severi formulated a definition of arithmetic genusgenusarithmetic valid in arbitrary dimension.

The notions of genus which we have studied until now for algebraic surfaces, as well as those formulated in all dimensions by Severi, are defined by algebraic means. None of them generalizes Riemann’s topological viewpoint, based on curves drawn on differentiable surfaces (see Chap. 14).

It is not clear from the initial definitions of Poincaré that the Betti numbers of a compact manifold are finite or that its fundamental group is finitely generated. In order to address in particular these two issues, PoincaréPoincaré proposed in 1899, in the sequel [149] to his article [148], an alternative definition of the Betti numbers, which used a polyhedral decomposition of the given manifold

The curves drawn on Riemann surfaces are supports of integration for the forms of degree 1. Similarly, the oriented submanifolds of dimension k of a given manifold V serve as supports for the computation of integrals of differential forms of degree k, as explained by Poincaré in [148, Chap. 7].

In Cartan’s CartanElie works discussed in the previous chapter, the operations on differential forms were used only for local studies, related to the resolution of PfaffPfaff systems. Only later, in the 1920s, did Cartan get interested in problems concerning the global behaviour of differential forms on manifolds. More specifically, he was interested in the topological structure of the underlying manifolds of Lie groups (which are, by definition, at the same time groups and differentiable manifolds).groupLie

Recall that RiemannRiemann used closed paths on his surfaces in order to integrate smooth forms of degree 1 along them. In fact, Riemann did not work with arbitrary such forms, but rather with those which may be written as f(z)dz in terms of a local holomorphic coordinate z, the function f(z) being also holomorphic.

We saw in Chap. 25 that since his student years, Weil was very interested in the analogies between arithmetic and classical algebraic geometry. That is, between the study of the sets of solutions of systems of Diophantine equations and that of algebraic varieties defined over the field of complex numbers. His most famous paper in this direction is probably [184], published in 1949. He formulated there several conjectures relating geometry over complex numbers and over finite fields, which we now reproduce

Since the beginnings of the development of the theory of algebraic surfaces, geometers have tried to prove a generalization of the Riemann–Roch theorem for algebraic curves (see Zariski’s account in [195, Chap. IV], Zariski as well as GrayGray’s article [84]). Max NoetherNoetherMax formulated such a generalization in his 1886 paper [144]. But he could only prove an inequality (I use the notations and the explanations from [32, Chaps. 7 and 35])

The ingredients appearing in Serre’s letter to Borel and which were very recent at that time are: complex fibrations and their Chern classes, sheafs of holomorphic sections and their cohomology groups. I will briefly explain the meaning of these notions and relate them to the objects discussed up to now.

The Chern classes appearing in Serre’s letter to Borel of Chap. 44 are particular characteristic classes, that is, certain cohomology classes measuring the way in which the fibres of a fibre bundle “fit together over the whole manifold”, as written by Whitney at the end of the next excerpt from the introduction to his 1937 paper [193]

In his 1758 paper [72], EulerEuler published the following theorem

In his 1876 paper [89], Harnack examined the following question: given a real algebraic curve of degree d in the projective plane, how many connected components can it have? He proved

Let us come back to the Riemann–Roch problem in higher dimensions.

Hirzebruch’s proof of Theorem 49.1 used in a fundamental way the usual topology of a complex variety, as well as the modern panoply of homology and cohomology groups, vector bundles and their characteristic classes. Therefore, it was not possible to derive from it a proof of an analogous theorem concerning algebraic varieties defined over other fields.