Italian school of algebraic geometry

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In relation to the history of mathematics, the Italian school of algebraic geometry refers to mathematicians and their work in birational geometry, particularly on algebraic surfaces, centered around Rome roughly from 1885 to 1935. There were 30 to 40 leading mathematicians who made major contributions, about half of those being Italian. 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

The

Foundational issues

Some proofs produced by the school are not considered satisfactory because of foundational difficulties. These included frequent use of birational models in dimension three of surfaces that can have non-singular models only when embedded in higher-dimensional projective space. In order to avoid these issues, 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 modern techniques were found, in embryonic form, and in some cases the articulation of these ideas exceeded the available technical language.

The geometers

According to Guerraggio & Nastasi (page 9, 2005),

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).

Advent of topology

In 1950 Henry Forder mentioned the Italian school in connection with algebraic curves.^[2]

Further development of the theory of plane curves is only fruitful when it is connected with the theory of

Abelian functions. This has been a favorite study during the last fifty years, of the Italian geometers, and they have also made contributions of great beauty to a similar theory of surfaces and of “Varieties” of higher dimensions. Herein a combination of the theory of integrals

on the varieties, and of their topology, yields decisive results. The theory of curves and surfaces is thus connected with modern algebra and topology...

, with the traditional material.

Collapse of the school

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In the earlier years of the Italian school under Castelnuovo, the standards of rigor were as high as most areas of mathematics. Under Enriques it gradually became acceptable to use somewhat more informal arguments instead of complete rigorous proofs, such as the "principle of continuity" saying that what is true up to the limit is true at the limit, a claim that had neither a rigorous proof nor even a precise statement. At first this did not matter too much, as Enriques's intuition was so good that essentially all the results he claimed were in fact correct, and using this more informal style of argument allowed him to produce spectacular results about algebraic surfaces. Unfortunately, from about 1930 onwards under Severi's leadership the standards of accuracy declined further, to the point where some of the claimed results were not just inadequately proved, but were incorrect. For example, in 1934 Severi claimed that the space of rational equivalence classes of cycles on an algebraic surface is finite-dimensional, but

By about 1950 it had become too difficult to tell which of the results claimed were correct, and the informal intuitive school of algebraic geometry collapsed due to its inadequate foundations.^{[citation needed]} From about 1950 to 1980 there was considerable effort to salvage as much as possible, and convert it into the rigorous algebraic style of algebraic geometry set up by