Griffiths, P., and Harris, J. Principles of algebraic geometry. Wiley-interscience, London-New-York, 1978.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(2.4) along a path of k dimensional subspaces, it has to be restricted to decomposable k forms. We will give the details of this restriction for n = 4 and k = 2 and then mention aspects of the case for general k and n. A 2 form U 2 ) is decomposable if U U = 0 (cf. Griffiths Harris [17]) Expanding U in terms of the standard basis (2.7) of 0 = U U = U i U j i j = U 1 U 6 U 2 U 5 U 3 U 4 ) e 1 e 2 e 3 e 4 : 2.12) De ne I : C by I(U) U 1 U 6 U 2 U 5 U 3 U 4 : 2.13) For a path of the equation (2.4) with k = 2 and n = 4 to be a path of 2 ....

....ne I : C by I(U) U 1 U 6 U 2 U 5 U 3 U 4 : 2.13) For a path of the equation (2.4) with k = 2 and n = 4 to be a path of 2 dimensional subspaces, the function (2.13) has to be preserved. The surface de ned by I(U) 0 is G 2 (C ) the Grassmannian manifold of 2 planes in C (cf. [17]) Alternatively, the form of the invariant (2.13) can be derived using the identity, T = 2.14) where is a 6 6 symmetric orthogonal matrix associated with Hodge duality, de ned in equation (A.11) in Appendix A. The identity (2.14) can be veri ed by direct calculation using ....

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P. Griffiths & J. Harris, Principles of Algebraic Geometry, Wiley: New York, 1978.

....order to invoke results from intersection theory [6] it is important to understand the intersection at the boundary of L. What is needed is a good compactification of L. It turns out that Problem 1.1 induces in a natural way a compactification and we will explain this in the sequel. Recall [8, 9] that the Grassmannian Grass(m, K ) m q is the set of all m dimensional subspaces in K . For each X ) let X = span then # 2 is a vector in the exterior product = K m ) and #m = k(# 1 #m ) for some k 0 in K. Therefore by considering the components of ....

P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley & Sons, New York, 1978.

....Thus the geometric logarithmic derivative lemma implies that f # # for every holomorphic map f : C X. Because X is one dimensional, it easily follows in this case that f must be constant. Allowing logarithmic singularities, this method also works for the complement of 3 points in P . In higher dimensions, even in the case of Abelian varieties, the di#culty lies in concluding from f # # that f is algebraically degenerate. In this paper, we establish a non Archimedean version of the geometric logarithmic derivative lemma in characteristic zero. However, because the structure of the ....

.... A [r 1 , # i ] together with the analytic functions f i define an e#ective Cartier divisor D on A [r 1 , r 2 ) Depending on F and r 2 , we cannot necessarily represent D as the zeros of an analytic function on all of [r 1 , r 2 ) However, we can define N(D, r) by choosing i large enough that # i r and then setting N(D, r) N(f i , 0, r) The choice of f i make this well defined. As usual, we get the group of Cartier divisors by taking formal di#erences of e#ective Cartier divisors, and we extend the definition of N(D, r) to the full group of Cartier divisors in the natural ....

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P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley & Sons, 1978.

....Schubert variety X F q (s) where j j = 1 and s is not among fs 1 ; s n g. This gives a rational curve violating Proposition 1.1. Thus the intersection is zero dimensional. Let N( be its degree, which may be computed using the classical Schubert calculus of enumerative geometry [12]. Thus we deduce the following Corollary of Proposition 1.1. Corollary 1.2. Given d r, rami cation data , and distinct points s 1 ; s n 2 P , the number of nondegenerate rational curves in P at point s i for each 1 i n is N( counted with multiplicity. Let ....

