# Italian school of algebraic geometry

﻿
Italian school of algebraic geometry

In relation with the history of mathematics, the Italian school of algebraic geometry refers to the work over half a century or more (flourishing roughly 1885-1935) done internationally in birational geometry, particularly on algebraic surfaces. There were in the region of 30 to 40 leading mathematicians who made major contributions; about half of those being in fact Italian. There is no question that the leadership fell to the group in Rome of Guido Castelnuovo, Federigo Enriques and Francesco Severi; who were involved in some of the deepest discoveries, as well as setting the style.

Algebraic surfaces

The emphasis on algebraic surfaces &mdash; algebraic varieties of dimension two &mdash; followed on from an essentially complete geometric theory of algebraic curves (dimension 1). The position in around 1870 was that the curve theory had incorporated with Brill-Noether theory the Riemann-Roch theorem in all its refinements (via the detailed geometry of the theta-divisor).

The classification of algebraic surfaces was a bold and successful attempt to repeat the division of curves by their genus "g". It corresponds to the rough classification into the three types: "g"= 0 (projective line); "g" = 1 (elliptic curve); and "g" > 1 (Riemann surfaces with independent holomorphic differentials). In the case of surfaces, the Enriques classification was into five similar big classes, with three of those being analogues of the curve cases, and two more (elliptic fibrations, and K3 surfaces, as they would now be called) being with the case of two-dimension abelian varieties in the 'middle' territory. This was an essentially sound, breakthrough set of insights, recovered in modern complex manifold language by Kunihiko Kodaira in the 1950s, and refined to include mod p phenomena by Zariski, the Shafarevich school and others by around 1960. The form of the Riemann-Roch theorem on a surface was also worked out.

Foundational issues

Qualification of what was actually proved is necessary because of the foundational difficulties. These included intensive use of birational models in dimension 3 of surfaces that can have non-singular models only when embedded in higher-dimensional projective space. That is, the theory wasn't posed in an intrinsic way. To get round that, a sophisticated theory of handling a linear system of divisors was developed (in effect, a line bundle theory for hyperplane sections of putative embeddings in projective space). Many of the modern techniques were found, in embryo form, and in some cases the articulation of those exceeded the available technical language.

The geometers

The roll of honour of the school includes the following major Italians: Giacomo Albanese, Bertini, Campedelli, Guido Castelnuovo, Oscar Chisini, Federigo Enriques, Michele De Franchis, Pasquale del Pezzo, Beniamino Segre, Corrado Segre, Francesco Severi, Guido Zappa (with contributions also from Luigi Cremona, Gino Fano, Rosati, Torelli, Giuseppe Veronese).

Elsewhere it involved H. F. Baker and Patrick du Val (UK), A. B. Coble and Oscar Zariski (USA), Charles Émile Picard (France), Lucien Godeaux (Belgium), G. Humbert, Hermann Schubert and Max Noether, and later Erich Kähler (Germany), H. G. Zeuthen (Denmark).

These figures were all involved in algebraic geometry, rather than the pursuit of projective geometry as synthetic geometry, which during the period under discussion was a huge (in volume terms) but secondary subject (when judged by its importance as research).

The new algebraic geometry that would succeed the Italian school was distinguished also by the intensive use of algebraic topology. The founder of that tendency was Henri Poincaré; during the 1930s it was developed by Lefschetz, Hodge and Todd. The modern synthesis brought together their work, that of the Cartan school, and of W.L. Chow and Kunihiko Kodaira, with the traditional material.

From the 1950s

The fashion and foundational attitude changed in algebraic geometry from 1950 onwards, leading to an axiomatisation and some acrimony as to the status of some results. For a while it may have seemed that the tradition of the Italian school would possibly be lost, in the sense that the old papers had become hard to read for the new generation of geometers.

The essentials were in fact transmitted, in particular through Zariski's students. Some of the areas opened up, such as moduli spaces for curves, have been at the centre of recent work related to physics. Very many of the fundamental concepts in algebraic geometry still bear the names of those of the Italian school.

References

* [http://www.mat.uniroma1.it/~rendicon/2005(2)/185-193.pdf Beniamino Segre and Italian geometry (PDF)] , article by Edoardo Vesentini

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Algebraic geometry — This Togliatti surface is an algebraic surface of degree five. Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It… …   Wikipedia

• List of algebraic geometry topics — This is a list of algebraic geometry topics, by Wikipedia page. Contents 1 Classical topics in projective geometry 2 Algebraic curves 3 Algebraic surfaces 4 …   Wikipedia

• Algebraic surface — In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface is therefore of complex dimension two (as a complex manifold, when it is non singular)… …   Wikipedia

• Geometry — (Greek γεωμετρία ; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences. Initially a body of… …   Wikipedia

• Projective geometry — is a non metrical form of geometry, notable for its principle of duality. Projective geometry grew out of the principles of perspective art established during the Renaissance period, and was first systematically developed by Desargues in the 17th …   Wikipedia

• Birational geometry — In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. In the case of dimension two, the birational geometry of… …   Wikipedia

• Minimal model (birational geometry) — In algebraic geometry, more specifically in the field of birational geometry, the theory of minimal models is part of the birational classification of algebraic varieties. Its goal is to construct, given a variety satisying certain restrictions,… …   Wikipedia

• List of interactive geometry software — Interactive geometry software (IGS, or dynamic geometry environments, DGEs) are computer programs which allow one to create and then manipulate geometric constructions, primarily in plane geometry. In most IGS, one starts construction by putting… …   Wikipedia

• Guido Castelnuovo — (14 August 1865 ndash; 27 April1952) was an Italian Jewish mathematician. His father, Enrico Castelnuovo, was a novelist and campaigner for the unification of Italy. Castelnuovo is best known for his contributions to the field of algebraic… …   Wikipedia

• Linear system of divisors — A linear system of divisors algebraicizes the classic geometric notion of a family of curves, as in the Apollonian circles. In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of… …   Wikipedia