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at P is TP.V / D fP 0 2 V.kŒ"/ j P 0 7! P under V.kŒ"/! V.k/g: Thus an element of TP.V / is a homomorphism of k-algebras ˛W kŒV! kŒ" whose "7!0! k is the point P. To say that kŒV! k is the point P means that composite with kŒ" its kernel is mP, and so mP D ˛1.."//. PROPOSITION 4.26. Let V be an algebraic subset of A e... |
an embedding V,! A n, and let P map to.a1; : : : ; an/. Then the point.a1; : : : ; an/ C.b1; : : : ; bn/" n.kŒ"/ is an element of TP.V / (definition 4.25) if and only if.b1; : : : ; bn/ is an element of A of TP.V / (definition 4.22). PROPOSITION 4.27. Let V be an affine variety, and let P 2 V. There is a canonical iso... |
and the map f 7! df defD f f.m/ mod m2 is a k-derivation A! m=m2 because, mod m2, 0 D.f f.m//.g g.m// D fg C f.m/g.m/ C f.g g.m// C g.f f.m// D d.fg/ C f dg C g df: PROPOSITION 4.29. Let.A; m/ be as above. There are canonical isomorphisms Homlocal k-algebra.A; kŒ"/! Derk.A; k/! Homk-linear.m=m2; k/: c7!c! A PROOF. The... |
a derivation by composition f 7!df! m=m2! k: A 92 4. LOCAL STUDY Tangent spaces and differentials We now summarize the above discussion in the context of affine algebraic varieties. 4.30. Let V be an affine algebraic variety, and let P be a point on V. Write mP for the corresponding maximal ideal in kŒV and nP for the... |
). 4.31. A regular map 'W V! W defines a map '.kŒ"/W V.kŒ"/! W.kŒ"/. If Q D '.P /, then'maps the fibre over P to the fibre over Q, i.e., it defines a map d'W TP.V /! TQ.W /: This map of tangent spaces is called the differential of'at P. (a) When V and W are embedded as closed subvarieties of A p. 89. TP.V / V.kŒ"/ d''T... |
/. Obviously, CP.V / TP.V /. CAUTION. If a is principal, say a D.F /, then a D.F/, but if a D.F1; : : : ; Fr /, then it need not be true that a D.F1; : : : ; Fr/. Consider for example a D.XY; XZ C Z.Y 2 Z2//. One can show that this is an intersection of prime ideals, and hence is radical. As the polynomial Y Z.Y 2 Z2/ ... |
ine algebraic variety V and P 2 V, we define the geometric tangent cone P / by its P /red is the quotient of gr. CP.V / of V at P to be Spm.gr. nilradical, and we define the tangent cone to be.CP.V /; gr. P /red/, where gr. P //. O O O As in the case of a curve, the dimension of the geometric tangent cone at P is the s... |
ible variety of dimension d, and let P be a nonsingular point on V. Then there exist d regular functions f1; : : : ; fd defined in an open neighbourhood U of P such that P is the only common zero of the fi on U. PROOF. Suppose that P is nonsingular. Let f1; : : : ; fd generate the maximal ideal nP in P. Then f1; : : : ... |
nonsingular. O A point P on V is nonsingular if and only if there exists an open affine neighbourhood U of P and functions f1; : : : fd on U such that.f1; : : : ; fd / is the ideal of all regular functions on U zero at P. THEOREM 4.37. The set of nonsingular points of an affine algebraic variety is dense and open. PRO... |
lynomials F ; @F @X1 ; : : : ; @F @Xd C1 ; and so it will be proper unless the polynomials @F=@Xi are identically zero on V. As in the proof of 4.7, if @F=@Xi is identically zero on V.F /, then it is the zero polynomial, and so F is a polynomial in X1; : : : ; Xi 1; Xi C1; : : : Xd C1 (characteristic zero) or in X1; : ... |
in the singular set of the singular set.3 The intersection of the surface with the surface Y D c is the cuspidal curve X 2 D Z3=c: 3, and let W be the zero set of 4.43. Let V be the union of the coordinate axes in A XY.X Y / in A 2. Each of V and W is a union of three lines meeting at the origin. Are they isomorphic a... |
terms of its points functor. For example, let Mn be the set of n n matrices, and let I be the identity matrix. Write e for I when it is to be regarded as the identity element of GLn. 4.46. A matrix I C "A has inverse I "A in Mn.kŒ"/, and so lies in GLn.kŒ"/. In fact, Te.GLn/ D fI C "A j A 2 Mng'Mn.k/: 4.47. On expandi... |
