# User:Maximilian Janisch/latexlist/Algebraic Groups/Algebraic geometry

The branch of mathematics dealing with geometric objects connected with commutative rings: algebraic varieties (cf. Algebraic variety) and their various generalizations (schemes, algebraic spaces, etc., cf. Scheme; Algebraic space).

Algebraic geometry may be "naively" defined as the study of solutions of algebraic equations. The geometrical intuition appears when every "set of solutions" is identified with a "set of points in a coordinate space" . If the coordinates are real numbers, and if the space is two- or three-dimensional, the situation may be clearly visualized; however, the language of geometry is also used in more general situations. This language implies problems, constructions and considerations which do not follow from the point of view of pure algebra. In turn, algebra provides a flexible and powerful apparatus which is just as suitable for the conversion of a tentative reasoning into a proof as for the formulation of such proofs in their most obvious and most general form.

In the case of algebraic geometry over the field of complex numbers, every algebraic variety is simultaneously a complex-analytic, differentiable and topological space in the ordinary Hausdorff topology. This fact makes it possible to introduce a large number of classical structures which induce such invariants of algebraic varieties that can be obtained by purely algebraic means only with great difficulty or perhaps not all. The concepts and the results of algebraic geometry are extensively used in number theory (Diophantine equations and the evaluation of trigonometric sums), in differential topology (both with respect to singularities and differentiable structures), in group theory (algebraic groups and simple finite groups connected with Lie groups), in the theory of differential equations ($K$-theory and the index of elliptic operators), in the theory of complex spaces, in the theory of categories (topoi, Abelian categories), and in functional analysis (representation theory). Conversely, the ideas and methods of these disciplines are utilized in algebraic geometry.

The genesis of algebraic geometry dates back to the 17th century, with the introduction of the concept of coordinates into geometry. Nevertheless, its crystallization into an independent branch of science only began in the mid-19th century. The utilization of coordinates in projective geometry created a situation in which algebraic methods could compete with synthetic methods. However, the achievements of this branch of algebraic geometry still are in the domain proper of projective geometry, but as a result it left to algebraic geometry the traditional study of projective algebraic varieties. Modern algebraic geometry arose as the theory of algebraic curves (cf. Algebraic curve). Historically, the first stage of development of the theory of algebraic curves consisted in the clarification of the fundamental concepts and ideas of this theory, using elliptic curves as examples. The system of concepts and results which one now calls the theory of elliptic curves arose as a part of analysis (rather than geometry) — the theory of integrals of rational functions on an elliptic curve. At first, these were the integrals that received the name "elliptic" , and it was only later that the meaning was extended to include functions and curves (cf. Elliptic integral).

At the end of the 17th century, Jakob and Johann Bernoulli noted a new interesting property of elliptic integrals. They studied integrals expressing the arc lengths of certain curves. They found methods for transforming one curve into another of the same arc length, even though the respective arcs could not be brought into correspondence. From the analytic point of view, this was tantamount to the conversion of an integral into another integral, and in certain cases the result was a transformation of an integral into itself. C.G. Fagnano, in the first half of the 18th century, gave numerous examples of such transformations.

L. Euler studied an arbitrary polynomial $f ( x )$ of the fourth degree and posed the problem of the relations between $\pi$ and $y$ that satisfy the equation

\begin{equation} \frac { d x } { \sqrt { f ( x ) } } = \frac { d y } { \sqrt { f ( y ) } } \end{equation}

He considered (1) as a differential equation, connecting $\pi$ and $y$. The desired relation is the ordinary integral of this equation. The reason for the existence of the integral of (1) and all its special cases, discovered by Fagnano and the Bernoulli's, is the presence of a group law on the elliptic curve $s ^ { 2 } = f ( t )$, and the invariance of the everywhere-regular differential form $s ^ { - 1 } d t$ with respect to shifts by elements of the group. Euler's relations, which interconnect $\pi$ with $y$ by (1), may be written as

\begin{equation} ( x , \sqrt { f ( x ) } ) \oplus ( c , \sqrt { f ( c ) } ) = ( y , \sqrt { f ( y ) } ) \end{equation}

where $\theta$ denotes the addition of points on the elliptic curve.

Thus it is seen that these results contain both the group law on an elliptic curve and the existence of an invariant differential form on this curve (cf. Elliptic curve).

