Italian school of algebraic geometry
Encyclopedia
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 surface
s. There were in the region of 30 to 40 leading mathematicians who made major contributions, about half of those being in fact 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.
s — algebraic varieties
of dimension
two — followed on from an essentially complete geometric theory of algebraic curve
s (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 surface
s 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 surface
s, 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.
. 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 embryonic form, and in some cases the articulation of these ideas exceeded the available technical language.
is "considered the founder of the Italian school of algebraic geometry". Later they explain that in Turin
the collaboration of D'Ovidio and Corrado Segre
"would bring, either by their own efforts or those of their students, Italian algebraic geometry to full maturity". A one-time student of Segre, H.F. Baker wrote (1926, page 269), [Corrado Segre] "may probably be said to be the father of that wonderful Italian school which has achieved so much in the birational theory of algebraical loci." On this topic, Brigaglia & Ciliberto (2004) say "Segre had headed and maintained the school of geometry that Luigi Cremona had established in 1860." Reference to the Mathematics Genealogy Project
shows that, in terms of Italian doctorates, the real productivity of the school began with Guido Castelnuovo
and Federigo Enriques
. In the USA Oscar Zariski
inspired many Ph.D.s.
The roll of honour of the school includes the following other Italians: Giacomo Albanese
, Bertini, Campedelli, Oscar Chisini
, Michele De Franchis, Pasquale del Pezzo
, Beniamino Segre
, Francesco Severi
, Guido Zappa
(with contributions also from Gino Fano
, Rosati, Torelli, Giuseppe Veronese
).
Elsewhere it involved H. F. Baker and Patrick du Val
(UK), Arthur Byron Coble
(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 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.
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 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Italian school of algebraic geometry
Algebraic geometry
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 occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
refers to the work over half a century or more (flourishing roughly 1885–1935) done internationally in birational geometry
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 algebraic surfaces was largely worked out by the Italian...
, particularly on algebraic surface
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 has complex dimension two and so of dimension four as a smooth manifold.The theory of algebraic surfaces is much more complicated than that...
s. There were in the region of 30 to 40 leading mathematicians who made major contributions, about half of those being in fact Italian. The leadership fell to the group in Rome
Rome
Rome is the capital of Italy and the country's largest and most populated city and comune, with over 2.7 million residents in . The city is located in the central-western portion of the Italian Peninsula, on the Tiber River within the Lazio region of Italy.Rome's history spans two and a half...
of Guido Castelnuovo
Guido Castelnuovo
Guido Castelnuovo was an Italian mathematician. His father, Enrico Castelnuovo, was a novelist and campaigner for the unification of Italy...
, Federigo Enriques
Federigo Enriques
Federigo Enriques was an Italian mathematician, now known principally as the first to give a classification of algebraic surfaces in birational geometry, and other contributions in algebraic geometry....
and Francesco Severi
Francesco Severi
Francesco Severi was an Italian mathematician.Severi was born in Arezzo, Italy. He is famous for his contributions to algebraic geometry and the theory of functions of several complex variables. He became the effective leader of the Italian school of algebraic geometry...
, who were involved in some of the deepest discoveries, as well as setting the style.
Algebraic surfaces
The emphasis on algebraic surfaceAlgebraic 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 has complex dimension two and so of dimension four as a smooth manifold.The theory of algebraic surfaces is much more complicated than that...
s — algebraic varieties
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
of dimension
Dimension of an algebraic variety
In mathematics, the dimension of an algebraic variety V in algebraic geometry is defined, informally speaking, as the number of independent rational functions that exist on V.For example, an algebraic curve has by definition dimension 1...
two — followed on from an essentially complete geometric theory of algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
s (dimension 1). The position in around 1870 was that the curve theory had incorporated with Brill–Noether theory the Riemann–Roch theorem
Riemann–Roch theorem
The Riemann–Roch theorem is an important tool in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles...
in all its refinements (via the detailed geometry of the theta-divisor
Theta-divisor
In mathematics, the theta divisor Θ is the divisor in the sense of algebraic geometry defined on an abelian variety A over the complex numbers by the zero locus of the associated Riemann theta-function...
).