terms of left invariant derivations), and G 7! Lie.G/ is a functor from the category of linear group varieties to that of Lie algebras. This functor is not fully faithful, for example, every ´etale homomorphism G! G0 defines an isomorphism Lie.G/! Lie.G0/, but it is nevertheless very useful. Assume that k has characte... |
Show that P is a nonsingular point on each irreducible component of V \ H on which it lies. (Each irreducible component has codimension 1 in V — you may assume this.) Give an example with H TP.V / and P singular on V \ H. Must P be singular on V \ H if H TP.V /? 4-3. Given a smooth point on a variety and a tangent vec... |
W 2 D 0. Let V be an affine algebraic variety over k. Show that the elements of V.RW / defD Homk-algebra.kŒV ; RW / are in natural one-to-one correspondence with the pairs.P; t / with P 2 V and t 2 W ˝ TP.V / (cf. Mumford, Lectures on curves..., 1966, p25). CHAPTER 5 Algebraic Varieties An algebraic variety is a ringe... |
jU / O O V / is an algebraic prevariety over k if there exists a finite open Thus, a ringed space.V; covering V D S Vi such that.Vi ; V jVi / is an affine algebraic variety over k for all i. An algebraic variety will be defined to be an algebraic prevariety satisfying a certain separation condition. O O An open subset... |
the map.a0W : : : W an/ 7! a0 ai ; : : : ; bai ai ; : : : ; an ai W Ui ui! A n (the term ai =ai is omitted) is a bijection. In Chapter 6 we shall show that there is a unique n for which each Ui is an open affine structure of a (separated) algebraic variety on P subvariety of P n and each map ui is an isomorphism of al... |
'j'1.U / \ Vj i is regular for each j; i (this is our assumption). It follows that f ı'is regular on '1.U / (sheaf condition 3.1(c)). Thus'is regular. The converse is even easier. c. Algebraic varieties 101 ASIDE 5.5. A differentiable manifold of dimension n is locally isomorphic to an open subset of n. In particular,... |
closed subset of A Now xi ı '1 and xi ı '2 are regular functions on Z, and the set where '1 and '2 agree is Tn iD1 V.xi ı '1 xi ı '2/, which is closed. DEFINITION 5.7. An algebraic prevariety V is said to be separated if it satisfies the following additional condition: Separation axiom: for every pair of regular maps ... |
an affine k-algebra A to the k-algebra of regular functions on V. For any P 2 V, f 7! ˛.f /.P / is a k-algebra homomorphism A! k, and so its kernel '.P / is a maximal ideal in A. In this way, we get a map O 'W V! spm.A/ which is easily seen to be regular. Conversely, from a regular map 'W V! Spm.A/, we get a k-algebra... |
U be an open subset of V. Then U is a union of open affines, and it follows that V jU / is a variety, called an open subvariety of V. A regular map 'W W! V is an open.U; immersion if '.W / is open in V and'defines an isomorphism W! '.W / of varieties. O Closed subvarieties Let Z be a closed subset of V. A function f o... |
orders). 104 5. ALGEBRAIC VARIETIES Application PROPOSITION 5.14. A prevariety V is separated if and only if two regular maps from a prevariety to V agree on the whole prevariety whenever they agree on a dense subset of it. PROOF. If V is separated, then the set on which a pair of regular maps '1; '2W Z V agree is clo... |
Vi 2 (a). Define for all i. Again, one checks easily that V is a sheaf of k-algebras satisfying (b), and that it is the only such sheaf. O O O For the final statement, if each.Vi ; Vi / is a finite union of open affines, so also V /. Moreover, to give a map 'W V! W amounts to giving a family of maps is.V; 'i W Vi! W s... |
we have (underlying set of V W /'Mor.A'Mor.A'(underlying set of V / (underlying set of W /: 0; V W / 0; V / Mor.A 0; W / Hence, our problem can be restated as follows: given two prevarieties V and W, define on the set V W the structure of a prevariety such that (a) the projection maps p; qW V W V; W are regular, and (... |
bras, but with the arrows reversed. Because of the category antiequivalence (3.25), this shows that Spm.A ˝k B/ will be the product of Spm A and Spm B in the category of affine algebraic varieties once we have shown that A ˝k B is an affine k-algebra. PROPOSITION 5.17. Let A and B be k-algebras with A finitely generate... |