P. Griffiths and J. Harris, Principles of Algebraic Geometry, J. Wiley and Sons, 1978.

....to GF , # S = # S GF : GF . 3) Then # S is a finite regular map of projective varieties. When F = C this map has a degree, which can be defined in this case as the number of preimages of a generic point and is independent of S . This degree was computed by Schubert in 1886 (see [12, 9, 10] for modern treatment) Theorem A When F = C, the degree of # S is d(m, p) 1 2 . p 1) mp) m (m 1) m p 1) 4) Projective duality implies that d(m, p) d(p, m) Here are some values of d(m, p) m = 2 3 4 5 6 7 8 p = 2 2 5 14 42 132 429 1430 p = 3 42 462 6006 ....

Ph. Gri#ths and J. Harris, Principles of algebraic geometry, Willey, NY, 1978.

....jth row has length #fi j a i = jg such that and the map j XwFq X w 0 w 0 Fq 0 : XwF q has degree 1. Lemma 11 is the surprising connection to the classical Pieri s rule that was mentioned in the Introduction. A typical geometric proof of Pieri s rule for Grassmannians (see [13, 15]) involves showing a triple intersection of Schubert varieties mG is transverse and consists of a single point, when G q ; G q are in suitably general position. We would like to construct a proof of Theorem 1 along those lines, studying a triple intersection of Schubert ....

P. Griffiths and J. Harris, Principles of Algebraic Geometry, Joseph Wiley and Sons, 1978.

....of the Pieri type formula that we complete in Section 3. Also needed is Lemma 2.1, which identifies a particular subspace of H he 1 ; e n i for c . For Lemma 2.1, we work in the (classical) Grassmannian G k (V : he 1 ; e n i. For basic definitions and results see any of [8, 5, 4]. Schubert c of G k (V ) are indexed by partitions oe 2 Y k , that is, integer sequences oe = oe 1 ; oe k ) with n Gamma k oe 1 Delta Delta Delta oe k 0. For oe 2 Y k define j = n Gamma k Gamma oe k 1 Gammaj . For oe; 2 Y k , define : fH 2 G k (V ) j dimH he ....

....formula that we complete in Section A.3. Also needed is Lemma A.2.1, which identifies a particular subspace of H hf 1 ; f n i for H 2 Y c . For Lemma A.2.1, we work in the (classical) Grassmannian G k (W : hf 1 ; f n i. For basic definitions and results see any of [8, 5, 4]. Schubert c of G k (W ) are indexed by partitions oe 2 Y k . For oe 2 Y k define j = n Gamma k Gamma oe k 1 Gammaj . For oe; 2 Y k , define : fH 2 G k (W ) j dimH hf k 1 Gammaj j ; f n i j; 1 j kg c : fH 2 G k (W ) j dimH hf 1 ; f j oe ....

P. Griffiths and J. Harris, Principles of Algebraic Geometry, J. Wiley and Sons, 1978.

....compactification (the Grothendieck quot scheme) of the space of holomorphic maps of a fixed degree from P to a Grassmannian. Partially supported by NSF grant DMS 9218215 and a Sloan fellowship In order to fix notation and refresh the reader s memory, we begin with an overview (following [GH]) of the classical Schubert calculus before continuing with the introduction. Let: V be a vector space over C of dimension n, 0 = V 0 ae V 1 ae : ae V n = V be a full flag for V , G : G(n Gamma k; n) be the Grassmannian of n Gamma k dim l subspaces of V , x ae V be the subspace ....

P. Griffiths and J. Harris, Principles of Algebraic Geometry, WileyInterscience (1978).

....of dimension n over the ring R, with basis either Sw (x) or Sw (x; y) w 2 S n , Z[x 1 ; x n : 2.5 Residue pairing. Let I be an ideal in P n = R[x 1 ; x n ] R ae C, generated by a regular system of parameters 1 ; n , and A : P n =I. 13 Proposition 6 ([GH], EL] ffl dimR A 1. ffl H : det i 62 I. Let d 0 : deg H, where we assume that deg x i = 1 for all 1 i n. Proposition 7 ( EL] P n and deg f = d 0 , then there exists a non zero ff 2 R such that f j H (mod I) P n , f 6= 0, and deg f d 0 , then there exists g 2 P ....

