View source for Algebraic surface

A two-dimensional [[Algebraic variety|algebraic variety]]. Together with the class of algebraic curves (cf. [[Algebraic curve|Algebraic curve]]), the class of algebraic surfaces is the class of algebraic varieties which has been most thoroughly studied. The richness of problems and the variety of ideas used to solve them makes the theory of algebraic surfaces one of the most interesting fields of algebraic geometry.

Unlike algebraic curves, two birationally isomorphic algebraic surfaces need not be biregularly isomorphic. During the classical period of the development of algebraic geometry (1868–1920) algebraic surfaces were understood to mean a class of birationally equivalent surfaces, and its representations were considered to be imbedded in complex three-dimensional projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a0116401.png" />, in which they were defined by a single algebraic equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a0116402.png" />. The foundations of the theory of algebraic surfaces were laid by A. Clebsch and M. Noether (ca. 1870), who defined the first important invariants of algebraic surfaces: the geometric genus and the canonical class. The subsequent development of the theory of algebraic surfaces is due to the Italian school of algebraic geometers: C. Segre, L. Cremona, E. Bertini, G. Castelnuovo, F. Enriques, F. Severi, and others. Topological and transcendental methods were introduced into the theory of algebraic surfaces by E. Picard, H. Poincaré and S. Lefschetz. Many general facts arrived at in these studies proved to be valid for algebraic varieties and schemes of arbitrary dimension as well, while other results up till now have not been generalized to higher-dimensional cases. The latter include the classification of algebraic surfaces by numerical invariants, criteria of rationality and ruledness of algebraic surfaces, and the theory of minimal models. Most of these results were critically reviewed and demonstrated once again in the 1950s and 1960s using modern cohomological methods [[#References|[1]]], [[#References|[5]]], [[#References|[8]]].

==General features of algebraic surfaces.==
Unless otherwise stated, an algebraic surface will be understood to mean a projective algebraic surface over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a0116403.png" />.

a) Resolution of singularities. The first rigorous proofs of the existence of a non-singular model were given in the 1930s by R. Walker and O. Zariski. The previous geometrical proofs given by the Italian workers had contained lacunae [[#References|[9]]]. The proof of the existence of a non-singular model of an algebraic surface for the case of a ground field with positive characteristic was given in 1956 (cf. [[Resolution of singularities|Resolution of singularities]]).

Any non-singular algebraic surface can be birationally projected in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a0116404.png" /> on a surface with ordinary singularities — ordinary double curves with a finite number of ordinary cusp points and triple points, which are triplanar singular points of the surface. Such surfaces represented the principal object of studies during the classical stage of development of the theory of algebraic surfaces. Owing to the theorem on the resolution of singularities of algebraic surfaces, global methods of the theory of schemes may be applied to the local study of singular points of algebraic surfaces. The fundamental work on this subject was done by D. Mumford (1961), who deduced important invariants of two-dimensional singularities and proved that the topological space of a normal algebraic surface is a topological manifold if and only if the surface is non-singular.

b) Numerical invariants of algebraic surfaces. The concept of the geometric genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a0116405.png" /> of an algebraic surface is a generalization of the concept of the genus of an algebraic curve in the theory of algebraic surfaces. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a0116406.png" /> is the maximum number of linearly independent regular two-dimensional differentials on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a0116407.png" /> (cf. [[Geometric genus|Geometric genus]]). The virtual arithmetic genus of the divisors in the canonical system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a0116408.png" /> of an algebraic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a0116409.png" /> is known as the linear genus and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164010.png" />. For a non-singular algebraic surface the following equation is valid:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164011.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164012.png" /> is the first [[Chern class|Chern class]] of the tangent bundle to the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164013.png" />. The multiple canonical systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164014.png" /> are called pluri-canonical systems of the algebraic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164015.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164016.png" /> is referred to as the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164017.png" />-th (pluri-) genus and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164018.png" />; it is identical with the dimension of the space of everywhere-regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164019.png" />-tuple two-dimensional differential forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164020.png" />.

