Algebraic Geometry

By Solomon Lefschetz

This text for advanced undergraduate students is both an introduction to algebraic geometry and a bridge between its two parts — the analytical-topological and the algebraic. Because of its extensive use of formal power series (power series without convergency), the treatment will appeal to readers conversant with analysis but less familiar with the formidable techniques of modern algebra.
The book opens with an overview of the results required from algebra and proceeds to the fundamental concepts of the general theory of algebraic varieties: general point, dimension, function field, rational transformations, and correspondences. A concentrated chapter on formal power series with applications to algebraic varieties follows. An extensive survey of algebraic curves includes places, linear series, abelian differentials, and algebraic correspondences. The text concludes with an examination of systems of curves on a surface.

• Language: English
• Publisher: Dover Publications
• Release date: Sep 5, 2012
• ISBN: 9780486154725

Book preview

GEOMETRY

I. Algebraic Foundations

§1. PRELIMINARIES

1. The reader is expected to be familiar with the elementary concepts of modern algebra: groups, rings, ideals, fields, and likewise with the customary notations of the subject. Multiplication is supposed to be commutative throughout. To avoid certain awkward points appeal is made to the well known device of an all embracing field Ω which includes all the elements of rings,· · ·, under consideration.

(1.1) Notations. Aggregates such as x0, · · · , xn or α1, · · · , αm will often be written x or α, the range being generally clear from the context. Accordingly the ring or field extensions K[x0, · · · , xn] or K(α1 · · · , αm) will be written K[x] or K(α), with evident variants of these designations. Similarly for example for the functional notations: f(x) or φ(α.) for f(x0, · · ·, xn), or φ(α1 · · · , αm). In this connection the partial extensions K[x0, · · · , xr], K(α1, · · · , αs), will also be written Kr[x], Ks(α), with meaning generally clear from the context.

The following symbols of point-set theory will also be utilized throughout:

⊂: is contained in; ⊃: union; ∈ : is an element of.

(1.2) The groundfield. Very soon a certain fundamental field K, the groundfield will dominate the situation and all rings and fields will then be extensions of K. When K is of characteristic p the universal field Ω is also supposed to be of the same characteristic. The groundfield is always assumed to be infinite and perfect (irreducible polynomials have no multiple roots in an algebraic extension of K). Often also K is supposed to be algebraically closed (polynomials with coefficients in K have all their roots in K.

All rings will have a unit element and will always be integral domains (without zero divisors) and with unity element.

(1.3) Noetherian rings:

(a) Every sequence of distinct increasing ideals of : 2 · · · , is necessarily finite.

(b) Every ideal of has a finite base.

That is to say there is a finite set {α1, · · · , αn, the base of the ideal, such that every α satisfies a relation:

One refers to (a) as the ascending chain property, and to (b) as the Hilbert base property. And now:

(1.4) The ascending chain property and the Hilbert base property are equivalent.

A Noetherian ring is a ring which possesses one or the other of the two properties, and therefore both.

(1.5) The polynomial ring K[x] is Noetherian.

(1.6) Every ideal of a Noetherian ring and hence of K[x] admits a canonical decomposition = s where the i are primary ideals. If i is the prime ideal associated with i, then the i are all distinct and unique.

For a detailed treatment of the above properties see van der Waerden [1], II, Ch. XII.

(1.7) Homogeneous rings, ideals, and fields. By a form f(x0, x1 · · · , xn) is meant a homogeneous polynomial. Let the quantities x0, · · · , xn be such that the only relations between them are of type fα(x0, · · , xn) = 0, where the fα are forms with coefficients in KH of forms g ∈ K[x0, · · · , xn] such that if g, g′ H then: (a) gg′ H; (b) if moreover g and g′ have the same degree then g + g′ H H H as a homogeneous ring and denote it by KH[x0, · · · , xn]. Homogeneous ideals H H are defined in the usual way save that addition is again restricted to forms of equal degree. Homogeneous integral domains, Noetherian rings, prime ideals, primary ideals, are also defined in the usual way and properties (H of the same degree make up a subfield of K(x0, · · , xn), called a homogeneous field and written KH(x0, · · · , xn).

The extension to multiforms ƒ(···;xi;···) homogeneous separately in say n sets of variables · · · ; xi · · .

2. We shall now recall a certain number of properties of polynomials in indeterminates xi, referring mainly to their factorization.