.... f d 0 , then choose g 2 P n such that g j f (mod I) and deg g d 0 , and define Res I (f) Res I (g) We will use also notation hfi I instead of Res I (f ) Finally, let us define a residue pairing h; i I on P n using the Grothendieck hf; gi I = Res I (f; g) f; g 2 P n : Proposition 8 ([GH]) ffl If f 2 I, then Res I (f) 0. ffl The residue pairing h; i I induces a non degenerate pairing on A = P =I. 14 We will use this general construction of residue pairing in the following two cases: i) R = Z, I n ae P n is an ideal generated by elementary symmetric polynomials e 1 (x) ....

Griffiths Ph., Harris J., Principles of algebraic geometry, John Wiley& Sons, New York, 1978.

....f)A. The sign of the index differs from the [Eells Wood] formulas, which use the complexified bundle map T X T instead of (2) If f is a generic perturbation of a holomorphic map, with ramification v(q) 1 at finitely many branch points q 2 X , then the Riemann Hurwitz theorem applies ( Griffiths Harris] X = deg f)A Gamma (v(q) Gamma 1) so the number of points where f is C linear is 2X (v(q) Gamma 1) It is well known that in this two dimensional case, f is conformal (anglepreserving) at points where df is C linear, and indirectly conformal (anglereversing) at points where df is ....

P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley-Interscience, New York, 1978.

....= 0 : m j = 0: This is impossible. If F extends on p j and (Y; g) is not complete. 0. 22) Let X (E) be the sheaf of the meromorphic forms having polar divisor contained in E: Recall that X (E) can be identified with the sheaf of the holomorphic sections of a line bundle on X (see [1] or [7]) which is usually still denoted by X (E) We may consider i 2 H (X; X (E) The (0.8) extends and by abuse of language we write h : X C P ; h(X) Q: Repeating the square root extraction: h = Vf; f : X C P is the extended Gauss map. Set M = f (O CP 1(1) M = X (E) We ....

.... M(N) fx 2 G(2; N) x) 0g: Let S be the dual of S: We identify as a section of : The result follows by computing the top Chern class: c 3f ) c 3 (Sym = 4 c 2 (S (c 1 (S 6= 0: In fact c 1 (S ) is ample and c 2 (S 6= 0 if f e : c 2 (S e = 1 (see [7] Ch. 3 x3) iii) Fix a half exact element s of N; s 6= 0: Set P = f Pi 2 G(2; N) s 2 Pg; P M = M(s;N) Now P is isomorphic to the projective space of the quotient of N by the space generated by s: Let O P ( Gamma1) be the tautological bundle of P and S P = S Omega O P the restriction of S ....

P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley Interscience, New York, 1978.

....(the growth here corresponds to the pole at p 2 2 Sigma which we have introduced in our convention for the Abel Jacobi map) Existence of the section follows from the fact that the g 2 fold Abel Jacobi map has degree one (this is the Jacobi inversion theorem , see for instance p. 235 of [9]) Uniqueness follows from the fact that a difference of two such would give an L section, showing that A 2 actually lies in the theta divisor, which we assumed it could not. We show the restrictions of OE T to the Sigma 2 side come close to approximating Phi 2 or, more precisely, that its ....

P. Griffiths and J. Harris. Principles of algebraic geometry. John Wiley & Sons, Inc., 1978.

....The results have applications to the graphical display of multidimensional data via Asimov s grand tour method. 1. INTRODUCTION Although there is a considerable literature dealing with Grassmannian spaces, exemplified by [Chow 1949; Leichtweiss 1961; Wong 1967; James and Constantine 1974; Griffiths and Harris 1978, Section 1.5; MacKenzie and Morgan 1995; Zanella 1995] the problem of finding the best packings in such spaces seems to have received little attention. We have made extensive computations on this problem, and have found a number of putatively optimal packings. These computations have led us to ....

