en>NHSavage: /* History */ clarify use of AQI

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History: clarify use of AQI

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In mathematics, an '''arithmetic surface''' over a [[Dedekind domain]] ''R'' with [[Field of fractions|fraction field]] <math>K</math> is a geometric object having one conventional dimension, and one other dimension provided by the [[infinitude of the primes]]. When ''R'' is the [[ring of integers]] ''Z'', this intuition depends on the [[prime ideal spectrum]] Spec(''Z'') being seen as analogous to a line. Arithmetic surfaces arise naturally in [[diophantine geometry]], when an [[algebraic curve]] defined over ''K'' is thought of as having reductions over the fields ''R''/''P'', where ''P'' is a prime ideal of ''R'', for [[almost all]] ''P''; and are helpful in specifying what should happen about the process of reducing to ''R''/''P'' when the most naive way fails to make sense.

Such an object can be less informally defined as an [[Scheme (mathematics)|R-scheme]] with a non-singular, [[Connectedness|connected]] [[Algebraic variety|projective curve]] <math>C/K</math> for a [[Generic point|generic fiber]] and unions of curves (possibly [[Irreducible component|reducible]], [[Singular point of an algebraic variety|singular]], [[Glossary of scheme theory|non-reduced]] ) over the appropriate [[residue field]] for [[Generic point|special fibers]].

==Formal definition==
In more detail, an arithmetic surface <math>S</math> (over the Dedekind domain <math>R</math>) is a [[Scheme (mathematics)|scheme]] with a [[Scheme (mathematics)|morphism]] <math>p:S\rightarrow \mathrm{Spec}(R)</math> with the following properties: <math>S</math> is [[Glossary of scheme theory|integral]], [[Glossary of scheme theory|normal]], [[Excellent ring|excellent]], [[Flat morphism|flat]] and of [[Finite morphism|finite type]] over <math>R</math> and the generic fiber is a non-singular, connected projective curve over <math>\mathrm{Frac}(R)</math> and for other <math>t</math> in <math>\mathrm{Spec}(R)</math>,
:<math>S\underset{\mathrm{Spec}(R)}{\times}\mathrm{Spec}(k_t)</math>
is a union of curves over <math>R/t</math>.<ref>Silverman, J.H. ''Advanced Topics in the Arithmetic of Elliptic Curves''. Springer, 1994, p. 311.</ref>

==Over a Dedekind Scheme==
In even more generality, arithmetic surfaces can be defined over Dedekind schemes, a typical example of which is the spectrum of the [[ring of integers]] of a number field (which is the case above). An arithmetic surface is then a regular fibered surface over a Dedekind scheme of dimension one.<ref>Liu, Q. ''Algebraic geometry and arithmetic curves''. Oxford University Press, 2002, chapter 8.</ref> This generalisation is useful, for example, it allows for base curves which are smooth and projective over finite fields, which is important in positive characteristic.

==What makes them "arithmetic"?==
Arithmetic surfaces over Dedekind domains are the arithmetic analogue of fibered surfaces over algebraic curves.<ref>Silverman, J.H. ''Advanced Topics in the Arithmetic of Elliptic Curves''. Springer, 1994, p. 311.</ref> Arithmetic surfaces arise primarily in the context of number theory.<ref>Eisenbud, D. and Harris, J. ''The Geometry of Schemes''. Springer-Verlag, 1998, p. 81.</ref> In fact, given a curve <math>X</math>over a number field <math>S</math>, there exists an arithmetic surface over the ring of integers <math>O_K</math> whose generic fiber is isomorphic to <math>X</math>. In higher dimensions one may also consider arithmetic schemes.<ref>Eisenbud, D. and Harris, J. ''The Geometry of Schemes''. Springer-Verlag, 1998, p. 81.</ref>

==Properties==
===Dimension===
Arithmetic surfaces have dimension 2 and relative dimension 1 over their base.<ref>Silverman, J.H. ''Advanced Topics in the Arithmetic of Elliptic Curves''. Springer, 1994, p. 311.</ref>

===Divisors===
We can develop a theory of [[Divisor (algebraic geometry)|Weil divisors]] on arithmetic surfaces since every local ring of dimension one is regular. This is briefly stated as "arithmetic surfaces are regular in codimension one."<ref>Silverman, J.H. ''Advanced Topics in the Arithmetic of Elliptic Curves''. Springer, 1994, p. 311.</ref> The theory is developed in Hartshorne's Algebraic Geometry, for example.<ref>Hartshorne, R. ''Algebraic Geometry''. Springer-Verlang, 1977, p. 130.</ref>

