Algebraic surface
Encyclopedia
In mathematics
, an algebraic surface is an algebraic variety
of dimension
two. In the case of geometry over the field of complex number
s, an algebraic surface has complex dimension two (as a complex manifold
, when it is non-singular) and so of dimension four as a smooth manifold.
The theory of algebraic surfaces is much more complicated than that of algebraic curve
s (including the compact
Riemann surface
s, which are genuine surface
s of (real) dimension two). Many results were obtained, however, in the Italian school of algebraic geometry
, and are up to 100 years old.
Examples of algebraic surfaces include (κ is the Kodaira dimension
):
For more examples see the list of algebraic surfaces.
The first five examples are in fact birationally equivalent. That is, for example, a cubic surface has a function field
isomorphic to that of the projective plane
, being the rational function
s in two indeterminates. The cartesian product of two curves also provides examples.
The birational geometry
of algebraic surfaces is rich, because of blowing up
(also known as a monoidal transformation); under which a point is replaced by the curve of all limiting tangent directions coming into it (a projective line
). Certain curves may also be blown down, but there is a restriction (self-intersection number must be −1).
Basic results on algebraic surfaces include the Hodge index theorem
, and the division into five groups of birational equivalence classes called the classification of algebraic surfaces. The general type class, of Kodaira dimension
2, is very large (degree 5 or larger for a non-singular surface in P3 lies in it, for example).
There are essential three Hodge number invariants of a surface. Of those, h1,0 was classically called the irregularity and denoted by q; and h2,0 was called the geometric genus pg. The third, h1,1, is not a birational invariant
, because blowing up
can add whole curves, with classes in H1,1. It is known that Hodge cycle
s are algebraic, and that algebraic equivalence coincides with homological equivalence, so that h1,1 is an upper bound for ρ, the rank of the Néron-Severi group. The arithmetic genus
pa is the difference
In fact this explains why the irregularity got its name, as a kind of 'error term'.
The Riemann-Roch theorem for surfaces was first formulated by Max Noether
. The families of curves on surfaces can be classified, in a sense, and give rise to much of their interesting geometry.
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...
, an algebraic surface is an algebraic variety
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. In the case of geometry over the field of complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s, an algebraic surface has complex dimension two (as a 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....
, when it is non-singular) and so of dimension four as a smooth manifold.
The theory of algebraic surfaces is much more complicated than that 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 (including the compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
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, which are genuine surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
s of (real) dimension two). Many results were obtained, however, in 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...
, and are up to 100 years old.
Examples of algebraic surfaces include (κ is the Kodaira dimension
Kodaira dimension
In algebraic geometry, the Kodaira dimension κ measures the size of the canonical model of a projective variety V.The definition of Kodaira dimension, named for Kunihiko Kodaira, and the notation κ were introduced in the seminar.-The plurigenera:...
):
- κ=−∞: the projective plane, quadricQuadricIn 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 in P3, cubic surfaceCubic surfaceA 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, Veronese surfaceVeronese surfaceIn mathematics, the Veronese surface is an algebraic surface in five-dimensional projective space, and is realized by the Veronese embedding, the embedding of the projective plane given by the complete linear system of conics. It is named after Giuseppe Veronese...
, del Pezzo surfaceDel Pezzo surfaceIn mathematics, a del Pezzo surface or Fano surface is a two-dimensional Fano variety, in other words a non-singular projective algebraic surface with ample anticanonical divisor class...
s, ruled surfaceRuled surfaceIn 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 - κ=0 : K3 surfaceK3 surfaceIn 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, abelian surfaceAbelian surfaceIn 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, Enriques surfaceEnriques surfaceIn mathematics, Enriques surfaces, discovered by , are complex algebraic surfacessuch that the irregularity q = 0 and the canonical line bundle K is non-trivial but has trivial square...
s, hyperelliptic surfaceHyperelliptic surfaceIn mathematics, a hyperelliptic surface, or bi-elliptic surface, is a surface with an elliptic fibration over an elliptic curve. Any such surface can be written as the quotient of a product of two elliptic curves by a finite abelian group....
s - κ=1: Elliptic surfaceElliptic surfaceIn 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 - κ=2: surfaces of general typeSurface of general typeIn 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...
.
For more examples see the list of algebraic surfaces.
The first five examples are in fact birationally equivalent. That is, for example, a cubic surface has a function field
Function field
Function field may refer to:*Function field of an algebraic variety*Function field...
isomorphic to that of the projective plane
Projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...
, being the rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
s in two indeterminates. The cartesian product of two curves also provides examples.
The 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...
of algebraic surfaces is rich, because of 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...
(also known as a monoidal transformation); under which a point is replaced by the curve of all limiting tangent directions coming into it (a projective line
Projective line
In mathematics, a projective line is a one-dimensional projective space. The projective line over a field K, denoted P1, may be defined as the set of one-dimensional subspaces of the two-dimensional vector space K2 .For the generalisation to the projective line over an associative ring, see...
). Certain curves may also be blown down, but there is a restriction (self-intersection number must be −1).
Basic results on algebraic surfaces include the Hodge index theorem
Hodge index theorem
In mathematics, the Hodge index theorem for an algebraic surface V determines the signature of the intersection pairing on the algebraic curves C on V...
, and the division into five groups of birational equivalence classes called the classification of algebraic surfaces. The general type class, of Kodaira dimension
Kodaira dimension
In algebraic geometry, the Kodaira dimension κ measures the size of the canonical model of a projective variety V.The definition of Kodaira dimension, named for Kunihiko Kodaira, and the notation κ were introduced in the seminar.-The plurigenera:...
2, is very large (degree 5 or larger for a non-singular surface in P3 lies in it, for example).
There are essential three Hodge number invariants of a surface. Of those, h1,0 was classically called the irregularity and denoted by q; and h2,0 was called the geometric genus pg. The third, h1,1, is not a birational invariant
Birational invariant
In algebraic geometry, a birational invariant is a quantity or object that is well-defined on a birational equivalence class of algebraic varieties...
, because 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...
can add whole curves, with classes in H1,1. It is known that Hodge cycle
Hodge cycle
In differential geometry, a Hodge cycle or Hodge class is a particular kind of homology class defined on a complex algebraic variety V, or more generally on a Kähler manifold. A homology class x in a homology group...
s are algebraic, and that algebraic equivalence coincides with homological equivalence, so that h1,1 is an upper bound for ρ, the rank of the Néron-Severi group. The arithmetic genus
Arithmetic genus
In mathematics, the arithmetic genus of an algebraic variety is one of some possible generalizations of the genus of an algebraic curve or Riemann surface.The arithmetic genus of a projective complex manifold...
pa is the difference
- geometric genus − irregularity.
In fact this explains why the irregularity got its name, as a kind of 'error term'.
The Riemann-Roch theorem for surfaces was first formulated by 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:...
. The families of curves on surfaces can be classified, in a sense, and give rise to much of their interesting geometry.
External links
- A gallery of algebraic surfaces
- SingSurf an interactive 3D viewer for algebraic surfaces.
- Some beautiful algebraic surfaces
- Some more, with their respective equations
- Page on Algebraic Surfaces started in 2008
- Overview and thoughts on designing Algebraic surfaces