P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, New York, 1978

....hence h(i) acts as ( Gamma1) l Gammap on V p . As Psi(v; h(i)v) is positive definite, Q is positive definite on V p if l Gamma p is even and negative definite otherwise. 1.9. Example. The cohomology groups H k (X; Q) as in 1. 2) have a polarization see [W] Th eor eme IV.7 and corollaire or [GH], p. 123. 2. The Hodge conjecture 2.1. Hodge cycles. The space of Hodge classes in a rational Hodge structure V of weight k is the Q subvector space: B(V ) 0 if k is odd; V V p;p if k = 2p: Note that V , VC = V Omega Q C; v 7 v Omega 1 and the intersection V V p;p takes ....

P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley and Sons, New York (1978).

....the conference Variedas abelianas y functiones theta for the oppertunity to present these lectures and for providing the pleasent working conditions in Morelia. 2 The Schottky problem Introduction. We recall the basic results on period matrices of Riemann surfaces. References are [ACGH] C] [GH] and [CGV] Then we briefly discuss modular forms, a reference is [Ig1] 1 December 6, 1999 2 2.1 Period matrices 2.1.1 Let C be a Riemann surface of genus g (we consider only compact Riemann surfaces in these lectures) On the homology group H 1 (C; Z) Z 2g there is an (alternating, ....

P. Griffiths and J. Harris, Principles of Algebraic Geometry. J. Wiley & Sons (1978).

....i ) i; 1 i rg: For example, when has a unique non zero part 1 = k, we get a special Schubert variety X k (V ffl ) fW 2 G r;s ; W V s 1 Gammak 6= 0g: We will use the notations Omega and X when the reference flag V ffl does not matter. The following facts are well known (see e.g. [10, 8, 19]) 1. The Schubert variety X is a closed subvariety of G r;s , defined locally by the vanishing of minors of the composite maps W ae C n C n =V i . 2. Let W 2 G r;s . The sequence dim (W V j ) goes from 0 to r, increasing at most by one at each step. So it increases strictly at exactly r ....

.... deduce the following fact: if and are partitions such that jj jj = rs, then Omega (V ffl ) and Omega (V 0 ffl ) meet transversely in a unique point if = where = s Gamma r ; s Gamma 1 ) is the complementary partition of , and have empty intersection otherwise (see [10], p. 198 or [19] 3.2.7) This implies that the cup product oe oe = ffi ; oe r Thetas ; where oe r Thetas is the class of a point. This means that the basis of the cohomology of the Grassmannian given by Schubert classes is, up to complementarity, self dual relatively to Poincar e duality. ....

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Griffiths P., Harris J.: Principles of algebraic geometry, Second edition, Wiley 1994.

.... : xu yv zw ts = 0 g: Note that we can rewrite the equation to obtain: T q P 3 = fp 2 (P 3 ) X i (ffl i (q) p) 0 g: 16 This implies that the lines in T q P 3 with X i = 0 (which form a linear line complex) are exactly the lines passing through the point ffl i (q) cf. [GH], p. 759 760) In particular, if T q P 3 S is a smooth quartic curve, and l i ; l 0 i are a pair of bitangents as before, then both lines have X i = 0 and thus they must intersect in ffl i (q) Let now p 2 T q P 3 ; p 6= ffl i (q) The condition that p 2 l i [ l 0 i is equivalent ....

....is a CIHS. It seems reasonable to expect that the CIHS defined by these H a i is actually Hitchin s system, but we could not establish that. 17 6 Quadratic Line Complexes 6.1 In this section we recall how the equations for the bitangents are determined. We summarize the results we need from [GH], Chapter 6 and follow [Hu] Let G ae P 5 be the Grassmannian of lines in P 3 , so G is viewed as a quadric in P 5 . For x 2 G we denote by l x the corresponding line in P 3 . For p 2 P 3 and h ae P 3 a plane we define oe(p) fx 2 G : p 2 l x g; oe(h) fx 2 G : l x ae hg: Both ....