After the work of Euler, the theory of elliptic integrals was mainly developed by A. Legendre. His considerations, beginning in 1786, are collected in a three-volume treatise Traité des fonctions elliptiques et intégrales Euleriennes ( "Traité des fonctions elliptiques et intégrales Euleriennes of LegendreA treatise on elliptic functions and Euler integrals" ).

The studies carried out by N.H. Abel on the theory of elliptic functions appeared in 1827–1829. His starting point was the elliptic integral

\begin{equation} \theta = \int _ { 0 } ^ { \lambda } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } } \end{equation}

where $i$ and $E$ are complex numbers. Abel considered this integral as a function of the upper limit and introduced the inverse function $\lambda ( \theta )$ and the function

\begin{equation} \Delta ( \theta ) = \sqrt { ( 1 - c ^ { 2 } \lambda ^ { 2 } ) ( 1 - e ^ { 2 } \lambda ^ { 2 } ) } \end{equation}

Both these functions have two periods, $2 \omega$ and $2 \pi$, in the complex domain:

\begin{equation} \omega = 2 \int _ { 0 } ^ { 1 / c } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } } \end{equation}

\begin{equation} \overline { w } = 2 \int _ { 0 } ^ { 1 / \varepsilon } \frac { d x } { \sqrt { ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } ) } } \end{equation}

so that the mapping $x = \lambda ( \theta ) , y = \Delta ( \theta )$ defines a uniformization of the elliptic curve $y ^ { 2 } = ( 1 - c ^ { 2 } x ^ { 2 } ) ( 1 - e ^ { 2 } x ^ { 2 } )$ by elliptic functions.

Somewhat later than Abel, but independently, C.G.J. Jacobi studied the function inverse to the elliptic integral; he showed that it has two independent periods, and obtained several results in the problem of transformations. By transforming Abel's expressions of elliptic functions in series into the form of products, Jacobi arrived at the concept of $6$-functions (cf. Theta-function) and discovered their numerous applications not only in the theory of elliptic functions, but also in number theory and in mechanics.

While studying the transformations of elliptic functions, Abel was the first to investigate the group of homomorphisms of a one-dimensional Abelian variety.

Finally, as a result of the publication of the inheritance of C.F. Gauss and, in particular, of his diary, it became clear that to some extent he possessed some of these ideas long before the work of Abel and Jacobi. The transition to the study of arbitrary algebraic curves still took place within an analytic framework: Abel had shown the way in which the basic properties of elliptic integrals can be generalized to include integrals of arbitrary algebraic functions. Such integrals were subsequently named Abelian integrals (cf. Abelian integral).

The work published by Abel in 1826 became the starting point of the general theory of algebraic curves. It contains the concepts of the genus of an algebraic curve and of the equivalence of divisors, and presents equivalence criteria in terms of integrals. It leads to the theory of the Jacobi variety of an algebraic curve.

In his dissertation, published in 1851, B. Riemann adopted a novel principle of studying functions of a complex variable. He assumed that such a function is given not on the plane of the complex variable, but on a "many-sheeted" surface above this plane.

The surfaces introduced by Riemann (cf. Riemann surface) are close to the modern concept of a one-dimensional analytic manifold: A manifold on which analytic functions are defined. Riemann posed and solved the question of the connection between his concept and the concept of an algebraic curve; the corresponding result is nowadays called Riemann's existence theorem. After having studied the possible position of branching points of the surface he proved that the set of classes depends on $3 p - 3$ independent parameters if $p > 1$, which he called the moduli (cf. Moduli problem).

The work of Riemann was the starting point for studies on the topology of algebraic curves; this study explains the topological meaning of the dimension $D$ of the space $\Omega ^ { \tau } [ X ]$ as one-half of the dimension of the one-dimensional homology group of the space $X ( C )$. Analytic methods led to the inequality $l ( D ) \geq \operatorname { deg } ( D ) - p + 1$. The Riemann–Roch equality was proved by E. Roch, a student of Riemann (cf. Riemann–Roch theorem). Finally, this study constitutes the first appearance of the field $k ( X )$ as the primary object connected with a curve $x$, and of the concept of a birational isomorphism. The problem of inversion of integrals of arbitrary algebraic functions had been posed much earlier by Abel. Another part of Riemann's study on Abelian functions deals with the relations between $6$-functions and the inversion problem in general, in particular with the series (in $D$ variables)

\begin{equation} \theta ( v ) = \sum _ { m } e ^ { F ( m ) + 2 ( m , v ) } \end{equation}

where $m = ( m _ { 1 } , \dots , m _ { p } )$ runs through all integral $D$-dimensional vectors

\begin{equation} v = ( v _ { 1 } , \ldots , v _ { p } ) , \quad ( m , v ) = \sum m _ { i } v _ { i } \end{equation}

\begin{equation} F ( m ) = \sum \alpha _ { j k } m _ { j } m _ { k } , \quad \alpha _ { j k } = \alpha _ { k j } \end{equation}