A ˝k B be such that i with the sets fb1; b2; : : :g 2; : : :g each linearly independent over k. For each maximal ideal m of A, we know i / D 0. Thus either all ˛˛0 D 0. As before, we can write ˛ D P ai ˝bi and ˛0 D P a0 i 1; b0 and fb0.P Nai bi /.P Na0 the ai 2 m or all the a0 i i / D 0 in B, and so either.P Nai bi / ... |
˝ a/ D 0:.because a 2 k/ Thus k0 ˝k k0 is not reduced, even though k0 is a field. (b) Let K be a finite separable extension of k and let ˝ be a second field containing k. By the primitive element theorem (FT 5.1), K D kŒ˛ D kŒX=.f.X//; for some ˛ 2 K and its minimal polynomial f.X/. Assume that ˝ is large enough to sp... |
second statement then follows by the argument on p. 105. (b) This follows from 5.17(b) and 2.27. COROLLARY 5.21. Let V and W be affine varieties. For every prevariety T, a map 'W T! V W is regular if p ı'and q ı'are regular. PROOF. If p ı'and q ı'are regular, then 5.20 implies that'is regular when restricted to any op... |
by patching the.Vi ; condition (5.15). Similarly, write W as a union of open affines W D S Wj. Then O V W D [ Vi Wj and the.Vi Wj ; W; O O V W / to be the variety obtained by patching the.Vi Wj ; Vi Wj /. O Vi Wj / satisfy the patching condition. Therefore, we can define.V V W just defined, V W becomes the PROPOSITION... |
'2/W Z! V V; z 7!.'1.z/; '2.z// is regular because its components '1 and '2 are regular (see p. 105). In particular, it is continuous, and so.'1; '2/1.V / is closed, but this is exactly the subset on which '1 and '2 agree. Conversely, V is the set on which the two projection maps V V! V agree, and so it is closed if V... |
˝k kŒU 0! kŒU \ U 0 is surjective; 5Recall that the topology on V V is not the product topology. Thus the statement does not contradict the fact that V is not Hausdorff. VΓϕvWϕ(v)(v,ϕ(v)) i. Fibred products 111 (c) the condition in (b) holds for the sets in some open affine covering of V. PROOF. Let U and U 0 be open ... |
0 is a sum of functions of the form P 7! f.P /g.P / with f and g regular functions on U and U 0. EXAMPLE 5.30. (a) Let V D P 5.3). Then U0 \ U1 D A U0 \ U1,! Ui are 1, and let U0 and U1 be the standard open subsets (see 1 X f0g, and the maps on rings corresponding to the inclusions f.X/ 7! f.X/W kŒX! kŒX; X 1 f.X/ 7! ... |
diagram: '0 V S W 0'V W S: The system.V S W; '0; 0/ has the following universal property: for any regular maps ˛W T! V, ˇW T! W such that '˛ D ˇ, there is a unique regular map.˛; ˇ/W T! V S W such that the following diagram T ˇ.˛; ˇ / V S W ˛ 0 V '0'W S commutes. In other words, Hom.T; V S W /'Hom.T; V / Hom.T;S/ Hom.... |
.B/'Spm.A ˝R B=N/; (25) where N is the ideal of nilpotent elements in A ˝R B. To prove this, note that for any algebraic variety T, Mor.T; Spm.A ˝R B=N//'Hom.A ˝R B=N; T.T // O T.T //'Hom.A ˝R B;'Hom.A; O T.T // O'Mor.T; Spm.A// Hom.R;OT.T // Mor.T;Spm.R// Hom.B; T.T // O Mor.T; Spm.B// (5.12).5.12). For the second iso... |
ŒU \ U 0 k.U /; where k.U / is the field of fractions of kŒU, and so k.U / is also the field of fractions of kŒU \ U 0 and of kŒU 0. Thus, attached to V there is a field k.V /, called the function field of V or the field of rational functions on V, which is the field of fractions of kŒU for any open affine U in V. The ... |
the elements x1 ˝1; : : : ; xd ˝1; 1˝y1; : : : ; 1˝ye are algebraically independent in kŒV ˝k kŒW. Obviously kŒV W is generated as a k-algebra by the elements xi ˝ 1, 1 ˝ yj, 1 i m, 1 j n, and all of them are algebraic over kŒx1; : : : ; xd ˝k kŒy1; : : : ; ye. Thus the transcendence degree of k.V W / is d C e. We ext... |
R-submodules of M and N, then M 0 ˝R N 0 is an Rsubmodule of M ˝R N. However, this is true if R is a field, because then M 0 and N 0 will be direct summands of M and N, and tensor products preserve direct summands. k. Dominant maps 115 (b) Proposition 5.36 becomes false if A n is replaced by an arbitrary affine variet... |