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P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, 1978.

....1. 1. Generalities on rational normal scrolls Let F be a locally free sheaf of rank r on P 1 generated by its global sections and : P(F) P 1 be the natural projection. Let W ae P N = P(H 0 (P 1 ; F) be the r dimensional scroll such that OW (1) F . 3 It is well known (see [G H], pg.516) that F decomposes as F = Phi i O(a i ) where O denotes O P 1; moreover F is very ample if and only if a i 0 for all i. Clearly in this case W = P(F) Recall the Riemann Roch theorem for vector bundles on P 1 h 0 (W; OW (1) deg(F) rk(F) 2) or, in our notation, deg(W ) ....

P.A. Griffiths, J. Harris, Principles of Algebraic Geometry, Wiley--Interscience, New York, 1978.

....can be removed by dividing the polynomials f0 ; f1 ; f2 by the gcd of g1 ; gn . So we will only consider the relevant case where G is a local complete intersection of codimension 2. In the following we will use techniques from algebraic geometry and hence its language. We refer to [20] and [19] for complete references. We recall that OX is the sheaf of rings of regular functions over a variety (or a scheme) X and that, if L is a sheaf on X, H 0 (X; L) denotes the zeroth cohomology group of L, that is its global sections. 2.1 The residual resultant Hereafter X denotes the projective ....

Griffiths, P., and Harris, J. Principles of algebraic geometry. Wiley-interscience, London-New-York, 1978.

....i ( d 1 ; d 2 ; dK ] 7) In fact the explicit form of Todd polynomials is given in terms of elementary symmetric functions of exponents [d 1 ; d 2 ; dK ] That is why the arguments are put inside square brackets. One formal definition of Todd polynomials is given by the relation [37] det A det (I Gamma e GammatA ) Gamma1) n t Gamman ( X i Todd i (P 1 (A) P i (A) t i ) 8) 11 where A is n Theta n matrix with eigenvalues 1 ; n and P i (A) is the i th elementary symmetric function of the eigenvalues. The formal way to find the ....

....deformation quantization [41] especially in the presence of symmetry [42] is not yet fully explored. 4. 1 Topological quantum numbers for energy bands The formal topological classification of complex vector bundles over CPN can be done by using rang and Chern classes of complex vector bundles [38, 39, 40, 37, 36]. These Chern numbers play the role of topological quantum numbers for energy bands [35] From the point of view of practical calculations it is quite useful to characterize the vector bundle by the Chern polynomial 1 c 1 x c 2 x 2 c 3 x 3 Delta Delta Delta c s x s ; 12) where ....

P. Griffiths, J. Harris. Principles of Algebraic Geometry. (WileyInterscience, New York, 1978).

....subspaces of dimension p are real, that is defined by equations with real coe#cients, the d(m, p) subspaces of dimension m need not be real. So the problem of counting real solutions of (P) arises in this case. For the general background on Schubert calculus and Grassmann varieties we refer to [9, 10]. Let F be one of the fields R (real numbers) or C (complex numbers) We denote by GF (m, n) 1 # m # n 1, the Grassmannian, that is the set of all linear subspaces of dimension m in F n . Such subspaces can be described as row spaces of mn matrices K of maximal rank. Two such matrices K 1 ....

Ph. Gri#ths and J. Harris, Principles of algebraic geometry, Willey, NY, 1978.

....When the 2m given lines are real, that is defined by equations with real coe#cients, the u(m 1) subspaces of codimension 2 need not be real. So the problem of counting real solutions of (P) arises in this case. For the general background on Schubert calculus and Grassman varieties we refer to [8, 9]. Let F be one of the fields R (real numbers) or C (complex numbers) We denote by GF (m, n) 1 # m # n 1, the Grassmanian, that is the set of all linear subspaces of dimension m in F n . Such subspaces can be described as row spaces of mn matrices K of maximal rank. Two such matrices K ....