According to the Cayley–Noether postulation formula, the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164021.png" /> of linearly independent forms of a sufficiently high degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164022.png" /> passing through a space curve of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164023.png" />, genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164024.png" /> and with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164025.png" /> triple points and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164026.png" /> double points, is given by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164027.png" /></td> </tr></table>

The concept of the arithmetic genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164028.png" />, defined by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164029.png" /></td> </tr></table>

(where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164030.png" /> is the degree, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164031.png" /> is the number of double points, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164032.png" /> is the genus and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164033.png" /> is the number of triple points of the double curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164034.png" />; for non-singular surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164036.png" />), was introduced in 1875 for algebraic surfaces with ordinary singularities. Other equivalent definitions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164037.png" /> were given subsequently (cf. [[Arithmetic genus|Arithmetic genus]]).

The difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164038.png" /> is always non-negative; it is called the irregularity of the algebraic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164039.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164040.png" />. An algebraic surface is called regular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164041.png" /> and irregular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164042.png" />. The first examples of irregular surfaces date back to the end of the 19th century (A. Cayley, Castelnuovo). Irregularity can be described as the maximal insufficiency of the linear system which is cut out by the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164043.png" /> on the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164044.png" /> (Enriques, 1896, see [[#References|[9]]]). A cohomological definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164046.png" /> for non-singular algebraic surfaces is given by the formulas:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164047.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164048.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164049.png" /></td> </tr></table>

The arithmetic genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164050.png" /> of a non-singular algebraic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164051.png" /> is expressed in terms of the Chern classes of the tangent bundle to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164052.png" /> by the formula:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164053.png" /></td> </tr></table>

called Noether's formula. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164054.png" /> is known as the Zeuten–Segre invariant.

c) The Riemann–Roch theorem for algebraic surfaces. The generalization of the Riemann–Roch theorem for algebraic curves to algebraic surfaces is due to Castelnuovo (1897). For a non-singular algebraic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164055.png" /> and a complete linear system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164056.png" /> of divisors of virtual degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164057.png" /> and virtual genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164058.png" />, the Riemann–Roch theorem asserts the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164059.png" /></td> </tr></table>

The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164060.png" /> is known as the speciality index of the divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164061.png" />. The cohomological proof of this theorem (cf. [[Riemann–Roch theorem|Riemann–Roch theorem]]) permits the stipulation of a necessary and sufficient condition such that this inequality becomes an equality. This condition is the vanishing of the cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164062.png" />. It has been proved that, for any divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164063.png" /> and all sufficiently large numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164064.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164065.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164066.png" /> and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164067.png" /> is a hyperplane section of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164068.png" /> (J.P. Serre, 1955). If the characteristic of the ground field is zero, then the inequality becomes an equality for any ample divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164069.png" />. This result is a generalization of the classical Picard–Severi theorem concerning the regularity of the adjoint system [[#References|[9]]].

d) Systems of curves on an algebraic surface. The principal method used by Italian geometers was their theory of linear and algebraic systems of curves on an algebraic surface. The concepts of algebraic and linear equivalence of divisors are closely connected with this theory. These concepts only coincide for regular algebraic surfaces (Castelnuovo, 1896). Any sufficiently general curve on an algebraic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164070.png" /> with irregularity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164071.png" /> is contained in a maximal algebraic family which fibrates into linear systems of equal dimension, the base of the fibration being a variety of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164072.png" /> (Enriques, 1904). This variety is an Abelian variety, and is known as the Picard variety of the algebraic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164073.png" /> (cf. [[Picard scheme|Picard scheme]]). The proof of Enriques' result (known as the  "fundamental theorem in the theory of irregular surfacesfundamental theorem in the theory of irregular surfaces" , or the  "theorem of completeness of characteristic seriestheorem of completeness of characteristic series" ), as well as subsequent proofs by Severi, contain lacunae. The first rigorous transcendental proof of this theorem was given by Poincaré in 1910, and the algebraic proof came 50 years later (A. Grothendieck, see [[#References|[5]]]). If the characteristic of the ground field is positive, this theorem is not valid in the general case (J. Igusa, 1955). It has been shown that, in the general case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164074.png" /> is not less than the dimension of the Picard variety.

The non-identity of algebraic and linear equivalence on an algebraic surface implies that everywhere-regular one-dimensional differentials exist on it. The dimension of the space of such differentials in the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164075.png" /> coincides with the irregularity (Castelnuovo, Severi, 1905). This fact is a special case of the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164076.png" />, which is valid for an arbitrary compact [[Kähler manifold|Kähler manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164077.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164078.png" />, this result is invalid.