U; 'U / and.U 0; 'U 0/ are said to be equivalent if 'U and 'U 0 agree on U \ U 0. An equivalence class of pairs is called a rational map 'W V Ü W. A rational map'is said to be defined at a point v of V if v 2 U for some.U; 'U / 2 '. The set U1 of v at which'is defined is open, and there is a regular map '1W U1! W such ... |
an open subset U0 of U. Now '1.U0/ '! U0 is the required map. A rational (or regular) map 'W V Ü W is birational if there exists a rational map '0W W Ü V such that '0 ı'D idV and'ı '0 D idW as rational maps. Two varieties V and V 0 are birationally equivalent if there exists a birational map from one to the other. In ... |
smooth) if it lies on a single irreducible component W and dim TP.V / D dim W. A point P is nonsingular if and only if the local ring P is regular. The singular points form a proper closed subvariety, called the O singular locus. 5.43. A variety is nonsingular (or smooth) if every point is nonsingular. n. ´Etale maps ... |
AŒX=.f / D kŒX1; : : : ; Xn=.f1; : : : ; fr ; f /: The tangent spaces to W and V at.a; b/ and a respectively are the null spaces of the matrices 0 B B B B @.a/ : : : @f1 @X1 ::: @f1 @Xn :::.a/ 0 @fr @X1 @f @X1.a/.a/ : : : : : : @fr @Xn @f @Xn.a/.a/ 0 @f @X.a; bf1 @X1 ::: @fr @X1.a/ : : :.a/ : : : 1 C C A.a/ @f1 @Xn ::... |
ALGEBRAIC VARIETIES be ´etale at P 2 W, and assume V to be normal; then there exist a map '0W W 0! V 0 with kŒW 0 D kŒV 0ŒX=.f.X//, and a commutative diagram W'V U1 ´etale U2 U 0 1 ´etale U 0 2 W 0 '0 V 0 with all the U open subvarieties and P 2 U1. The failure of the inverse function theorem for the Zariski topology ... |
viewed as a Riemann surface, V.C/ consists of two sheets joined at a single point O. As a point on the surface moves around O, it shifts from one sheet to the other. Thus the true picture is more complicated. To get a section to ', it is necessary to remove a line in C from 0 to infinity, which is not closed for the Z... |
this becomes the statement: n! A n is an isomorphism. If we.a/ is never zero (for a 2 kn), then'has an inverse. never zero, implies that det @Pi @Xj @Pi @Xj is a nonzero constant (by the Null- stellensatz 2.11 applied to the ideal generated by det ). This conjecture, which is known as the Jacobian conjecture, has not ... |
map to.d˛/a is.dXi /o 7!.d Qfi /a. ˛ D. Qf1; : : : ; Qfd /W U 0! A The next lemma then shows that ˛ is ´etale on an open neighbourhood U of P. d is ´etale. LEMMA 5.54. Let W and V be nonsingular varieties. If ˛W W! V is ´etale at P, then it is ´etale at all points in an open neighbourhood of P. PROOF. The hypotheses i... |
; : : : ; 0/ 2 A d. PROOF. This is a restatement of the Proposition. ASIDE 5.56. Note the similarity to the definition of a differentiable manifold: every point P on a nonsingular variety of dimension d has an open neighbourhood that is also a “neighbourhood” of d. There is a “topology” on algebraic varieties for which... |
of nonsingular varieties of dimensions m and n respectively, and let P 2 V. If rank.TP.'// D n, then there exists a commutative diagram 'jUP.x1;:::;xm/7!.x1;:::;xn/ UP ´etale m A U'.P / ´etale n A 122 5. ALGEBRAIC VARIETIES in which UP and U'.P / are open neighbourhoods of P and '.P / respectively and the vertical map... |
U into A 1 to A 1 to V. p. Smooth maps DEFINITION 5.60. A regular map 'W V! W of nonsingular varieties is smooth at a point P of V if.d'/P W TP.V /! T'.P /.W / is surjective;'is smooth if it is smooth at all points of V. THEOREM 5.61. A map 'W V! W is smooth at P 2 V if and only if there exist open neighbourhoods UP a... |
rank theorem. q. Algebraic varieties as functors Let R be an affine k-algebra, and let V be an algebraic variety. We define a point of V with coordinates in R (or an R-point of V ) to be a regular map Spm.R/! V. For example, if V D V.a/ A n, then V.R/ D f.a1; : : : ; an/ 2 Rn j f.a1; : : : ; an/ D 0 all f 2 ag; which ... |
algebra R, such that for every homomorphism ˛W R! S of affine k-algebras, rhe diagram '.R/ V.R/ W.R/ V.˛/ V.ˇ / '.S/ V.S/ W.S/ (*) commutes. Every family of maps with this property arises from a unique morphism of algebraic varieties. Let Vark (resp. Affk) denote the category of algebraic varieties over k (resp. affine... |