Ph. Gri#ths and J. Harris, Principles of algebraic geometry, Willey, NY, 1978.

....) extends analytically into the disk Delta(a; ae i ) and if N deg P Delta deg Q, then f is an analytic rational function with poles outside of Delta ae N : Proof. Since P and Q are assumed coprime, the intersection ZP ZQ consists of at most deg P Delta deg Q points by the Bezout theorem [GH]. By the condition for N there exists j = 1; N such that ae 2 j 6= z Gamma a) w Gamma a) for any (z ; w ) 2 ZP ZQ : Hence P (z; w) and Q(z; w) have no common zero on a;ae j = f(z Gamma a) w Gamma a) ae 2 j ; 0 jz Gamma aj ae j g so all the conditions of Theorem 3.1 ....

P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, 1978.

.... at each a 2 supp P , we have r 0 = ramification index of f n: Now let f k : C P 1 Gamma P( k 1 C m ) defined by f k = f f 0 Delta Delta Delta f (k) in C m be the k th associated curve of f , and use the Plucker formulas (an extension of the Riemann Hurwitz formula see [13]) which on C P 1 are Gammad k Gamma1 2d k Gamma d k 1 Gamma 2 = r k ; THE SPINOR REPRESENTATION OF SURFACES IN SPACE 21 where d k is the degree of f k , and r k is the ramification index of f k . In the table below, multiplying the numbers on the left by the inequalities on the right and ....

Griffiths, P., and Harris, J. Principles of algebraic geometry. New York: Wiley-Interscience, 1978.

....p;q (M; E) is 1 2 (p q Gamma n) hence [ 3 ; Sigma ] Sigma Sigma . Set 1 = 1 2 ( Gamma ) and 2 = Gamma p Gamma1 2 ( Gamma Gamma ) Then a (a = 1; 2; 3) satisfy the standard su (2) commutation relations [ a ; b ] p Gamma1ffl abc c : 2. 1) See for example [10]. So there is a unitary representation of SU(2) on Omega ; M; E) let S a (ff) e p Gamma1ff a be the corresponding group elements. We now introduce a slightly more generalized setup. Definition 2.1. Let oe 2 Omega 1;1 (M; E) be a real valued (1; 1) form. Set (oe) oe Delta , ....

....3 (oe)S 3 (ff) 3 (oe) 2.4) Proof. A straightforward calculation using the above anti commutation relations shows that [ a ; b (oe) p Gamma1ffl abc c (oe) This means that f a (oe)g is an SU(2) triplet. Hence the result. q.e.d. It is clear that the Hodge relations (see for example [10]) Gamma ; p Gamma1 ; Gamma ; Gamma p Gamma1 (2.5) p Gamma1 ; Gamma p Gamma1 (2.6) still hold after coupling to the vector bundle E. Moreover, we have the Bochner Kodaira Nakano identities Delta = Gamma = 2 3 ( p ....

P. Griffiths & J. Harris, Principles of algebraic geometry, John Wiley, New York, 1978, x0.7.

....in X is equivalent to the k very ampleness of some adjoint linear systems. In this paper, we show that the result remains true for the zero dimensional subscheme defined by the zero set of a global section of a rank n vector bundle (Theorem 7) generalizing a theorem of Griffiths and Harris [8], p.677. Due to the Bogomolov inequality for rank 2 semistable vector bundles [4] 12] we can establish the Cayley Bacharach theorem for codimension 2 subschemes defined by global sections of rank 2 vector bundles (Theorem 8 and Corollary 9) This result can be used to reprove Paoletti s theorem ....