(2.1) If f(x) =f(x1 · , xn) ∈ K[x1 · · , xn] and ƒ ≠ 0, there exists an infinite number of sets α = (α1 · · · , αn), αi ∈ K, such that f (α) ≠ 0.

(2.2) Factorization in the ring K[x1, · · · , xn] is unique to within a factor in K.

(2.3) Let f, g ∈ K[x1 · · · ,xn, y] = K[x; y]. If f is divisible by g in K(x1 · · · , xn)[y] = K(x)[y], i.e., when both are considered as polynomials in y with coefficients in K(x), and if g has no factor free from y (i.e. in K[x]) then f is divisible by g in K[x; y], or ƒ = gh, h ∈ K[x; y].

(2.4) Forms. The following properties may be stated: Let us describe the sets (α0, · · · , αn) and (β0, · · · , βη) of numbers of K as essentially distinct whenever not all the αi, nor all the βi are zero and there is no number p ∈ K such that βi = ραi, i = 0, 1, · · · , п. Then:

(2.5) Property (2.1) for n > 1 holds for forms when the infinite sets (α0, ··· , αn) under consideration are restricted to sets essentially distinct in pairs.

(2.6) The factorization properties (2.2) and (2.3) hold when all the polynomials are forms.

A polynomial or form f(x1 · · · , xn) of degree s is said to be regular in xi .

(2.7) Given a polynomial or form f(x0, · · · , xn) ∈ K[x0, , xn] it is always possible to find a non-singular linear transformation

which changes ƒ into a new polynomial or form g(y0, · · · , yn) regular in some or all the variables yj.

§2. RESULTANTS AND ELIMINATION

3. We shall recall some elementary properties of resultants and elimination theory. For further elaboration and proofs the reader is referred to treatises on algebra, and notably to van der Waerden, [1], II, Chapter XI, and E. Netto, [1], II.

Consider first two polynomials in one variable x:

where the ai, bj are indeterminates. The resultant R(f, g) is a doubly homogeneous form in the ai·, bj, be the associated doubly homogeneous rings of the ai, bi. Then the only pertinent facts as to the resultant are:

(3.1) R(f, g) is of degree n in the ai and m in the bj. One of its terms is

(3.2) R(f, g) is irreducible in every ring . (This is so-called absolute irreducibility.)

(3.3) There exist unique polynomials A and B of degrees at most n — 1 and m — 1 in x, with coefficients in such that

(3.4) Let the coefficients ai, bj and the roots ξi of f and ηj of g be elements of a field K. Then

(3.5) Let ƒ and g have their coefficients in a field K. If they have a common factor then R = 0. Conversely if R = 0 and a0 or b0 ≠ 0 then f and g have a common f actor .

Let now ƒ, g be forms of degrees m, n in x0, · · · , xr. Let R(f; g; xi) denote the resultant as to xi, i.e. as if ƒ and g were polynomials in xi. Then:

(3.6) Let f or g have indeterminate coefficients. Then R(f, g; xr) is a form of degree mn in x0, · · · , хr–1, and in (3.3a) A and B are forms in all the xi and of degrees ≤ n – 1 and m – 1 in xr. Moreover a n.a.s.c. in order that f, g ∈ KH[x0, · · ·, xr] both containing xr and one of them regular in xr have a common factor containing xr is that R(f, g; xr) = 0. When it exists the common factor is in KH[x0, · · , xr].

Consider now r + 1 forms in x0, · · , xr with indeterminate coefficients and let mi be the degree of fi and m = ∏mi. There exists a multiform Rh(f0, · · ·, fr) in the sets of coefficients of the fi, whose coefficients are integers, the resultant of the fi, and with the following properties:

(3.7) RH is of degree m/mi in the coefficients of fi.

(3.8) RH is absolutely irreducible in the ring of multiforms with integral coefficients.

(3.9) There take place identities

where the are multiforms with integral coefficients in the coefficients of the fj and in the xi.

(3.10) If one takes for the fi forms of KH[x0, · · , xr] then RH = 0 is a n.a.s.c. in order that the system

admit a solution with the хi not all zero and in .

(3.11) If is the highest degree term in xi of fi then RH contains a term .

More generally given any set of forms f0, · · · , fp with indeterminate coefficients there exists a resultant system RiH(f0, · · , fp), i = 1, 2, · · · , q where each RiH is an irreducible multiform such as RH above and now:

(3.12) Same as (3.10) with RiH = 0, i = 1, 2, · · · , q as the n.a.s.c.

