In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an algebraic surface is an
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
of
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
two. In the case of geometry over the field of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
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 curves (including the
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
s, which are genuine
surfaces of (real) dimension two). Many results were obtained, however, in the
Italian school of algebraic geometry
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 ...
, and are up to 100 years old.
Classification by the Kodaira dimension
In the case of dimension one varieties are classified by only the
topological genus, but dimension two, the difference between the
arithmetic genus and the
geometric genus
In algebraic geometry, the geometric genus is a basic birational invariant of algebraic varieties and complex manifolds.
Definition
The geometric genus can be defined for non-singular complex projective varieties and more generally for complex ...
turns to be important because we cannot distinguish birationally only the topological genus. Then we introduce the
irregularity
Irregular, irregulars or irregularity may refer to any of the following:
Astronomy
* Irregular galaxy
* Irregular moon
* Irregular variable, a kind of star
Language
* Irregular inflection, the formation of derived forms such as plurals in ...
for the classification of them. A summary of the results (in detail, for each kind of surface refers to each redirection), follows:
Examples of algebraic surfaces include (κ is the
Kodaira dimension):
* κ = −∞: the
projective plane,
quadrics in P
3,
cubic surface
In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather tha ...
s,
Veronese surface,
del Pezzo surfaces,
ruled surfaces
* κ = 0 :
K3 surfaces,
abelian surface In mathematics, an abelian surface is a 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 via the Riemann bili ...
s,
Enriques surfaces,
hyperelliptic surface In mathematics, a hyperelliptic surface, or bi-elliptic surface, is a surface whose Albanese morphism is an elliptic fibration. Any such surface can be written as the quotient of a product of two elliptic curves by a finite abelian group.
Hyperel ...
s
* κ = 1:
elliptic surfaces
* κ = 2:
surfaces of general type.
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 functions in two indeterminates. The Cartesian product of two curves also provides examples.
Birational geometry of surfaces
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).
Castelnuovo's Theorem
One of the fundamental theorems for the birational geometry of surfaces is
Castelnuovo's theorem. This states that any birational map between algebraic surfaces is given by a finite sequence of blowups and blowdowns.
Properties
The
Nakai criterion says that:
:A Divisor ''D'' on a surface ''S'' is ample if and only if ''D
2 > 0'' and for all irreducible curve ''C'' on ''S'' ''D•C > 0.
Ample divisors have a nice property such as it is the pullback of some hyperplane bundle of projective space, whose properties are very well known. Let
be the abelian group consisting of all the divisors on ''S''. Then due to the
intersection theorem
In projective geometry, an intersection theorem or incidence theorem is a statement concerning an incidence structure – consisting of points, lines, and possibly higher-dimensional objects and their incidences – together with a pair of object ...
:
is viewed as a
quadratic form. Let
:
then
becomes to be a numerical equivalent class group of ''S'' and
:
also becomes to be a quadratic form on
, where
is the image of a divisor ''D'' on ''S''. (In the below the image
is abbreviated with ''D''.)
For an ample line bundle ''H'' on ''S'', the definition
:
is used in the surface version of the Hodge index theorem:
:for
, i.e. the restriction of the intersection form to
is a negative definite quadratic form.
This theorem is proven using the Nakai criterion and the Riemann-Roch theorem for surfaces. The Hodge index theorem is used in Deligne's proof of the
Weil conjecture.
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 Classification is a process related to categorization, the process in which ideas and objects are recognized, differentiated and understood.
Classification is the grouping of related facts into classes.
It may also refer to:
Business, organizat ...
. The ''general type'' class, of
Kodaira dimension 2, is very large (degree 5 or larger for a non-singular surface in P
3 lies in it, for example).
There are essential three
Hodge number invariants of a surface. Of those, ''h''
1,0 was classically called the irregularity and denoted by ''q''; and ''h''
2,0 was called the geometric genus ''p''
''g''. The third, ''h''
1,1, is not a
birational invariant In algebraic geometry, a birational invariant is a property that is preserved under birational equivalence.
Formal definition
A birational invariant is a quantity or object that is well-defined on a birational equivalence class of algebraic varieti ...
, because
blowing up can add whole curves, with classes in ''H''
1,1. It is known that
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
:H_k(V, \Complex) = H ...
s are algebraic, and that
algebraic equivalence coincides with
homological equivalence, so that ''h''
1,1 is an upper bound for ρ, the rank of the
Néron-Severi group. The
arithmetic genus ''p''
''a'' is the difference
:geometric genus − irregularity.
In fact this explains why the irregularity got its name, as a kind of 'error term'.
Riemann-Roch theorem for surfaces
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.
References
*
*
External links
Free program SURFERto visualize algebraic surfaces in real-time, including a user gallery.
an interactive 3D viewer for algebraic surfaces.
Overview and thoughts on designing Algebraic surfaces
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