....this is just saying that jKX Lj is (k Gamma 1) very ample. ut If L = GammaK X , then obviously jKX Lj is base point free. Hence the first part of Theorem 7 is true for k = 1. This is the Cayley Bacharach Theorem due to Griffiths and Harris without the assumption that Z(s) is reduced (cf. [8], p.677) Since OP n(k) is k very ample, one obtains a Cayley Bacharach theorem on P n [17] In fact, this theorem is sharp. 4. Rank 2 Vector Bundle Case A divisor L is called numerically effective (nef) if the intersection number L:C is non negative, for all irreducible curves C on X . ....

Griffiths, P., Harris, J., Principles of Algebraic Geometry. Wiley-Interscience (1978)

.... Since f ramifies at each a 2 supp P , we have r 0 = ramification index of f n: Now let f k : P 1 Gamma P( k 1 C m ) defined by f k = f f 0 : f (k) in C m be the k th associated curve of f , and use the Plucker formulas (an extension of the Riemann Hurwitz formula see [9]) which on CP 1 are Gammad k Gamma1 2d k Gamma d k 1 Gamma 2 = r k ; where d k is the degree of f k , and r k is the ramification index of f k . In the table below, multiplying the numbers on the left by the inequalities on the right and adding yields d 0 (m n) m Gamma 1) m: But n ....

Griffiths, P., and Harris, J. Principles of algebraic geometry. New York: Wiley-Interscience, 1978.

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Griffiths, P., and Harris, J. Principles of algebraic geometry. Wiley-interscience, London-New-York, 1978.

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Ph. Griffiths, J. Harris, Principles of Algebraic Geometry, J.Wiley and Sons, New York 1978.

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P. Griffiths and J. Harris. Principles of Algebraic Geometry. Wiley Classics Library. John Wiley and Sons, inc, 1 edition, 1994.

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Griffiths, P., Harris, J.: Principles of algebraic geometry. New York: Wiley 1978.

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Ph. GriOEths and J. Harris. Principles of Algebraic Geometry. Wiley Interscience, New York, 1978.

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Griffiths P., and Harris J., Principles of Algebraic Geometry (Wiley Interscience, Newyork, N.Y) (1978).

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Griffiths, P.; Harris, J.: Principles of algebraic geometry. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York, 1978.

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Griffiths, P.A., and Harris, J., Principles of Algebraic Geometry, John Wiley & Sons, New York, 1978. 26

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P. Griffiths, J. Harris, Principles of Algebraic Geometry, Wiley-Interscience, 1978.

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P. Griffiths and J. Harris, Principles of Algebraic Geometry, J. Wiley and Sons, 1978.

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Ph. Griffiths, J. Harris. Principles of Algebraic Geometry. Wiley, N.Y., 1978.

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P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley and Sons, 1978.

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Ph. Gri#ths and J. Harris, Principles of algebraic geometry, Willey, NY, 1978.

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Griffiths, P.A., and Harris, J., Principles of Algebraic Geometry, John Wiley & Sons, New York, 1978.

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Griffiths, P.A., and Harris, J., Principles of Algebraic Geometry, John Wiley & Sons, New York, 1978.

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P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley & Sons, New York, 1978

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P. Griffiths, J. Harris, Principles of Algebraic Geometry, Wiley-Interscience, 1978.

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P. Griffiths, J. Harris, Principles of Algebraic Geometry, John Wiley and Sons (1978).

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P. Griffiths and J. Harris, Principles of algebraic geometry, John Wiley & Sons, 1978.

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P. Griffiths and J. Harris. Principles of Algebraic Geometry. John Wiley & Sons, New York, 1978.

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P.A. Griffiths, J. Harris. Principles of Algebraic Geometry. WileyInterscience, New York, 1978.

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Griffiths, P. and Harris, J., Principles of Algebraic Geometry, Wiley, N.Y., 1978.

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P.A. Griffiths, J. Harris, Principles of Algebraic Geometry, John Wiley & Sons (1978)

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