§3. ALGEBRAIC DEPENDENCE. TRANSCENDENCY

4. Let Ф be a field over K. The elements α1, · · · ,αp of Ф are said to be algebraically dependent over K whenever they satisfy a relation P(α1 · · · , αp) — 0, Ρ(α1 · · · , αp) ∈ K[α1, · · · , αp]. If the term α1 is actually present in P we say that α1 is algebraically dependent on α2, · · , αp over K. As the groundfield K is generally clear from the context the mention "over K" is usually omitted.

A transcendence base {αi} for Ф over K is a set of elements of Ф such that: (a) no finite subset of the αi is algebraically dependent; (b) every element α

(4.1) If the number p of elements in one transcendence base is finite (only such cases arise in the sequel) then it is the same for all other such bases.

The number p of elements in a transcendence base is called the transcendency of Ф over K, written transcK Ф, or merely transe Ф when the particular K is clear from the context.

One may manifestly define the transcendency ρ over K of any set {α1 · · · , αs} of elements of Ф as the maximum number of elements which are algebraically independent over K. Let ψ = K(α1 · · · , αs), so that Ψ is a field between K and Ф. It is readily shown that transc Ψ = ρ.

(4.2) If K ⊂ L ⊂ Ф, where all three are fields then transeK Ф = transcL Ф + transcK L.

(4.3) Rational and homogeneous bases. These two concepts will be found very convenient later. Given a field L over K we will say that a set {αl, · · · , αn} of elements of L is a rational base for L over K whenever L = K(α1,· · ·, αn). A set of elements {β0, · · · , βτ} of some field ψ over L is known as a homogeneous base for L over K whenever

If βh ≠ 0 then this condition is equivalent to L = K({βi/βη}) where βh is now fixed.

(4.4) If L has a finite rational base then r = transcK L is finite.

Another noteworthy property is:

(4.5) Let K have zero characteristic. Then if {α1, · · · , αr} is a transcendence base for L over K and L is a finite extension of K(α1, · · · , αr) there exists an element β of L such that {α1, · · , αr, β} is a rational base for L. Hence if M is a field over L and α0, · · · , αr ∈ M are such that {αi/α0} is a transcendence base f or L and L is a finite extension of K({αi/α0}), there exists an αr+1 ∈ M with αr+1/α0 ∈ L such that {α0, · · · , αr+1} is a homogeneous base for L.

§4. EXTENSION OF THE GROUNDFIELD

5. In many questions arising naturally in the study of algebraic varieties it is necessary to replace the groundfield K by a finite pure transcendental extension, that is to say by an extension K(u1, ··· , ur) by a finite number of indeterminates.

= 1, consisting of all polynomials of the ring K[x] = K[x1, ···,xn≠ l.

has the base {f1(x), · , fr(x)}. Upon replacing K by any field L ⊃ K the fi will span in L[x* referred to as the extension . Let in particular L = K(u1 · · · , us) be an extension by indeterminates uj.. If f(x; u1,···, us) ∈ L[x] and disregarding a common denominator ∈ L, we may write

* is equivalent to: every ƒα ... β . If the extension is by a single variable u we will write

* has the following properties:

(5.2) It preserves the relations of inclusion, sum, intersection and product.

(5.3) If are a prime and a primary ideal then so are *.

(5.4) Iƒis the prime ideal of then * is the prime ideal of * and if p then *p *.

(5.5) The factorization into primary ideals is preserved.

Observe at the outset that it is sufficient to consider a simple extension K(u). Moreover since elements of the groundfield may be multiplied in without affecting our arguments we may always assume our polynomials to be polynomials in u also. Finally (5.2) and the derivation of (5.4), (5.5) from (5.2), (5.3) are elementary. Thus we only need to take up the proof of (5.3). The case of prime ideals is simple and indeed it reduces essentially to Eisenstein’s classical lemma. We consider it first.

prime and let

where ai, bj ∈ K[x]. Suppose now that ab *. We may manifestly suppress in a and b the terms whose coefficients ai bj . If as a consequence say a *. Suppose that neither a nor b reduces to zero. Thus we will have (5.6) with a0, b. Since ab * all the coefficients of the powers of u in ab . Hence a0bis prime one of the factors say a. This contradiction proves that, say in a*. Hence a * is prime.

is much more difficult. Following E. Snapper, its treatment will be made to rest upon a noteworthy lemma due to Dedekind.

