Federigo Enriques
Encyclopedia
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
.
He was born in Livorno
, and brought up in Pisa
, in a Sephardi Jewish family of Portuguese
descent. He became a student of Guido Castelnuovo
, and became an important member of the Italian school of algebraic geometry
. He also worked on differential geometry. He collaborated with Castelnuovo, Corrado Segre
and Francesco Severi
. He had positions at the University of Bologna
, and then the University of Rome La Sapienza
. He lost his position in 1938, when the Fascist government enacted the "leggi razziali" (racial laws), which in particular banned Jews from holding professorships in Universities.
The Enriques classification, of complex algebraic surface
s up to birational equivalence, was into five main classes, and was background to further work until Kunihiko Kodaira reconsidered the matter in the 1950s. The largest class, in some sense, was that of surfaces of general type
: those for which the consideration of differential form
s provides linear system
s that are large enough to make all the geometry visible. The work of the Italian school had provided enough insight to recognise the other main birational classes. Rational surface
s and more generally ruled surface
s (these include quadric
s and cubic surface
s in projective 3-space) have the simplest geometry. Quartic surface
s in 3-spaces are now classified (when non-singular) as cases of K3 surface
s; the classical approach was to look at the Kummer surfaces, which are singular at 16 points. Abelian surface
s give rise to Kummer surfaces as quotients. There remains the class of elliptic surface
s, which are fiber bundle
s over a curve with elliptic curve
s as fiber, having a finite number of modifications (so there is a bundle that is locally trivial actually over a curve less some points). The question of classification is to show that any surface, lying in projective space
of any dimension, is in the birational sense (after blowing up
and blowing down
of some curves, that is) accounted for by the models already mentioned.
No more than other work in the Italian school would the proofs by Enriques now be counted as complete and rigorous. Not enough was known about some of the technical issues: the geometers worked by a mixture of inspired guesswork and close familiarity with examples. Oscar Zariski
started to work in the 1930s on a more refined theory of birational mappings, incorporating commutative algebra
methods. He also began work on the question of the classification for characteristic p, where new phenomena arise. The schools of Kunihiko Kodaira and Igor Shafarevich
had put Enriques' work on a sound footing by about 1960.
Italy
Italy , officially the Italian Republic languages]] under the European Charter for Regional or Minority Languages. In each of these, Italy's official name is as follows:;;;;;;;;), is a unitary parliamentary republic in South-Central Europe. To the north it borders France, Switzerland, Austria and...
mathematician, now known principally as the first to give a classification of algebraic surfaces 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...
, and other contributions in 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...
.
He was born in Livorno
Livorno
Livorno , traditionally Leghorn , is a port city on the Tyrrhenian Sea on the western edge of Tuscany, Italy. It is the capital of the Province of Livorno, having a population of approximately 160,000 residents in 2009.- History :...
, and brought up in Pisa
Pisa
Pisa is a city in Tuscany, Central Italy, on the right bank of the mouth of the River Arno on the Tyrrhenian Sea. It is the capital city of the Province of Pisa...
, in a Sephardi Jewish family of Portuguese
Portugal
Portugal , officially the Portuguese Republic is a country situated in southwestern Europe on the Iberian Peninsula. Portugal is the westernmost country of Europe, and is bordered by the Atlantic Ocean to the West and South and by Spain to the North and East. The Atlantic archipelagos of the...
descent. He became a student 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...
, and became an important member of the 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 done internationally in birational geometry, particularly on algebraic surfaces. There were in the region of 30 to 40 leading mathematicians who made major...
. He also worked on differential geometry. He collaborated with Castelnuovo, 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....
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...
. He had positions at the University of Bologna
University of Bologna
The Alma Mater Studiorum - University of Bologna is the oldest continually operating university in the world, the word 'universitas' being first used by this institution at its foundation. The true date of its founding is uncertain, but believed by most accounts to have been 1088...
, and then the University of Rome La Sapienza
University of Rome La Sapienza
The Sapienza University of Rome, officially Sapienza – Università di Roma, formerly known as Università degli studi di Roma "La Sapienza", is a coeducational, autonomous state university in Rome, Italy...
. He lost his position in 1938, when the Fascist government enacted the "leggi razziali" (racial laws), which in particular banned Jews from holding professorships in Universities.
