Arithmetic surface
In mathematics, an arithmetic surface over a Dedekind domain "R" with fraction field formula_1 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 defined more formally as an R-scheme with a non-singular, connected projective curve formula_2 for a generic fiber and unions of curves (possibly reducible, singular, non-reduced ) over the appropriate residue field for special fibers. Formal definition. In more detail, an arithmetic surface formula_3 (over the Dedekind domain formula_4) is a scheme with a morphism formula_5 with the following properties: formula_3 is integral, normal, excellent, flat and of finite type over formula_4 and the generic  fiber is a non-singular, connected projective curve over formula_8 and for other formula_9 in formula_10, is a union of curves over formula_12. 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. 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. Arithmetic surfaces arise primarily in the context of number theory. In fact, given a curve formula_13over a number field formula_3, there exists an arithmetic surface over the ring of integers formula_15 whose generic fiber is isomorphic to formula_13. In higher dimensions one may also consider arithmetic schemes. Properties. Dimension. Arithmetic surfaces have dimension 2 and relative dimension 1 over their base. Divisors. We can develop a theory of 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." The theory is developed in Hartshorne's Algebraic Geometry, for example. Examples. Projective line. The projective line over Dedekind domain formula_4 is a smooth, proper arithmetic surface over formula_4. The fiber over any maximal ideal formula_19 is the projective line over the field formula_20 Regular minimal models. Néron models for elliptic curves, initially defined over a global field, are examples of this construction, and are much studied examples of arithmetic surfaces. There are strong analogies with elliptic fibrations. 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. 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. 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. 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.