(6.1) Lemma. Let a, b be as in (5.6) and let

If are the ideals of K[x] spanned respectively by the ai, bj, ck then nn.

n n. All that is necessary then is to show that the inclusion may be reversed.

nn+1 consists of all the products αj of any n + 1 of the coefficients ah. Now αj = αr0 αr1–1 ··· αrn−n, r0 < r1 < ··· < rn.. Let all the products αj be ordered lexicographically, and let us agree to set αi = 0 wherever i > m. Suppose also the elements of the base α0, α1, · · · , αs written in increasing order.

Now corresponding to αj above we may introduce the determinant

In its expansion αj, the diagonal term, is the term of highest order: δj = αj + terms αh preceding αj.·. Taking then successively j = s, s – 1, · · · , this relation will enable us to replace in succession, in the base {αj}, the elements αs, αs–1, · · · by δs, δs–1 · · ·. In other words Δ = {δjn+1.

Consider now the following system obtained by equating the powers of u in ab = c:

We may view (6.2) as a set of linear equations in the bi. The equations beginning with ar0, · · · , arп have δj for determinant of the b’s. Hence if δj ≠ 0, i.e. if δj· does figure in the base Δ, then the subsystem of the equations just mentioned yields relations

where the δji are minors of order n of δj n. Since {δjbknп+1 n. This completes the proof of the lemma.

Returning now to our main problem the proof of (primary is immediate. Let a, b ∈ K(u)[x] where b *. If c = ab then c and b nn nσ , then in aσ every coefficient of u σ and so aσ * is a primary ideal and this completes the proof of (5.3).

§5. DIFFERENTIALS (CHARACTERISTIC ZERO)

7. We shall find it convenient to organise differentiation with differentials and not derivatives in the central position. The treatment, largely following Ernst Snapper, is confined to a field Φ of finite transcendency over a groundfield K.

Let Φ be of transcendency n over K. It is referred to as a differential field whenever there is: (a) an n with Φ as its scalar domain; (b) an operation d: such that if α, β ∈ Φ and k ∈ K then:

I. d(α + β) = dα + dβ; II. dαβ = αdβ + βdα; III. dk = 0; IV. the dα, dβ.

is the space of the differentials of Φ over K, and dα. is the differential of α over K.

Immediate consequences of I, II, III are

If γ = α/β, then α = βγ, hence quickly from II:

If R(α1, · · · , αp) ∈ K(α1, · · · , αp) denote by Rαi. the usual partial derivative (taken as if the αi were indeterminates). Then:

(7.3) If F(x1, · · · , xp) ∈ K[x1, · · · , xp] where the xi are indeterminates, then Fxi = 0 is a n.a.s.c. for F not to contain xi.

(7.4) If R(α1, · · · , αp) ∈ K(α1, · · · , αp), αi ∈ Φ then dR = ∑Rαidαi

It is first proved for a polynomial then by means of (7.2) for any R.

(7.5) If = {α1, · · · , αn} is a transcendence base for Ф then = {dαi} is a linear base for .

If β ∈ Φ there is a relation

where F(α; x) is irreducible as an element of K(α)[x]. Owing to this it has no common factor with

whose degree < r, and hence Fβ ≠ 0. Applying (7.4) we find

and hence

Therefore d .

(7.8) The ordinary or partial successive derivatives of various orders are defined in the obvious way. We merely recall:

(7.9) Let f(x) ∈ K[x], x indeterminate. N.a.s.c. in order that c be an n-tuple root of f(x) are:

(7.10) Remark. Ordinary or partial derivatives of any order may be defined for a groundfield of any characteristic and the formal properties (7.3) and (7.9) continue to hold.

8. Construction of a system of differentials. Take for the dαi independent vectors and compute dβ for any β ∈ Φ by (7.7). This defines d obeying rules I, II, III over a simple extension Φβ = K(α; β). Let us show that it is unique over Φβ. An element γ of Φβ may have various representations

and we must show that

In the last analysis we must prove that if S(α; β) ∈ K(α;β) and S(α; β) = 0 then dS(α; β) = 0. This follows however by rule III.

Suppose now β ∈ Фγ. Since d is uniquely defined throughout Φγ, dβ is the same whether obtained as element of Φβ or of Фγ. Hence d is unique throughout Ф.