n and some finite set S of polynomials in kŒX1; X2; : : : ; Xn, G is isomorphic to the functor sending R to the set of zeros of S in Rn. PROOF. Certainly an affine group variety defines such a functor. Conversely, the conditions imply that G D hV for an affine algebraic variety V (unique up to a unique isomorphism). T... |
ine subvariety of U U and the maps R U defined by the projections are open immersions; (b) the set R.k/ is an equivalence relation on U.k/, and the map U.k/! F.k/ realizes F.k/ as the quotient of U.k/ by R.k/. PROOF. Let F D hV for V an algebraic variety. Choose a finite open affine covering V D S Ui of V, and let U D ... |
be an ideal in kŒX1; : : :. If A has no nonzero nilpotent elements, then every k-algebra homomorphism kŒX1; : : :! A that is zero on a is also zero on rad.a/, and so Homk.kŒX1; : : :=a; A/'Homk.kŒX1; : : :=rad.a/; A/: This is not true if A has nonzero nilpotents. 126 5. ALGEBRAIC VARIETIES The functor defined by A.E/ ... |
Gabriel, Groupes alg´ebriques: g´eom´etrie alg´ebrique, g´en´eralit´es, groupes commutatifs. 1970. r. Rational and unirational varieties DEFINITION 5.72. Let V be an algebraic variety over k. (a) V is unirational if there exists a dominant rational map P (b) V is rational if there exists a birational map P n Ü V: n Ü ... |
. This led him to introduce “abstract” algebraic varieties, neither affine nor projective (in 1946). Weil first made use of the Zariski topology when he introduced fibre spaces into algebraic geometry (in 1949). For more on this, see my article: The Riemann hypothesis over finite fields: from Weil to the present day. E... |
; : : : ; an/ D c.b0; : : : ; bn/ for some c 2 k: Let.a0 W : : : W an/ denote the equivalence class of.a0; : : : ; an/, and let denote the map knC1 X f.0; : : : ; 0/g! P n: Let Ui be the set of.a0 W : : : W an/ 2 P n such that ai ¤ 0, and let ui be the bijection.a0W : : : W an/ 7! a0 ai ; : : : ; bai ai ; : : : ; an ai... |
: : : ; Xnd of kŒX0; : : : ; Xn, and kŒX0; : : : ; Xn D M d 0 kŒX0; : : : ; Xnd I 129 130 6. PROJECTIVE VARIETIES in other words, every polynomial F can be written uniquely as a sum F D P Fd with Fd homogeneous of degree d. Let P D.a0 W : : : W an/ 2 P n. Then P also equals.ca0 W : : : W can/ for any c 2 k, and so we ... |
we can speak of the zeros of (26) with coordinates in Q. They also form a group E.Q/, which Mordell showed to be finitely generated. It is easy to compute the torsion subgroup of E.Q/, but there is at present no known algorithm for computing the rank of E.Q/. More precisely, there is an “algorithm” which works in prac... |
) V.0/ D P (b) V.ab/ D V.a \ b/ D V.a/ [ V.b/I (c) V.P ai / D T V.ai /. PROOF. For the second statement in (a), note that V.a/ D ; ” V aff.a/ f.0; : : : ; 0/g ” rad.a/.X0; : : : ; Xn/ (strong Nullstellensatz 2.16). The remaining statements can be proved directly, as in (2.10), or by using the relation between V.a/ and ... |
fnonempty closed cones in knC1g V I fproper graded radical ideals in kŒX0; : : : ; Xng Here the top map sends S to the affine cone over S, and the maps V and I are in the sense of projective geometry and affine geometry respectively. The composite of any three of these maps is the identity map, which proves the first ... |
each polynomial f.X1; : : : ; Xn/, we attach the homogeneous polynomial of the same degree f.X0; : : : ; Xn/ D X deg.f / 0 X1 X0 f ; : : : ; Xn X0 ; and to each homogeneous polynomial F.X0; : : : ; Xn/, we attach the polynomial F.X1; : : : ; Xn/ D F.1; X1; : : : ; Xn/: PROPOSITION 6.5. Each subset Ui of P we endow it ... |
0/. Let U be a nonempty open subset of P n; then U \ Ui is open in Ui. For some i, U \ Ui is nonempty, and so must meet Ui \ Uj. Therefore U meets every Uj, and so is dense in every Uj. It follows that its closure is all of P n. c. Closed subsets of An and Pn We identify A n with U0, and examine the closures in P n of... |
X1; X 2 1 C X2/ D f.0; 0/gI then V.X0X1; X 2 1 (which is contained in H1).1 C X0X2/ consists of the two points.1W 0W 0/ (the closure of V ) and.0W 0W 1/ 6.10. For V D H1 D V.X0/, we have V D ; D V.1/ and.V/ D ; ¤ V. 1Of course, in this case a D.X1; X2/, a D.X1; X2/, and V D f.1W 0W 0/g, and so this example doesn’t cont... |
