Background on Curves and Jacobians
4.2 Function fields
52 Ch. 4 Background on Curves and Jacobians
4.2.1 Morphisms of affine varieties
We want to define maps between affine varieties that are continuous with respect to the Zariski topologies. We shall call such mapsmorphisms. We begin withV =An.
Definition 4.33 AmorphismϕfromAnto the affine lineA1is given by a polynomialf(x)∈K[x]
and defined by
ϕ:An → A1 P = (a1, . . . , an) → f
(a1, . . . , an)
=:f(P).
One sees immediately thatfis uniquely determined byϕ.
To ease notation we shall identifyf withϕ. Hence the set of morphisms fromAnto the affine line is identified withK[x]. In fact we can make the set of morphisms to aK-algebra in the usual way by adding and multiplying values. AsK-algebra it is then isomorphic toK[x].
As desired, the mapf is continuous with respect to the Zariski topology. It maps closed sets to closed sets, varieties to varieties, and for extension fieldsLofKwe getf
An(L)
⊂A1(L).
Definition 4.34 AmorphismϕfromAntoAm(for n, m∈N) is given by anm-tuple f1(x), . . . , fm(x)
of polynomials inK[x]mappingP ∈Anto
f1(P), . . . , fm(P) .
Sinceϕis determined byf1, . . . , fm, the set of morphisms fromAntoAmcan be identified with K[x]m. Again one checks without difficulty that morphisms are continuous with respect to the Zariski topology and map varieties to varieties.
LetV be an affine variety inAnwith corresponding prime idealI⊂K[x].
Definition 4.35 Amorphism fromV ⊂Anto a varietyW ⊂Amis given by the restriction toV of a morphism fromAntoAmwith image inW.
We denote the set of morphisms fromV toW byMorK(V, W).
Example 4.36 As basic example takeW =A1. ForV = A1we already haveMorK(A1,A1) = K[x]. For an arbitrary varietyV =VI one has thatMorK(V,A1)is asK-algebra isomorphic to K[V] =K[x]/I.
Remark 4.37 Takeϕ∈MorK(V, W)andf ∈MorK(W,A1) =K[W]. Thenf◦ϕis an element ofMorK(V,A1) =K[V], and so we get aninducedK-algebra morphism
ϕ∗:K[W]→K[V].
The morphismϕ∗is injective if and only ifϕis surjective. Ifϕ∗is surjective thenϕis injective.
Definition 4.38 The mapϕis an isomorphism if and only ifϕ∗is an isomorphism. This means that the inverse map ofϕis again a morphism, i.e., given by polynomials.
Two varietiesV andW are called isomorphic if there exists an isomorphism fromV toW, and we have seen that this is equivalent to the fact thatK[V]is isomorphic toK[W]asK-algebra.
Example 4.39 Assume thatchar(K) = p > 0. Then the exponentiation with pis an automor- phismφp ofKsinceKis assumed to be perfect. The mapφp is called the (absolute) Frobenius automorphism ofK(cf. Section 2.3.2).
§ 4.2 Function fields 53
We can extendφp to points of projective spaces overK by sending the point(X0, . . . , Xn) to (X0p, . . . , Xnp). We applyφpto polynomials overKby applying it to the coefficients.
IfV is a projective variety overKwith idealIwe can applyφptoIand get a varietyφp(V)with idealφp(I). The points ofV are mapped to points onφp(V).
The corresponding morphism fromV toφp(V)is called theFrobenius morphismand is again denoted byφp. We note thatφp isnotan isomorphism as the polynomial ringsK[V]/K[φp(V)]
form a proper inseparable extension.
4.2.2 Rational maps of affine varieties
LetV ⊂Anbe an affine variety with idealI=I(V)and takeϕ∈K[V]with representing element f ∈K[x].
By definition, the setDf consists of the pointsP inAn in whichf(P) = 0. It is open in the Zariski topology ofAn, and henceUϕ:=Df∩V is open inV. Its complementVϕinV is the zero locus ofϕ. It is not equal toV if and only ifUϕis not empty, and this is equivalent tof /∈I.
We assume now thatf /∈I. ForP∈Uϕdefine(1/ϕ)(P) :=f(P)−1.
