Italian School of Algebraic Geometry - Collapse of The School

Collapse of The School

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 hopelessly wrong. For example, in 1934 Severi claimed that the space of rational equivalence classes of cycles on an algebraic surface is finite dimensional, but Mumford (1968) showed that this is false for surfaces of positive geometric genus, and in 1946 Severi published a paper claiming to prove that a degree-6 surface in 3-dimensional projective space has at most 52 nodes, but the Barth sextic has 65 nodes. Severi did not accept that his arguments were inadequate, leading to some acrimonious disputes as to the status of some results.

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 simply collapsed due to its inadequate foundations. From about 1950 to 1980 there was considerable effort to salvage as much as possible from the wreckage, and convert it into the rigorous algebraic style of algebraic geometry set up by Weil and Zariski. In particular in the 1960s Kodaira and Shafarevich and his students rewrote the Enriques classification of algebraic surfaces in a more rigorous style, and also extended it to all compact complex surfaces, while in the 1970s Fulton and MacPherson put the classical calculations of intersection theory on rigorous foundations.