, and there is We can replace U0 with Un in the above discussion, and write P n D Un t H1 with H1 D f.a0W : : : W an1W 0/g, as in Example 6.1. Note that in this example the point at infinity on the elliptic curve Y 2 D X 3 C aX C b is the intersection of the closure of any vertical line with H1. e. Pn is an algebraic v... |
Œ X1 X0 U0; and U0 with Spm ; X2 X0 kŒ X1 X0 ; : : : ; Xn X0 ; : : : ; Xn X0. becomes identified with the ring of regular functions on Next consider the open subset of U0; U01 D f.a0 W : : : W an/ j a0 ¤ 0, a1 ¤ 0g: /, and is therefore an affine subvariety of.U0; It is D. X1 X0 corresponds to the inclusion of rings kŒ ... |
0 X1 /, and is therefore an affine 1/, and the inclusion U01,! U1 corresponds to the inclusion of rings ; X1 X0 subvariety of.U1; O kΠX0,! kΠX0. An element f. X0 ; : : : ; Xn ; : : : ; Xn X1 X1 X1 X1 X1 defines the function.a0 W : : : W an/ 7! f. a0 ; : : : ; an / on U01. a1 a1 ; X1 and kΠX0 of k.X0; X1; : : : ; Xn/... |
curve C W Y 2Z D X 3 and assume that char.k/ ¤ 2. For each a 2 k, there is an automorphism 'a! C:.x W y W z/ 7!.ax W y W a3z/W C Patch two copies of C A 1 together along C.A 1 f0g/ by identifying.P; a/ with.'a.P /; a1/, P 2 C, a 2 A 1 X f0g. One obtains in this way a singular surface that is not quasiprojective (see H... |
� G, H homogeneous of the same degree [ f0g: Write k.X0; : : : ; Xn/0 for this field (the subscript 0 is short for “subfield of elements of n/ D k.X0; : : : ; Xn/0. Note that for F D G degree 0”), so that k.P H in k.X0; : : : ; Xn/0;.a0 W : : : W an/ 7! G.a0; : : : ; an/ H.a0; : : : ; an/ W D.H /! k, is a well-defined ... |
is homogeneous of degree 0. LEMMA 6.14. Each element of khomŒV can be written uniquely in the form with fi homogeneous of degree i. f D f0 C C fd PROOF. Let F represent f ; then F can be written F D F0 C C Fd with Fi homogeneous of degree i ; when read modulo p, this gives a decomposition of f of the required type. Su... |
and denominator are multiplied by cdeg.g/ D cdeg.h/. We can write f in the form g h in many different ways,2 but if then f D g h D g0 h0 (in k.V /0), gh0 D g0h (in khomŒV ) and so g.a0; : : : ; an/ h0.a0; : : : ; an/ D g0.a0; : : : ; an/ h.a0; : : : ; an/: Thus, if h0.P / ¤ 0, the two representations give the same val... |
let U 0 be the union of all the lines through the origin that meet U, that is, U 0 D 1.U /. Then U 0 is again open in knC1 X foriging, because U 0 D S cU, c 2 k, and x 7! cx is an automorphism of knC1 X foriging. The complement Z of U 0 in knC1 X foriging is a closed cone, and the proof of (6.3) shows n; but.U / is th... |
; : : : ; bm// for all points.b0 W : : : W bm/ in some neighbourhood of P in V.a/. PROOF. Straightforward. EXAMPLE 6.21. We prove that the circle X 2 CY 2 D Z2 is isomorphic to P 1. This equation can be rewritten.X C iY /.X iY / D Z2, and so, after a change of variables, the equation of the circle becomes C W XZ D Y 2.... |
i Xi Xj P ci Xi for which cj ¤ 0 can be omitted. For a fixed P D.a0W : : : W an/ 2 P n, the set of c D.c0W : : : W cn/ such that Lc.P / defD X ci ai ¤ 0 is a nonempty open subset of P n (n > 0). Therefore, for any finite set S of points of P n, fc 2 P n j S D.Lc/g n is irreducible). In particular, S is contained in is ... |
/: D 1, which is correct. A general F.X0; X1; : : : ; Xn/ D F1.X1; : : : ; Xn/ C X0F2.X0; X1; : : : ; Xn/ with F1 homogeneous of degree m and F2 homogeneous of degree m 1. But mCn m D mCn1 m C mCn1 m1 because they are the coefficients of X m in.X C 1/mCn D.X C 1/.X C 1/mCn1; and this proves the induction. 142 6. PROJEC... |
that is regular. 2, because kŒX0; : : : ; Xn is a unique factorization domain, We shall see in the next chapter that the image of any projective variety under a regular n/ is defined by the system of map is closed, but in this case we can prove directly that.P equations: bi0:::inbj0:::jn D bk0:::knb`0:::`n; ih C jh D ... |
