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Algebraic surface
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{{Short description|Zero set of a polynomial in three variables}} {{multiple issues| {{technical|date=May 2023}} {{confusing|reason=it is unclear when the article is restricted to smooth algebraic surfaces over the complexes, and when more general cases are considered|date=May 2023}}}} In [[mathematics]], an '''algebraic surface''' is an [[algebraic variety]] of [[dimension of an algebraic variety|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 [[Algebraic curve#Singularities|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 space|compact]] [[Riemann surface]]s, which are genuine [[Surface (topology)|surfaces]] of (real) dimension two). Many results were obtained, but, in the [[Italian school of algebraic geometry ]], and are up to 100 years old. == Classification by the Kodaira dimension == {{Main|Enriques–Kodaira classification}} In the case of dimension one, varieties are classified by only the [[topological genus]], but, in dimension two, one needs to distinguish the [[arithmetic genus]] <math>p_a</math> and the [[geometric genus]] <math>p_g</math> because one cannot distinguish birationally only the topological genus. Then, [[Irregularity of a surface|irregularity]] is introduced for the classification of varieties. 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 [[complex projective plane|projective plane]], [[quadric]]s in '''P'''<sup>3</sup>, [[cubic surface]]s, [[Veronese surface]], [[del Pezzo surface]]s, [[ruled surface]]s * κ = 0 : [[K3 surface]]s, [[abelian surface]]s, [[Enriques surface]]s, [[hyperelliptic surface]]s * κ = 1: [[elliptic surface]]s * κ = 2: [[surface of general type|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 of an algebraic variety|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. == 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 contraction theorem|'''Castelnuovo's theorem''']]. This states that any birational map between algebraic surfaces is given by a finite sequence of blowups and blowdowns. == Properties == The [[ample line bundle#Intersection theorem|'''Nakai criterion''']] says that: :A Divisor ''D'' on a surface ''S'' is ample if and only if ''D<sup>2</sup> > 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 <math>\mathcal{D}(S)</math> be the abelian group consisting of all the divisors on ''S''. Then due to the [[intersection number|intersection theorem]] :<math>\mathcal{D}(S)\times\mathcal{D}(S)\rightarrow\mathbb{Z}:(X,Y)\mapsto X\cdot Y</math> is viewed as a [[quadratic form]]. Let :<math>\mathcal{D}_0(S):=\{D\in\mathcal{D}(S)|D\cdot X=0,\text{for all } X\in\mathcal{D}(S)\}</math> then <math>\mathcal{D}/\mathcal{D}_0(S):=Num(S)</math> becomes to be a '''numerical equivalent class group''' of ''S'' and :<math>Num(S)\times Num(S)\mapsto\mathbb{Z}=(\bar{D},\bar{E})\mapsto D\cdot E</math> also becomes to be a quadratic form on <math>Num(S)</math>, where <math>\bar{D}</math> is the image of a divisor ''D'' on ''S''. (In the below the image <math>\bar{D}</math> is abbreviated with ''D''.) For an ample line bundle ''H'' on ''S'', the definition :<math>\{H\}^\perp:=\{D\in Num(S)|D\cdot H=0\}.</math> is used in the surface version of the '''Hodge index theorem''': :for <math>D\in\{\{H\}^\perp|D\ne0\}, D\cdot D < 0</math>, i.e. the restriction of the intersection form to <math>\{H\}^\perp</math> 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 conjectures|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]]. The ''general type'' class, of [[Kodaira dimension]] 2, is very large (degree 5 or larger for a non-singular surface in '''P'''<sup>3</sup> lies in it, for example). There are essential three [[Hodge number]] invariants of a surface. Of those, ''h''<sup>1,0</sup> was classically called the '''irregularity''' and denoted by ''q''; and ''h''<sup>2,0</sup> was called the '''geometric genus''' ''p''<sub>''g''</sub>. The third, ''h''<sup>1,1</sup>, is not a [[birational invariant]], because [[blowing up]] can add whole curves, with classes in ''H''<sup>1,1</sup>. It is known that [[Hodge cycle]]s are algebraic and that [[algebraic equivalence]] coincides with [[homological equivalence]], so that ''h''<sup>1,1</sup> is an upper bound for ρ, the rank of the [[Néron-Severi group]]. The [[arithmetic genus]] ''p''<sub>''a''</sub> is the difference :geometric genus − irregularity. This explains why the irregularity got its name, as a kind of 'error term'. == Riemann-Roch theorem for surfaces == {{Main|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== *{{eom|title=Algebraic surface|first=I.V.|last= Dolgachev}} *{{Citation | last1=Zariski | first1=Oscar | author1-link=Oscar Zariski | title=Algebraic surfaces | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Classics in Mathematics | isbn=978-3-540-58658-6 | mr=1336146 | year=1995}} == External links == * [http://imaginary.org/program/surfer Free program SURFER] to visualize algebraic surfaces in real-time, including a user gallery. * [http://www.singsurf.org/singsurf/SingSurf.html SingSurf] an interactive 3D viewer for algebraic surfaces. * [http://www.bru.hlphys.jku.at/surf/index.html Page on Algebraic Surfaces started in 2008] * [http://maxwelldemon.com/2009/03/29/surfaces-2-algebraic-surfaces/ Overview and thoughts on designing Algebraic surfaces] {{Authority control}} [[Category:Algebraic surfaces| ]]
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