This series converges for all values of $v$ if the real part of the quadratic form $H ^ { \prime }$ is negative definite. The principal properties of the function $6$ are the relations

\begin{equation} \theta ( v + \pi i r ) = \theta ( r ) , \quad \theta ( v + \alpha _ { j } ) = e ^ { L _ { j } ( v ) } \theta ( v ) \end{equation}

where $N$ is an integer vector, $\alpha$ is a column of the matrix $( \alpha _ { j k } )$ and $L _ { j } ( v )$ is a linear function.

It was shown by Riemann that it is possible to select cuts $1 , \ldots , a _ { p } , b _ { 1 } , \ldots , b _ { p }$ which convert the surface he has introduced into a simply-connected surface, and to select $u _ { 1 } , \dots , u _ { p }$ of the everywhere-finite integrals on this surface such that the integrals $u _ { j }$ with respect to $\alpha _ { k }$ are 0 if $i \neq k$, and are $\pi i$ if $i = k$, while the integrals $u _ { j }$ with respect to $b _ { i }$ form a symmetric matrix $( \alpha _ { j k } )$ which satisfies conditions that ensure the convergence of the series (2). He considered the function $6$ corresponding to these coefficients $\alpha j k$. The periods of an arbitrary $2 n$-periodic function of $12$ variables satisfy relations analogous to those necessary for the convergence of the series which determine the $6$-functions. These relations between the periods were explicitly written out by G. Frobenius, who showed that they are necessary and sufficient for the existence of non-trivial functions which satisfy the functional equation (3) (cf. Theta-function; Abelian function). These relations are necessary and sufficient for the existence of a meromorphic function with given periods which cannot be reduced to a function in a smaller number of variables by linear transformations. This theorem was formulated by K. Weierstrass and proved by H. Poincaré. It was shown in 1921 by S. Lefschetz that, if the Frobenius relations are satisfied, the $6$-function defines an imbedding of the variety $C ^ { x } / \Omega$ into a projective space ($S$ is the lattice corresponding to the given matrix of periods; cf. Complex torus).

The concepts and the results which now form the base of the theory of algebraic curves were created under the influence and within the framework of the theory of algebraic functions and their integrals. The purely geometrical theory of algebraic curves developed as an independent branch. Thus, J. Plücker in 1834 arrived at formulas interconnecting the class of the curve, its order and the number of its double points. He also showed that a plane curve of order three has nine points of inflection. However, such studies were only of secondary importance at that time.

It was only after the studies of Riemann that the geometry of algebraic curves became an important field in mathematics, together with the theory of Abelian integrals and Abelian functions. This change was mainly due to the work of A. Clebsch. While Riemann based his work on functions, Clebsch based his work on algebraic curves. Clebsch and P. Gordan  deduced a formula for the number $D$ of linearly independent integrals of the first kind (i.e. of the same kind as the genus of the curve $x$), which expresses this number in terms of the order of the curve and the number of singular points. They also showed that if $p = 0$, the curve has a rational parametrization and, if $p = 1$, it becomes a plane curve of order three.

An error made by Riemann proved useful in the development of the algebraic-geometric aspect of the theory of algebraic curves. In proving his existence theorems, Riemann considered it self-evident that a variational problem — the "Dirichlet principle" — is solvable. It was soon shown by Weierstrass that this is not true in all cases. Accordingly, Riemann's results remained unfounded for a certain period of time. One way in which this difficulty could be overcome was to prove these theorems by an algebraic method; their formulation was essentially algebraic. These studies, which were undertaken by Clebsch, greatly facilitated the understanding of the algebraic-geometric nature of the results of Abel and Riemann, which had hitherto been masked by the analytic approach.

The studies initiated by Clebsch were continued and expanded by a student of his school — M. Noether. Noether's ideas were presented very clearly in a study published by himself and A. Brill. They posed the problem of the development of geometry on an algebraic curve lying in a projective plane as the set of results which are invariant with respect to one-to-one (i.e. birational) transformations (cf. Birational geometry).