The classification of algebraic surfaces was a bold and successful attempt to repeat the division of curves by their genus
Genus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
g. It corresponds to the rough classification into the three types: g= 0 (projective line); g = 1 (elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
); and g > 1 (Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
s 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 surface
K3 surface
In mathematics, a K3 surface is a complex or algebraic smooth minimal complete surface that is regular and has trivial canonical bundle.In the Enriques-Kodaira classification of surfaces they form one of the 5 classes of surfaces of Kodaira dimension 0....
s, as they would now be called) being with the case of two-dimension abelian varieties
Abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...
in the 'middle' territory. This was an essentially sound, breakthrough set of insights, recovered in modern complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
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 spaceProjective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
. 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
Linear system of divisors
In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family....
was developed (in effect, a line bundle
Line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these...
theory for hyperplane sections of putative embeddings in projective space). Many of the 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) Luigi CremonaLuigi Cremona
Luigi Cremona was an Italian mathematician. His life was devoted to the study of geometry and reforming advanced mathematical teaching in Italy. His reputation mainly rests on his Introduzione ad una teoria geometrica delle curve piane...
is "considered the founder of the Italian school of algebraic geometry". Later they explain that in Turin
Turin
Turin is a city and major business and cultural centre in northern Italy, capital of the Piedmont region, located mainly on the left bank of the Po River and surrounded by the Alpine arch. The population of the city proper is 909,193 while the population of the urban area is estimated by Eurostat...
the collaboration of D'Ovidio and Corrado Segre
Corrado Segre
Corrado Segre was an Italian mathematician who is remembered today as a major contributor to the early development of algebraic geometry....
"would bring, either by their own efforts or those of their students, Italian algebraic geometry to full maturity". A one-time student of Segre, H.F. Baker wrote (1926, page 269), [Corrado Segre] "may probably be said to be the father of that wonderful Italian school which has achieved so much in the birational theory of algebraical loci." On this topic, Brigaglia & Ciliberto (2004) say "Segre had headed and maintained the school of geometry that Luigi Cremona had established in 1860." Reference to the Mathematics Genealogy Project
Mathematics Genealogy Project
The Mathematics Genealogy Project is a web-based database for the academic genealogy of mathematicians. As of September, 2010, it contained information on approximately 145,000 mathematical scientists who contribute to "research-level mathematics"...
shows that, in terms of Italian doctorates, the real productivity of the school began with Guido Castelnuovo
Guido Castelnuovo
Guido Castelnuovo was an Italian mathematician. His father, Enrico Castelnuovo, was a novelist and campaigner for the unification of Italy...
and Federigo Enriques
Federigo Enriques
Federigo Enriques was an Italian mathematician, now known principally as the first to give a classification of algebraic surfaces in birational geometry, and other contributions in algebraic geometry....
. In the USA Oscar Zariski
Oscar Zariski
Oscar Zariski was a Russian mathematician and one of the most influential algebraic geometers of the 20th century.-Education:...
inspired many Ph.D.s.
The roll of honour of the school includes the following other Italians: Giacomo Albanese
Giacomo Albanese
Giacomo Albanese was an Italian mathematician known for his work in algebraic geometry. He took a permanent position in São Paulo, Brazil, in 1936.-External links:...
, Bertini, Campedelli, Oscar Chisini
Oscar Chisini
Oscar Chisini was an Italian mathematician. He introduced the Chisini mean in 1929.-Biography:Chisini was born in Bergamo....
, Michele De Franchis, Pasquale del Pezzo
Pasquale del Pezzo
Pasquale del Pezzo, Duke of Cajanello and Marquis of Campodisola , was a Neapolitan mathematician.He was born in Berlin on 2 May 1859. He died in Naples on 20 June 1936...
, Beniamino Segre
Beniamino Segre
Beniamino Segre was an Italian mathematician who is remembered today as a major contributor to algebraic geometry and one of the founders of combinatorial geometry....
, Francesco Severi
Francesco Severi
Francesco Severi was an Italian mathematician.Severi was born in Arezzo, Italy. He is famous for his contributions to algebraic geometry and the theory of functions of several complex variables. He became the effective leader of the Italian school of algebraic geometry...
, Guido Zappa
Guido Zappa
Guido Zappa is an Italian mathematician and a noted group theorist: his other main research interests are geometry and complex analysis in one and several variables, and also the history of mathematics. Zappa is particularly known for some examples of algebraic curves that strongly influenced the...
(with contributions also from Gino Fano
Gino Fano
Gino Fano was an Italian mathematician. He was born in Mantua, Italy and died in Verona, Italy.Fano worked on projective and algebraic geometry; the Fano plane, Fano fibration, Fano surface, and Fano varieties are named for him....