The group of divisor classes with respect to algebraic equivalence is finitely generated (see [[Néron–Severi group|Néron–Severi group]]). Its rank is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164079.png" /> and is called the Picard number of the algebraic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164080.png" />. The order of the torsion subgroup of this group is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164081.png" /> and is known as the Severi divisor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164082.png" /> [[#References|[9]]].

Severi also laid the foundations of the theory of zero-dimensional cycles on an algebraic surface; this theory contains many interesting problems that have not yet been solved.

e) Topological properties of an algebraic surface. A non-singular projective algebraic surface over the field of complex numbers is a compact four-dimensional oriented real manifold; in particular, the integral homology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164083.png" /> and the integral cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164084.png" />, satisfying [[Poincaré duality|Poincaré duality]], have been defined for such an algebraic surface. Each divisor on an algebraic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164085.png" /> specifies a two-dimensional cycle, the divisors which are algebraically equivalent to zero correspond to cycles homologous to zero. The intersection index of divisors is identical with the topological intersection index of cycles. The following equalities are valid: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164086.png" /> is equal to the order of the torsion group of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164087.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164088.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164089.png" /> is the first Betti number of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164090.png" /> [[#References|[4]]], [[#References|[9]]].

A method, based on the study of a linear pencil of curves on a surface and on the study of the topology of the curves which vary in this pencil, was developed for the investigation of the topology of algebraic surfaces. It was proved in this way that non-singular hyperplanes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164091.png" /> are simply connected [[#References|[6]]]. This method was subsequently applied to studies on the topology of manifolds of higher dimension.

The study of étale cohomology of schemes makes it possible to formulate and prove algebraic analogues of many classical statements in the topology of algebraic surfaces for surfaces defined over a field of arbitrary characteristic. In particular, the [[L-adic-cohomology|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164092.png" />-adic cohomology]] groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164093.png" />, the [[Fundamental group|fundamental group]] and the [[Homotopy type|homotopy type]] of an algebraic surface have all been defined [[#References|[9]]].

E. Picard [[#References|[6]]] initiated the theory of integrals on an algebraic surface. His results were subsequently generalized to manifolds of higher dimension (W.V.D. Hodge, ca. 1940). These results yielded, in particular, the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164094.png" />. It was proved, moreover, that the number of positive squares in the quadratic form which is defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164095.png" /> by the intersection index equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164096.png" />. The analogues of these results for fields of finite characteristic are not true. The development of the transcendental theory of algebraic surfaces is due to the work of P.A. Griffiths.

f) Projective immersions of an algebraic surface. In the context of the abstract concept of an algebraic variety there arose the problem of the existence of (abstract) algebraic surfaces that are not immersible in a projective space. It was proved that a non-singular complete algebraic surface is always projective [[#References|[8]]]. There exist complete singular non-projective algebraic surfaces. There are various numerical criteria for the projectivity of an algebraic surface (cf. [[Ample sheaf|Ample sheaf]]). Any non-singular two-dimensional algebraic space is an algebraic surface.

==Minimal models of algebraic surfaces.==
A non-singular algebraic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164097.png" /> is called a minimal model if each birational morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164098.png" /> onto a non-singular algebraic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a01164099.png" /> is an isomorphism. A minimal model exists in each class of birational equivalence. Except for classes of ruled surfaces, this minimal model is unique up to an isomorphism [[#References|[1]]], [[#References|[2]]], [[#References|[8]]]. For an algebraic surface to constitute a minimal model it is necessary and sufficient for it not to contain exceptional curves of the first kind, i.e. irreducible curves which do contract to a non-singular point for some birational morphism. Such curves were first studied by M. Noether in 1895. They are characterized by the following requirements: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640100.png" /> must be a rational non-singular curve and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640101.png" /> ([[#References|[1]]], [[#References|[2]]]). Minimal models of ruled surfaces have been completely classified.