By the 1850s many special properties of algebraic varieties of dimension higher than one (mostly surfaces) had been discovered. For instance, a detailed study was made of surfaces of the third degree; in particular, it was shown by G. Salmon and A. Cayley in 1849 that any cubic surface without singular points contains 27 different straight lines. However, these results were not connected with any general principles for a long time, and remained unrelated to the fundamental ideas in the theory of algebraic curves which were developed at the same time.

The development of algebraic geometry was strongly affected by the Italian school, in particular by L. Cremona, C. Segre and E. Bertini. The principal representatives of this school were G. Castelnuovo, F. Enriques and F. Severi. One of the principal achievements of the Italian school was the classification of algebraic surfaces (cf. Algebraic surface). The first result in this direction was reported by Bertini in 1877; he gave the classification of involutive plane transformations or, in modern terms, a classification up to conjugation in the group of birational automorphisms of the plane of all elements of order two in this group. This classification is very simple and, in particular, it can be readily deduced that the quotient of the plane by a group of order two is a rational surface. In other words, if a surface $x$ is unirational and a morphism $f : P ^ { 2 } \rightarrow X$ is of degree two, then $x$ is rational. Castelnuovo (1893) solved (positively) the general case of the Lüroth problem for algebraic surfaces. He also posed and solved the problem of the characterization of rational surfaces by numerical invariants. A classification of surfaces was achieved by Enriques in a series of studies, which ended only in the first decade of the 20th century.

The principal tool of the Italian school was the study of families on a surface; the families were taken linear or algebraic (the latter also being known as "continuous" ). This led to the concept of linear and algebraic equivalence. The connection between these concepts was first studied by Castelnuovo. An important contribution to the subsequent development of the problem was made by Severi. A large portion of the concepts was defined in an analytic form, and the algebraic significance was clarified in the course of time. Nevertheless, there are still many concepts and results which are essentially analytic, at least from the modern point of view.

The studies of F. Klein and H. Poincaré on the problem of uniformization of algebraic curves by automorphic functions appeared in the early 1880s. Their objective was the uniformization of all curves by functions now known as automorphic, analogous to the uniformization of curves of order one by elliptic functions. Klein's starting point was the theory of modular functions. The field of modular functions is isomorphic to the field of rational functions, but it is possible to consider functions that are invariant with respect to various subgroups of the modular group and to obtain more complicated fields in this manner. In particular, Klein considered functions that are automorphic with respect to the group consisting of all transformations $z \rightarrow ( \alpha z + b ) f ( c z + d )$, where $a , b , c$ and $a$ are integers, $a d - b c = 1$, and

\begin{equation} \left( \begin{array} { l l } { \alpha } & { b } \\ { c } & { d } \end{array} \right) \equiv \left( \begin{array} { l l } { 1 } & { 0 } \\ { 0 } & { 1 } \end{array} \right) ( \operatorname { mod } 7 ) \end{equation}

He showed that these functions uniformize the curve $x _ { 0 } ^ { 3 } x _ { 1 } + x _ { 1 } ^ { 3 } x _ { 2 } + x _ { 2 } ^ { 3 } x _ { 0 } = 0$ of genus three. It is possible in this manner to distort the fundamental polygon of this group and to obtain new groups which uniformize curves of genus three. A similar reasoning underlies the studies of Klein and Poincaré, the latter constructing automorphic functions with the aid of series which are now named after him. They both correctly conjectured that any algebraic curve can be uniformized by a corresponding group, and made considerable advances in their attempts to prove this result. The complete proof was only obtained in 1907 by Poincaré and, independently, by P. Koebe. A major contribution to this proof was made by the studies of Poincaré on the concepts of the fundamental group and a universal covering.

The topology of algebraic curves is very simple, and was exhaustively investigated by Riemann. E. Picard studied the topology of algebraic surfaces by a method based on the study of the fibres of a morphism $f : X \rightarrow P ^ { 1 }$. He studied the variation of the topology of the fibre $f ^ { - 1 } ( a )$ with the variation of the point $\alpha \in P ^ { 1 }$ and, in particular, conditions under which this fibre contains a singular point. For instance, it was proved in this way that smooth surfaces in $p 3$  are simply connected. Poincaré also made an important contribution to the topology of algebraic surfaces.

