## Solution of Algebraic Curves: An Introduction to Algebraic Geometry

Plane conics. Cubic plane curves elliptic curves and the group law. Affine algebraic sets and the Nullstellensatz. Rings of functions on varieties.

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Projective varieties and birational equivalence. Salmon and A. Cayley in 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. The principal representatives of this school were G.

Castelnuovo, F. Enriques and F. 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 ; 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.

## Algebraic Geometry

In other words, if a surface is unirational and a morphism is of degree two, then is rational. 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. 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.

## Introduction to Algebraic Curves, Autumn

In particular, Klein considered functions that are automorphic with respect to the group consisting of all transformations , where and are integers, , and. He showed that these functions uniformize the curve 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. 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 topology of algebraic curves is very simple, and was exhaustively investigated by Riemann. Picard studied the topology of algebraic surfaces by a method based on the study of the fibres of a morphism. He studied the variation of the topology of the fibre with the variation of the point and, in particular, conditions under which this fibre contains a singular point. For instance, it was proved in this way that smooth surfaces in  are simply connected. In S. Lefschetz started to use a new science — topology — in the study of algebraic varieties over the field of complex numbers.

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 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. Cartan, J.

Barry Mazur - New Rational Points of Algebraic Curves over Extension Fields

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.

• Tombs of the South Asasif Necropolis: Thebes, Karakhamun (TT 223), and Karabasken (TT 391) in the Twenty-Fifth Dynasty.

Hirzebruch . In the middle of the s 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.

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 s 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.

In particular, he developed the theory of intersections in a smooth projective variety.

### Lecture notes

Weil, in , 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 i. 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. Zariski, P. Samuel, C. Chevalley and J. Serre introduced powerful methods of commutative and, in particular, local algebra into algebraic geometry in the early s.

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 s algebraic geometry underwent a further radical transformation, due to A. Grothendieck's work on the foundation of the concept of a scheme. If this causes problems, contact me. Exercises will be assigned on a bi-weekly basis.

These are to be handed in to Johan Commelin j. Handing in by email is possible only if you write your solutions using La TeX; in that case, send the pdf output. You are allowed to collaborate with other students but what you write and hand in should be your own work.