Arithmetic surface

Informally, an arithmetic surface over a Dedekind domain $R$ with fraction field $K$ is an R-scheme with a non-singular, connected projective curve $C/K$ for a generic fiber and unions of curves (possibly reducible, singular, non-reduced ) over the appropriate residue field for special fibers. In more detail, an arithmetic surface $S$ is a scheme with a morphism $p:S\rightarrow \mathrm {Spec} (R)$ with the following properties: $S$ is integral, normal, excellent, flat and of finite type over $R$ and the generic fiber is a non-singular, connected projective curve over $\mathrm {Frac} (R)$ and for other $t$ in $\mathrm {Spec} (R)$ ,

S{\underset {Spec(R)}{\times }}\mathrm {Spec} (k_{t})

is a union of curves over $R/t$ .^[1]

What makes them "arithmetic?"

Arithmetic surfaces are the arithmetic analogue of fibred surfaces with the spectrum of a Dedekind domain replacing the base curve.^[2] Such surfaces arise primarily in the context of number theory.^[3] In higher dimensions one may also consider arithmetic schemes.^[4]

Properties

Dimension

Arithmetic surfaces have dimension 2 and relative dimension 1 over their base.^[5]

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."^[6] The theory is developed in Hartshorne's Algebraic, for example.^[7]

Examples

Projective line

The projective line over Dedekind domain $R$ is a smooth, proper arithmetic surface over $R$ . The fiber over any maximal ideal ${\mathfrak {m}}$ is the projective line over the field $R/{\mathfrak {m}}.$ ^[8]

Regular minimal models

Neron models are an example of this construction, and a non-trivial example of an arithmetic surface.^[9]

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.^[10] 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 because an arithmetic surface is not complete and a linear equivalence may move an intersection point out to infinity.^[11] The simplest 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.^[12]

Arakelov theory

Arakelov theory offers a more satisfactory 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.^[13]

Notes

^ Silverman, J.H. Advanced Topics in the Arithmetic of Elliptic Curves. Springer, 1994, p. 311.
^ Silverman, J.H. Advanced Topics in the Arithmetic of Elliptic Curves. Springer, 1994, p. 311.
^ Eisenbud, D. and Harris, J. The Geometry of Schemes. Springer-Verlag, 1998, p. 81.
^ Eisenbud, D. and Harris, J. The Geometry of Schemes. Springer-Verlag, 1998, p. 81.
^ Silverman, J.H. Advanced Topics in the Arithmetic of Elliptic Curves. Springer, 1994, p. 311.
^ Silverman, J.H. Advanced Topics in the Arithmetic of Elliptic Curves. Springer, 1994, p. 311.
^ Hartshorne, R. Algebraic Geometry. Springer-Verlang, 1977, p. 130.
^ Silverman, J.H. Advanced Topics in the Arithmetic of Elliptic Curves. Springer, 1994, p. 312.
^ Silverman, J.H. Advanced Topics in the Arithmetic of Elliptic Curves. Springer, 1994, Chapter IV.
^ Silverman, J.H. Advanced Topics in the Arithmetic of Elliptic Curves. Springer, 1994, p. 339.
^ Silverman, J.H. Advanced Topics in the Arithmetic of Elliptic Curves. Springer, 1994, p. 340.
^ Silverman, J.H. Advanced Topics in the Arithmetic of Elliptic Curves. Springer, 1994, p. 341.
^ Silverman, J.H. Advanced Topics in the Arithmetic of Elliptic Curves. Springer, 1994, p. 344.

References

Robin Hartshorne (1997). Algebraic Geometry. Springer-Verlag. ISBN 0-387-90244-9.
Qing Liu (2002). Algebraic Geometry and Arithmetic Curves. Oxford University Press. ISBN 0-19-850284-2.
David Eisenbud (1998). The Geometry of Schemes. Springer-Verlag. ISBN 0-387-98637-5. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
Joseph Silverman (1994). Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag. ISBN 0-387-94328-5.