In 1921 S. Lefschetz started to use a new science — topology — in the study of algebraic varieties over the field of complex numbers. His first aim was to obtain a simplification of the results of Poincaré and Picard and to obtain generalizations of these results to higher-dimensional cases. However, he did considerably more and his works opened a new domain of research in algebraic geometry. A further application of Lefschetz to algebraic geometry is connected with the theory of algebraic cycles on algebraic varieties. He proved that a two-dimensional cycle on an algebraic variety is homologous to a cycle representable by an algebraic curve if and only if the regular double integral $\int \int R ( x , y , z ) d x d y$ has a zero period over this cycle. Lefschetz' studies laid the foundations of the modern theory of complex manifolds. Such manifolds were subsequently studied by powerful tools, including the theory of harmonic integrals (W. Hodge, G. de Rham) and the theory of sheaves (H. Cartan, J. Leray). It could be shown by using these techniques that non-singular algebraic varieties form part of an important class of complex manifolds — Kähler manifolds (cf. Kähler manifold).

The theory of sheaves, and the related theory of vector bundles on complex manifolds, supplied a new interpretation and considerable generalization of many classical invariants of algebraic surfaces (of both the arithmetic and geometric genus, and of the canonical system). One of the most important achievements of this theory was the creation of the theory of Chern classes (cf. Chern class) and a considerable generalization of the classical Riemann–Roch theorem by F. Hirzebruch .

In the middle of the 1920s much work was done on broadening the scope of algebraic geometry from the aspect of set theory and from the axiomatic aspect. The scope of application of algebraic geometry was greatly extended to include complex manifolds and algebraic varieties over arbitrary fields. Interest in algebraic geometry over "non-classical" fields originated in the theory of congruences, interpreted as equations over a finite field. It was claimed by Poincaré in his address at the International Congress of Mathematicians in 1908 that methods of the theory of algebraic curves may be used for studying equations with two unknowns. The ground for a systematic construction of algebraic geometry had been prepared in the first decade of the 20th century by the general development of the theory of fields and the theory of rings.

In the 1930s H. Hasse and his school attempted to prove the Riemann hypothesis (cf. Riemann hypotheses), which may be formulated for any algebraic curve over a finite field; this involved the development of a theory of algebraic curves over an arbitrary field. The hypothesis itself was proved by Hasse for elliptic curves. Advances in the construction of algebraic geometry over arbitrary fields are also due to the studies of B.L. van der Waerden in the period between 1931 and 1939. In particular, he developed the theory of intersections in a smooth projective variety.

A. Weil, in 1940, succeeded in formulating a proof of the Riemann hypothesis for an arbitrary algebraic curve over a finite field. He found two ways of proving the hypothesis: one based on the theory of correspondences of the curve $x$ (i.e. of divisors on the surface $X \times X$), while the other was based on the study of its Jacobi variety. Thus, higher-dimensional varieties are used in both cases. Accordingly, Weil's book  contains the construction of algebraic geometry over an arbitrary field: the theory of divisors, cycles and intersections. For the first time "abstract" (not necessarily quasi-projective) varieties were defined by pasting together affine pieces. O. Zariski, P. Samuel, C. Chevalley and J.-P. Serre introduced powerful methods of commutative and, in particular, local algebra into algebraic geometry in the early 1950s.

Serre gave a definition of varieties based on the concept of a sheaf. He also established the theory of coherent algebraic sheaves, modelled on the theory of coherent analytic sheaves, which had been introduced only a short time before (cf. Coherent algebraic sheaf; Coherent analytic sheaf).

In the late 1950s algebraic geometry underwent a further radical transformation, due to A. Grothendieck's work on the foundation of the concept of a scheme. By using the language of category theory, Grothendieck succeeded in generalizing and clarifying many important classical constructions in algebraic geometry, the abstract definitions of which had been until now only little geometric. He also founded many new branches of algebraic geometry (cf. Abstract algebraic geometry). The language of the theory of schemes has now become a part of the routine of modern algebraic geometry, has united it naturally with commutative algebra, and has made a significant contribution to the advances made in the study of arithmetical problems of algebraic varieties (cf. Algebraic varieties, arithmetic of). The theory of schemes directly affected one of the most important recent achievements of algebraic geometry — the solution (by H. Hironaka) of the problem on the existence of a non-singular birational model of an algebraic variety defined over a field of characteristic zero (cf. Resolution of singularities).