: n, defines an isomorphism of the hypersurface Thus for any closed subvariety W of P section W \ H of V onto the hyperplane section.W / \ L of.W /. This observation often allows one to reduce questions about hypersurface sections to questions about hyperplane sections. As one example of this, note that maps the comple... |
will contain n exactly deg.F / points. Thus, the hyperplanes are exactly the closed subvarieties H of P such that (a) dim.H / D n 1; (b) #.H \ L/ D 1 for all lines L not contained in H. These are geometric conditions, and so any automorphism of P hyperplanes. But on an open subset of P n, such an automorphism takes th... |
7!.x0W W xn). Somewhat surprisingly, there are surjective regular n! P n. However, there is a surjective regular map A n. Consider the map n! A.x0W : : : W xn/ 7!.x2 0 W W x2 n/W P n! P n: It is mW 1 with m > 1 except over the points.0W W 1W W 0/. If H is a general hyperplane avoiding these points, then P n. For examp... |
. If Mm is of constant rank r, then we say that M has rank r. See CA 12. Let P n.R/ D fdirect summands of rank 1 of RnC1g. Then P n is a functor from k-algebras to sets. When K is a field, every K-subspace of KnC1 is a direct summand, and so P n.K/ consists of the lines through the origin in KnC1. Let Hi be the hyperpl... |
S/ in P n d 1. m. Grassmann varieties 147 PROPOSITION 6.29. The map S 7! P.S/W Gd.E/! P closed subset of P 1 n d. n d 1 is injective, with image a n d 1 We give the proof below. The maps P defined by different bases of E differ by an automorphism of P, and so the statement is independent of the choice of the basis — la... |
0 ”.s; s0/ 2 S: Conversely, the graph of any homomorphism S0! S 0 lies in Gd.V /S 0. Thus, Gd.V /S 0 Hom.S0; S 0/ Hom.E=S 0; S 0/: (27) The isomorphism Gd.V /S 0 Hom.E=S 0; S 0/ depends on the choice of S0 — it is the element of Gd.V /S 0 corresponding to 0 2 Hom.E=S 0; S 0/. The decomposition E D S0 ˚ S 0 gives a deco... |
, e 2 E. The elements of Vd E are called (exterior) d -vectors:The exterior algebra of E is a finite-dimensional graded algebra over k wedge with V0 E D k, V1 E D E; if e1; : : : ; en form an ordered basis for V, then the n products d 0 d ei1 ^ : : : ^ eid.i1 < < id / form an ordered basis for Vd E. In particular, Vn E... |
ive. For each i, 1 i d, let n d 1 e0 i D ei C P d <j n aij ej (30) denote the unique element of S projecting to ei. Then e0 d is a basis for S. Conversely, for any.aij / 2 kd.nd /, the e0 i defined by (30) span an S 2 Gd.E/S 0 and project to the ei. Therefore, S $.aij / gives a one-to-one correspondence Gd.E/S 0 $ kd.n... |
35. Let w be a nonzero d -vector and let M.w/ D fv 2 E j v ^ w D 0gI then dimk M.w/ d, with equality if and only if w is pure. PROOF. Let e1; : : : ; em be a basis of M.w/, and extend it to a basis e1; : : : ; em; : : : ; en of V. Write w D X ai1:::id ei1 ^ : : : ^ eid ; ai1:::id 2 k. 1i1<:::<id If there is a nonzero t... |
d.E/ D P.Vd E/ \ W: 150 6. PROJECTIVE VARIETIES with Ei a subspace of E of dimension di. The map Gd.E/ F 7!.E i /! Q i Gdi.E/ Q i P.Vdi E/ realizes Gd.E/ as a closed subset8 Q i Gdi.E/, and so it is a projective variety, called a flag variety. The tangent space to Gd.E/ at the flag F consists of the families of homomor... |
ities. PROOF. Decompose C and D into their irreducible components. Clearly it suffices to prove the theorem for each irreducible component of C and each irreducible component of D. We can therefore assume that C and D are themselves irreducible. We know from 2.62 that C \ D is of dimension zero, and so is finite. After... |
; ˛mn be the roots of R (some of them may be multiple). Each such root can be written ˛i D bi, and R.ai ; bi / D 0. According to ai 7.28 this means that the polynomials F.ai ; bi ; Z/ and G.ai ; bi ; Z/ have a common root ci. Thus.ai W bi W ci / is a point on C \ D, and conversely, if.a W b W c/ is a point on C \ D (s... |
the second half of the twentieth century. o. Hilbert polynomials (sketch) Recall that for a projective variety V P n, khomŒV D kŒX0; : : : ; Xn=b D kŒx0; : : : ; xn; where b D I.V /. We observed that b is graded, and therefore khomŒV is a graded ring: khomŒV D M m0 khomŒV m; where khomŒV m is the subspace generated by... |