, Rosati, Torelli, Giuseppe Veronese
Giuseppe Veronese
Giuseppe Veronese was an Italian mathematician. He was born in Chioggia, near Venice.Although his work was severely criticised as unsound by Peano, he is now recognised as having priority on many ideas that have since become parts of transfinite numbers and model theory, and as one of the...
).
Elsewhere it involved H. F. Baker and Patrick du Val
Patrick du Val
Patrick du Val was a British mathematician, known for his work on algebraic geometry, differential geometry, and general relativity. The concept of Du Val singularity of an algebraic surface is named after him....
(UK), Arthur Byron Coble
Arthur Byron Coble
Arthur Byron Coble was an American mathematician. He did research on finite geometries and the group theory related to them, Cremona transformations associated with the Galois theory of equations, and the relations between hyperelliptic theta functions, irrational binary invariants, the Weddle...
(USA), Charles Émile Picard
Charles Émile Picard
Charles Émile Picard FRS was a French mathematician. He was elected the fifteenth member to occupy seat 1 of the Académie Française in 1924.- Biography :...
(France), Lucien Godeaux
Lucien Godeaux
Lucien Godeaux was a prolific Belgian mathematician. His total of more than 1000 papers and books, 669 of which are found in Mathematical Reviews, made him one of the most published mathematicians. He was the sole author of all but one of his papers, and his Erdős number is undefined.He is best...
(Belgium), G. Humbert, Hermann Schubert
Hermann Schubert
Hermann Cäsar Hannibal Schubert was a German mathematician.Schubert was one of the leading developers of enumerative geometry, which considers those parts of algebraic geometry that involve a finite number of solutions. In 1874, Schubert won a prize for solving a question posed by Zeuthen...
and Max Noether
Max Noether
Max Noether was a German mathematician who worked on algebraic geometry and the theory of algebraic functions. He has been called "one of the finest mathematicians of the nineteenth century".-Biography:...
, and later Erich Kähler
Erich Kähler
was a German mathematician with wide-ranging geometrical interests.Kähler was born in Leipzig, and studied there. He received his Ph.D. in 1928 from the University of Leipzig. He held professorial positions in Königsberg, Leipzig, Berlin and Hamburg...
(Germany), H. G. Zeuthen (Denmark).
These figures were all involved in algebraic geometry, rather than the pursuit of projective geometry
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
as synthetic geometry
Synthetic geometry
Synthetic or axiomatic geometry is the branch of geometry which makes use of axioms, theorems and logical arguments to draw conclusions, as opposed to analytic and algebraic geometries which use analysis and algebra to perform geometric computations and solve problems.-Logical synthesis:The process...
, 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
The new algebraic geometry that would succeed the Italian school was distinguished also by the intensive use of algebraic topologyAlgebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
. The founder of that tendency was Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
; during the 1930s it was developed by Lefschetz, Hodge
W. V. D. Hodge
William Vallance Douglas Hodge FRS was a Scottish mathematician, specifically a geometer.His discovery of far-reaching topological relations between algebraic geometry and differential geometry—an area now called Hodge theory and pertaining more generally to Kähler manifolds—has been a major...
and Todd
J. A. Todd
John Arthur Todd FRS was a British geometer. He was born in Liverpool, and went to Trinity College of the University of Cambridge in 1925. He did research under H.F. Baker, and in 1931 took a position at the University of Manchester. He became a lecturer at Cambridge in 1937...
. The modern synthesis brought together their work, that of the Cartan
Henri Cartan
Henri Paul Cartan was a French mathematician with substantial contributions in algebraic topology. He was the son of the French mathematician Élie Cartan.-Life:...
school, and of W.L. Chow and Kunihiko Kodaira, with the traditional material.
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 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
Intersection theory
In mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring. The theory for varieties is older, with roots in Bézout's theorem on curves and...
on rigorous foundations.
External links
- David Mumford email about the errors of the Italian algebraic geometry school under Severi
- Kevin Buzzard what mistakes did the Italian algebraic geometers actually make?
- A. Brigaglia, C. Ciliberto, & E. Sernesi Geometria algebraica italiana at University of PalermoUniversity of PalermoThe University of Palermo is a university located in Palermo, Italy, and founded in 1806. It is organized in 12 Faculties.-History:The University of Palermo was officially founded in 1806, although its earliest roots date back to 1498 when medicine and law were taught there...
.