==Examples==
===Projective line===
The [[projective line]] over Dedekind domain <math>R</math> is a [[Glossary of scheme theory|smooth]], [[Glossary of scheme theory|proper]] arithmetic surface over <math>R</math>. The fiber over any maximal ideal <math>\mathfrak{m}</math> is the projective line over the field <math>R/\mathfrak{m}.</math><ref>Silverman, J.H. ''Advanced Topics in the Arithmetic of Elliptic Curves''. Springer, 1994, p. 312.</ref>

===Regular minimal models===
[[Néron model]]s for [[elliptic curve]]s, initially defined over a [[global field]], are examples of this construction, and are much studied examples of arithmetic surfaces.<ref>Silverman, J.H. ''Advanced Topics in the Arithmetic of Elliptic Curves''. Springer, 1994, Chapter IV.</ref> There are strong analogies with [[elliptic fibration]]s.

==Intersection theory==
Given two distinct irreducible divisors and a closed point on the special fiber of an arithmetic surface, we can define the local intersection index of the divisors at the point as you would for any algebraic surface, namely as the dimension of a certain quotient of the local ring at a point.<ref>Silverman, J.H. ''Advanced Topics in the Arithmetic of Elliptic Curves''. Springer, 1994, p. 339.</ref> The idea is then to add these local indices up to get a global intersection index. The theory starts to diverge from that of algebraic surfaces when we try to ensure linear equivalent divisors give the same intersection index, this would be used, for example in computing a divisors intersection index with itself. This fails when the base scheme of an arithmetic surface is not "compact". In fact, in this case, linear equivalence may move an intersection point out to infinity.<ref>Silverman, J.H. ''Advanced Topics in the Arithmetic of Elliptic Curves''. Springer, 1994, p. 340.</ref> A partial resolution to this is to restrict the set of divisors we want to intersect, in particular forcing at least one divisor to be "fibral" (every component is a component of a special fiber) allows us to define a unique intersection pairing having this property, amongst other desirable ones.<ref>Silverman, J.H. ''Advanced Topics in the Arithmetic of Elliptic Curves''. Springer, 1994, p. 341.</ref> A full resolution is given by Arakelov theory.

===Arakelov theory===
[[Arakelov theory]] offers a solution to the problem presented above. Intuitively, fibers are added at infinity by adding a fiber for each [[archimedean absolute value]] of K. A local intersection pairing that extends to the full divisor group can then be defined, with the desired invariance under linear equivalence.<ref>Silverman, J.H. ''Advanced Topics in the Arithmetic of Elliptic Curves''. Springer, 1994, p. 344.</ref>

==Notes==
{{Reflist}}

== References ==
*{{cite book
| author = [[Robin Hartshorne]]
| year = 1997
| title = Algebraic Geometry
| publisher = Springer-Verlag
| isbn = 0-387-90244-9
}}
*{{cite book
| author = Qing Liu
| year = 2002
| title = Algebraic Geometry and Arithmetic Curves
| publisher = [[Oxford University Press]]
| isbn = 0-19-850284-2
}}
*{{cite book
| author = [[David Eisenbud]]
| coauthors = [[Joe Harris (mathematician)|Joe Harris]]
| year = 1998
| title = The Geometry of Schemes
| publisher = [[Springer Science+Business Media|Springer-Verlag]]
| isbn = 0-387-98637-5
}}
*{{cite book | first=Serge | last=Lang | authorlink=Serge Lang | title=Introduction to Arakelov theory | publisher=[[Springer-Verlag]] | place=New York | year=1988| isbn=0-387-96793-1 | mr=0969124 | zbl=0667.14001 }}
*{{cite book
| author = [[Joseph H. Silverman]]
| year = 1994
| title = Advanced Topics in the Arithmetic of Elliptic Curves
| publisher = [[Springer-Verlag]]
| isbn = 0-387-94328-5
}}

==See also==
*[[Glossary of arithmetic and Diophantine geometry]]
*[[Arakelov theory]]
*[[Néron model]]


[[Category:Articles created via the Article Wizard]]

[[Category:Diophantine geometry]]
[[Category:Surfaces]]

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en>NHSavage: /* History */ clarify use of AQI