==Classification of algebraic surfaces.==
According to the classification of Enriques [[#References|[2]]], each non-singular algebraic surface over a field of characteristic zero belongs to one of the following types, up to a birational equivalence: a) ruled surfaces (cf. [[Ruled surface|Ruled surface]]); b) two-dimensional Abelian varieties (cf. [[Abelian variety|Abelian variety]]); c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640102.png" />-surfaces (cf. [[K3-surface|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640103.png" />-surface]]); d) elliptic surfaces (cf. [[Elliptic surface|Elliptic surface]]); or e) general-type algebraic surfaces (cf. [[General-type algebraic surface|General-type algebraic surface]]). This classification is highly analogous to the classification of algebraic curves. Moreover, rational curves correspond to ruled surface, surfaces of the types b)–d) correspond to elliptic curves, and surfaces of general type correspond to curves of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640104.png" />. The type of the minimal model of an algebraic surface is determined by the values of its numerical invariants. Ruled surfaces are characterized by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640105.png" />; type-b) algebraic surfaces are characterized by the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640107.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640108.png" />; type-c) algebraic surfaces are characterized by the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640110.png" />; type-d) algebraic surfaces are characterized by the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640111.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640112.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640113.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640114.png" />. Finally, algebraic surfaces of general type are characterized by the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640115.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640116.png" />.

The class of ruled surfaces comprises rational surfaces (cf. [[Rational surface|Rational surface]]), which are characterized by the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640117.png" /> (this makes it possible to give a definite answer to the [[Lüroth problem|Lüroth problem]] in the two-dimensional case).

Many different results in the theory of the classification of algebraic surfaces concern the problem of the construction of an algebraic surface with given numerical invariants. Algebraic surfaces are most frequently constructed by representing them as a double covering of the projective plane with a specially selected branch curve (cf. [[Double plane|Double plane]]). The values that can be assumed by the invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640118.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640119.png" /> of a general-type surface are not known (1986). Some of the results obtained in connection with the classification have recently been extended to include fields of arbitrary characteristic. K. Kodaira generalized the classification results obtained by Enriques to include complex-analytic surfaces (cf. [[Analytic surface (in algebraic geometry)|Analytic surface (in algebraic geometry)]]).

==The moduli problem for algebraic surfaces.==
This is the problem of the classification of algebraic surfaces up to an isomorphism. The first formula for the number of parameters (moduli) which determine an algebraic surface with given invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640120.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640121.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640122.png" /> for one class of algebraic surfaces was given by Noether in 1888. Non-ruled surfaces with given invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640123.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640124.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640125.png" /> depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640126.png" /> moduli, where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640127.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640128.png" /> is a birational invariant of the algebraic surface under consideration [[#References|[2]]], [[#References|[9]]].

The modern theory of deformations gives the following interpretation of this classical result. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640129.png" /> of moduli is identical with the dimension of the Zariski space tangent to the local (or, in the algebraic case, the formal) moduli scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640130.png" /> for the polarized surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640131.png" /> of this class of algebraic surfaces. Moreover, if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640132.png" /></td> </tr></table>

is the canonical extension of the tangent sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640133.png" /> to the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640134.png" /> by the structure sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640135.png" />, defined by the fundamental class of the polarization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640136.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640137.png" /></td> </tr></table>

Formula (*) is a consequence of this exact sequence and the Riemann–Roch theorem. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640138.png" /> turns out to be the sum

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640139.png" /></td> </tr></table>

Also, the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640140.png" /></td> </tr></table>

is valid [[#References|[9]]]. The local moduli scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640141.png" /> can be singular, even if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640142.png" />. This shows that the  "actual"  number of moduli, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640143.png" />, can be less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640144.png" />. The difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640145.png" /> is called the number of obstructions to the deformation; the estimate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640146.png" /> is known. The existence of a global moduli variety of an algebraic surface has been proved only for certain cases. The moduli variety for general-type surfaces and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640147.png" />-surfaces exists as an [[Analytic space|analytic space]] or as an [[Algebraic space|algebraic space]].