9. We shall now show that the system (d) is essentially unique. Let Ф, Ф′ be isomorphic fields over K under an isomorphism τ: Ф → Ф′ preserving Kand d′′ have their natural meaning. A differential isomorphism of onto ′ such that if V, Vand α ∈ Ф then

′ are differentially isomorphic. Hence in a given field differentiation is unique to within a differential isomorphism.

, β′, R′ for ταi, τβ, τR then under our rules d′β′ is given by (which can be done since {dαj.

10. (10.1) If α1, · · · , αK are algebraically independent elements of Ф then dα1, · · · , dαk are linearly independent elements of and conversely.

The algebraic independence of the αi, i ≤ k, implies k ≤ n = transe Ф. Hence one may then augment the set by elements αk+1, · · , αn= {αj}. Since the dαj, j ≤ n(7.5) the same holds for those with j ≤ k.

Conversely let dα1, · · · , dα. This implies k ≤ n . Suppose now α1, · · · αk algebraically dependent. Thus there is a relation F(α1, · · · , αk) = 0, F(α) ∈ K[α], where not every Fαi. = 0. Now F(α) = 0 implies

Hence the dαi are linearly dependent. This contradiction shows that α1, · · , αk. are algebraically independent.

(10.2) Noteworthy special case: If dα = 0, α ∈ Φ, then α is algebraic over K. Hence if K is algebraically closed α ∈ K. Thus if K is algebraically closed the elements of K are those and only those elements of Ф whose differentials are zero.

11. Derivatives and Jacobians= {α1, · · · , αn} be a transcendence base for Ф, and β ∈ Φ. Then dβ is given by (7.7) in terms of the dαi. The coefficient of dαi is the partial derivative of β relative to αι and written ∂β/∂αi. We have explicitly

If β1, · · · , βm are elements of Φ then

is the usual Jacobian matrix. If m = n then the determinant of (11.2) is the Jacobian determinant or functional determinant written

From the relations

and the fact that {dαj} is a linear base for the space of the differentials together with (10.1) follows:

(11.4) A n.a.s.c. for β1, · · · , βm ∈ Φ to be algebraically independent is that the Jacobian matrix (11.2) be of rank m, or equivalently for m = n that the Jacobian determinant be ≠ 0.

Suppose in particular that {βj} = {β, α1, · · · , αi–1, αi+1, · · · , αn}. Then the Jacobian determinant reduces to ± ∂β/∂αi.. Hence if ∂β/∂αj = 0, β and α1, · · · , αi–1, αi+1, · · · , αn are algebraically dependent. Since the αi are algebraically independent β is then algebraic over the field K(α1, · · , . αi–1, αi+1, · · · ,αn). Conversely when β has this property dβ spanned by the dαj, j ≠ i, hence ∂β/∂αi = 0.

Thus:

(11.5) A n.a.s.c. for β ∈ Φ to be algebraic over the field K(α1, · · , αi+1 αi+1, · · · , αn) is that ∂β/∂αi = 0.

12. Subfields. Let K ⊂ Ψ ⊂ Ф, Тa subfield of Ф. Let transe Ψ = m ≤ п. If {α1, · · · ,αm} is a transcendence base for Ψ one may find in Ф – ψ elements αm+1, · · · , αn such that {α1, · · · , α′ be the vector space based on {dα1,· · · , dαm′ is spanned by dψ where d is the differential operator for Φ. Thus

(12.1) A differentiation in Φ induces one in every field Ψ between K and Ф.

(12.2) Application to homogeneous fields. A homogeneous field ψ = Kh(α0, · · , αn) is defined in terms of an ordinary field Ф = K(α0, · · · , αn) whose elements only satisfy homogeneous equations with coefficients in K in such manner that Ψ = K(αi/αj}), αj·≠ 0. Thus K ⊂ Ψ ⊂ Ф and so one may apply (12.1).

One of the αj will be ≠ 0. Let their numbering be such that α0 ≠ 0. Then ψ = K({(αi/α0}). If transe Ψ = r, r of the αi/α0 will be algebraically independent and the rest will be algebraically dependent on these. Let again the numbering of the αi be such that the αi/α0, i ≤ r, are algebraically independent. For the rest we will have algebraic relations

with coefficients in K or after clearing fractions

where fi ∈ Kh[α0, · · · , αr, αr+j] and actually contains αr+j. Notice also that α0, · · , αr are algebraically independent. For under our assumptions a relation between them must be of the type

Now if p is the degree of g :

which is ruled out since the