The Enriques classification, of complex 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 up to birational equivalence, was into five main classes, and was background to further work until Kunihiko Kodaira reconsidered the matter in the 1950s. The largest class, in some sense, was that of surfaces of general type
Surface of general type
In algebraic geometry, a surface of general type is an algebraic surface with Kodaira dimension 2. Because of Chow's theorem any compact complex manifold of dimension 2 and with Kodaira dimension 2 will actually be an algebraic surface, and in some sense most surfaces are in this...
: those for which the consideration of differential form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
s provides linear system
Linear system
A linear system is a mathematical model of a system based on the use of a linear operator.Linear systems typically exhibit features and properties that are much simpler than the general, nonlinear case....
s that are large enough to make all the geometry visible. The work of the Italian school had provided enough insight to recognise the other main birational classes. Rational surface
Rational surface
In algebraic geometry, a branch of mathematics, a rational surface is a surface birationally equivalent to the projective plane, or in other words a rational variety of dimension two...
s and more generally ruled surface
Ruled surface
In geometry, a surface S is ruled if through every point of S there is a straight line that lies on S. The most familiar examples are the plane and the curved surface of a cylinder or cone...
s (these include quadric
Quadric
In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial...
s and cubic surface
Cubic surface
A cubic surface is a projective variety studied in algebraic geometry. It is an algebraic surface in three-dimensional projective space defined by a single polynomial which is homogeneous of degree 3...
s in projective 3-space) have the simplest geometry. Quartic surface
Quartic surface
In mathematics, especially in algebraic geometry, a quartic surface is a surface defined by an equation of degree 4.More specifically there are two closely related types of quartic surface: affine and projective...
s in 3-spaces are now classified (when non-singular) as cases of 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; the classical approach was to look at the Kummer surfaces, which are singular at 16 points. Abelian surface
Abelian surface
In mathematics, an abelian surface is 2-dimensional abelian variety.One dimensional complex tori are just elliptic curves and are all algebraic, but Riemann discovered that most complex tori of dimension 2 are not algebraic...
s give rise to Kummer surfaces as quotients. There remains the class of elliptic surface
Elliptic surface
In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper connected morphism to an algebraic curve, almost all of whose fibers are elliptic curves....
s, which are fiber bundle
Fiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
s over a curve with 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...
s as fiber, having a finite number of modifications (so there is a bundle that is locally trivial actually over a curve less some points). The question of classification is to show that any surface, lying in projective space
Projective 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....
of any dimension, is in the birational sense (after blowing up
Blowing up
In mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with all the directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the point with the projectivized tangent space at that point...
and blowing down
Blowing down
In mathematics, blowing down is a type of geometric modification in algebraic geometry. It is the inverse operation of blowing up.On an algebraic surface, blowing down a curve lying on the surface is a typical effect of a birational transformation...
of some curves, that is) accounted for by the models already mentioned.
No more than other work in the Italian school would the proofs by Enriques now be counted as complete and rigorous. Not enough was known about some of the technical issues: the geometers worked by a mixture of inspired guesswork and close familiarity with examples. Oscar Zariski
Oscar Zariski
Oscar Zariski was a Russian mathematician and one of the most influential algebraic geometers of the 20th century.-Education:...
started to work in the 1930s on a more refined theory of birational mappings, incorporating commutative algebra
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
methods. He also began work on the question of the classification for characteristic p, where new phenomena arise. The schools of Kunihiko Kodaira and Igor Shafarevich
Igor Shafarevich
Igor Rostislavovich Shafarevich is a Soviet and Russian mathematician, founder of a school of algebraic number theory and algebraic geometry in the USSR, and a political writer. He was also an important dissident figure under the Soviet regime, a public supporter of Andrei Sakharov's Human Rights...
had put Enriques' work on a sound footing by about 1960.
Works
- Enriques F. Lezioni di geometria descrittiva. Bologna, 1920.
- Enriques F. Lezioni di geometria proiettiva. Italian ed. 1898 and german ed. 1903.
- Enriques F. Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche. Bologna, 1915-1934. Volume 1, Volume 2, Volume 3-4
- Severi F. Lezioni de geometria algebrica : geometria sopra una curva, superficie di Riemann-integrali abeliani. Italian ed. 1908 and german ed. 1921
- Castelnouvo G., Enriques F. Die algebraischen Flaechen// Encyklopädie der mathematischen Wissenschaften, III C 6
- Enriques F. Le superficie algebriche. Bologna, 1949 (scanned/djvu)