==Automorphisms of algebraic surfaces.==
The automorphism group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640148.png" /> of a complete algebraic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640149.png" /> is the group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640150.png" />-points of a certain group scheme, the connected component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640151.png" /> of which is an algebraic group. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640152.png" /> is not a ruled surface and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640153.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640154.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640155.png" /> or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640156.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640157.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640158.png" /> is an elliptic surface and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640159.png" /> is a one-dimensional Abelian variety. In the other cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640160.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640161.png" /> are Abelian surfaces [[#References|[2]]]. For general-type surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640162.png" /> is a finite subgroup of a projective group. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640163.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640164.png" />-surfaces over the field of complex numbers and for ruled surfaces has been thoroughly studied. If the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640165.png" /> is not ruled, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640166.png" /> coincides with the group of birational transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640167.png" />. This group has no algebraic structure for ruled surfaces and has not been thoroughly studied (cf. [[Cremona group|Cremona group]]). The automorphism group of affine algebraic surfaces (cf. [[Algebraic variety, automorphism of an|Algebraic variety, automorphism of an]]) is now under intensive study.

==Algebraic surfaces over algebraically non-closed fields.==
Number-theoretic problems in the theory of algebraic surfaces involve Diophantine problems (cf. [[Diophantine geometry|Diophantine geometry]]). The classification of rational surfaces over non-closed fields, begun by Enriques, Comessatti and Segre, has now been completed. The group of birational automorphisms has been studied for a certain class of such surfaces.

The results of the theory of algebraic surfaces are used in the study of algebraic curves over function fields (cf. [[Mordell conjecture|Mordell conjecture]]). The generalization of certain results of the theory of algebraic surfaces (minimal models, intersection theory) to the wider class of regular two-dimensional schemes [[#References|[7]]] permits the use of the geometrical language in the study of algebraic curves over number fields.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  "Algebraic surfaces"  ''Trudy Mat. Inst. Steklov.'' , '''75'''  (1965)  (In Russian)  {{MR|}} {{ZBL|0154.33002}} {{ZBL|0154.21001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. Enriques,   "Le superficie algebraiche" , Bologna  (1949)      {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H.W.E. Jung,   "Algebraische Flächen" , Hannover  (1925)  {{MR|}}  {{ZBL|51.0522.03}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Lefschetz,   "L'analysis situs et la géométrie algébrique" , Gauthier-Villars  (1924)  {{MR|0033557}} {{MR|1520618}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  D. Mumford,   "Lectures on curves on an algebraic surface" , Princeton Univ. Press  (1966)  {{MR|0209285}} {{ZBL|0187.42701}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E. Picard,   G. Simart,   "Théorie des fonctions algébriques de deux variables indépendantes" , '''1–2''' , Chelsea, reprint  (1971)      {{MR|0392468}} {{ZBL|37.0404.02}}  {{ZBL|28.0327.01}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  I.R. Shafarevich,   "Lectures on minimal models and birational transformations of two-dimensional schemes" , Tata Inst.  (1966)  {{MR|0217068}} {{ZBL|0164.51704}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  O. Zariski,   "Introduction to the problem of minimal models in the theory of algebraic surfaces" , Math. Soc. Japan  (1958)  {{MR|0097403}} {{ZBL|0093.33904}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  O. Zariski,   "Algebraic surfaces" , Springer  (1971)  {{MR|0469915}} {{ZBL|0219.14020}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  O. Zariski,   "An introduction to the theory of algebraic surfaces" , Springer  (1969)  {{MR|0263819}} {{ZBL|0177.49001}} </TD></TR></table>



====Comments====
One of the more important achievements is the proof of the Miyaoka–Yao–Bogomolov inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640168.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640169.png" /> denotes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011640/a011640170.png" />-th Chern class of an algebraic surface [[#References|[a1]]]. A recent reference is [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Y. Miyaoka,   "On the Chern numbers of surfaces of general type"  ''Invent. Math.'' , '''42'''  (1977)  pp. 225–237  {{MR|0460343}} {{ZBL|0374.14007}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. van de Ven,   "Compact complex surfaces" , Springer  (1984)  {{MR|}} {{ZBL|0718.14023}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Beauville,   "Surfaces algébrique complexes"  ''Astérique'' , '''54'''  (1958)  (Translated into English as a London Math. Soc. Lecture Notes)  {{MR|2757649}} {{MR|0485887}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P.A. Griffiths,   J.E. Harris,   "Principles of algebraic geometry" , '''1–2''' , Wiley (Interscience)  (1978)  {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR></table>