the map A space of dimension d m C 1, and so 2! A P.V; T / D d T C 1: Thus V has dimension 1 (which we certainly knew) and degree d. Macaulay knows how to compute Hilbert polynomials. REFERENCES: Hartshorne 1977, I.7; Harris 1992, Lecture 13. p. Dimensions The results for affine varieties extend to projective varietie... |
closed subvariety of V ; if codim.Z/ D r, then there exist homogeneous polynomials f1; : : : ; fr in kŒX0; : : : ; Xn such that Z is an irreducible component of V \ V.f1; : : : ; fr /. PROOF. Use the same argument as in the proof 3.47. PROPOSITION 6.46. Every pure closed subvariety Z of P i.e., I.Z/ D.f / for some f h... |
a hyperplane H containing none of the nonzero Ei ; consequently, H contains none of the irreducible components Vi of V, and so each Vi \ H is a pure variety of dimension r 1 (or is empty). By induction, there is an linear subvariety E0 not meeting V \ H. Take E D E0 \ H. LEMMA 6.49. Let W be a vector space of dimensio... |
an/. Then ˛ ı PROOF. Let W A is regular, and there exist polynomials F0; : : : ; Fm 2 kŒX0; : : : ; Xn such that ˛ ı is the map.a0; : : : ; an/ 7!.F0.a/ W : : : W Fm.a//: As ˛ ı factors through P n, the Fi must be homogeneous of the same degree. Note that ˛.a0 W : : : W an/ D.F0.a/ W : : : W Fm.a//: If m < n and the F... |
: : : ; Xn, f ˝ g is the function.v; a/ 7! f.v/ g.a/W V A nC1! k: n, a 2 A, has degree P ij — The ring B has an obvious grading — a monomial aX i0 and so we have the notion of a graded ideal b B. It makes sense to speak of the zero set V.b/ V P n of such an ideal. For any ideal a A, aB is graded, and V.aB/ D V.a/ P n.... |
.b/ is prime. Exercises 6-1. Show that a point P on a projective curve F.X; Y; Z/ D 0 is singular if and only if @F=@X, @F=@Y, and @F=@Z are all zero at P. If P is nonsingular, show that the tangent line at P has the (homogeneous) equation.@F=@X/P X C.@F=@Y /P Y C.@F=@Z/P Z D 0. Verify that Y 2Z D X 3 C aXZ2 C bZ3 is n... |
JECTIVE VARIETIES 6-7. Write 0, 1, 1 for the points.0W 1/,.1W 1/, and.1W 0/ on P (a) Let ˛ be an automorphism of P 1 such that 1. ˛.0/ D 0; ˛.1/ D 1; ˛.1/ D 1: Show that ˛ is the identity map. (b) Let P0, P1, P2 be distinct points on P 1. Show that there exists an ˛ 2 PGL2.k/ such that ˛.0/ D P0; ˛.1/ D P1; ˛.1/ D P2: ... |
map is compact, and hence is closed if the image space is Hausdorff. Moreover, a Hausdorff space V is compact if and only if, for all topological spaces T, the projection map qW V T! T is closed, i.e., maps closed sets to closed sets (see Bourbaki, N., General Topology, I, 10.2, Corollary 1 to Theorem 1). a. Definitio... |
of complete varieties are complete. Let V1; : : : ; Vn be complete varieties, and let T be a variety. The projection Q is the composite of the projections i Vi T! T V1 Vn T! V2 Vn T!! Vn T! T; all of which are closed. 7.6. If 'W W! V is surjective and W is complete, then V is complete. Let T be a variety, and let Z be... |
.) This proves the first statement, and the second follows from the first applied to the identity map. n! A 7.11. In order to show that a variety V is complete, it suffices to check that qW V T! T is a closed mapping when T is affine (or even an affine space A Every variety T can be written as a finite union of open af... |
! S is proper, and W is a closed subvariety of V, then W '! S is proper. PROPOSITION 7.18. A composite of proper maps is proper. PROOF. Let V3! V2! V1 be proper maps, and let T be a variety. Consider the diagram V3 V2 V1 V3 V2.V2 V1 T /'V3 V1 T closed V2 V1 T closed T: Both smaller squares are cartesian, and hence so a... |
in Theorem 7.31 below. THEOREM 7.22. A projective variety is complete. n itself; thus we n W! W is a closed mapping in the case that W PROOF. After 7.3, it suffices to prove the Theorem for projective space P have to prove that the projection map P is an irreducible affine variety (7.11). Write p for the projection W ... |
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