Definition 4.40 Assume thatUis a nonempty open set of an affine varietyV and let the maprUbe given by
rU : U → A1 P → (ψ/ϕ)(P)
for someψ, ϕ∈K[V]andU ⊂Uϕ. ThenrU is arational map fromV toA1withdefinition setU. We introduce an equivalence relation on rational maps: for givenV the rational maprUis equivalent torU if for all pointsP ∈U∩Uwe have:rU(P) =rU(P).
Definition 4.41 The equivalence class of a rational map fromV toA1is called arational function onV.
Proposition 4.42 LetV be an affine variety. The set of rational functions onV is equal toK(V).
The addition (resp. multiplication) inK(V)corresponds to the addition (resp. multiplication) of rational functions defined by addition (resp. multiplication) of the values.
LetV ⊂An. As in the case of morphisms we can extend the notion of rational maps from the case W =A1to the general case thatW ⊂Amis a variety:
Definition 4.43 Arational maprfromV toW is anm-tuple of rational functions(r1, . . . , rm) withri ∈K(V)having representativesRidefined on a nonempty open setU ⊂V withR(U) :=
R1(U), . . . , Rm(U)
⊂W.
A rational maprfromV toW isdominantif (with the notation from above)R(U)is dense in W, i.e., if the smallest closed subset inW containingR(U)is equal toW.
A rational mapr:V →W isbirationalif there exists an inverse rational mapr :W →V such thatr◦ris equivalent toIdV andr◦ris equivalent toIdW.
If there exists a birational map fromV toW the varieties are calledbirationally equivalent.
Example 4.44 Consider the rational maps
rij :An→An, rij= (rij1, . . . , rnij),
54 Ch. 4 Background on Curves and Jacobians
where (forij)
rijk(x1, . . . , xn) :=
⎧⎪
⎪⎨
⎪⎪
⎩
xk/xj, k < i 1/xj, k=i xk−1/xj, i < kj xk/xj, j < k.
The casei > jworks just the same. For fixedjand arbitraryithe mapsrijare defined onDxj. Using the embeddingsφiofAnintoPnone has the description
rij =φ−j1◦φi.
The inverse map is justrjiand sorijrepresents a birational map regular onDxj∩Dxi. It describes the coordinate transition of affine coordinates with respect toφi to affine coordinates with respect toφjonPn.
Proposition 4.45 Assume that the rational maprfromV toWis dominant. Then the composition ofrwith elements inK(W)induces a field embeddingr∗ofK(W)intoK(V)fixing elements in K, generalizing the definition made for morphisms in Remark 4.37.
Ifris birational thenK(V)is isomorphic toK(W)asK-algebra.
Example 4.46 A projective curveC corresponds to a function field of transcendence degree1.
SinceK is perfect, there are elements x1, x2 ∈ K(C)and an irreducible polynomialf(x1, x2) such thatK(C) = Quot
K[x1, x2]/
f(x1, x2)
. Hence,C(and every affine part of dimension 1 ofC) is birationally equivalent to the plane curve V(f) and of course to its projective closure V(f)⊂P2.
Example 4.47 We consider again the Frobenius morphismφpfrom Example 4.39. The map φ∗p:K
φp(V)
→K(V)
has as its imageK(V)psince the coordinate functions ofV are exponentiated bypunder the map φp.
4.2.3 Regular functions
We continue to assume thatV is an affine variety.
Definition 4.48 A rational functionf ∈K(V)isregular at a pointP ∈V iffhas as representative a rational mapf˜with set of definitionU containingP.
In other wordsf is regular atP ∈V if there is an open neighborhoodUofPwheref|U = (g/h)|U forg, h∈K[x]andP ∈Dh. If this is the case we say thatf is defined atP with valuef(P) = g(P)/h(P).
Definition 4.49 For two varietiesV ⊂An, W ⊂Ama rational mapr:V →Wis regular atPif there is a nonempty open setU ofV containingPsuch that the restriction ofrtoUis given by an m-tuple of rational maps defined onU.
In other words: a mapr is regular if locally it can be represented viam-tuples of quotients of polynomials inK[x]which are defined atP∈U.