Classical Algebraic Geometry: a Modern View Igor V

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Cambridge University Press 978-1-107-01765-8 - Classical Algebraic Geometry: A Modern View Igor V. Dolgachev Index More information

Symbol index

(123, 94), 118 V22, 264 (166), 522 W(EN ), 334 , r (l r), 152 Wg−1, 199 ∨ (a0,...,ar), 511 X , 29 k 2 , 402 X0(n), 85 A f , 36 Δ, 533 Ak, 34 Δk, 533 Bg, 193 Γ(3), 121 C(d), 212 Γ(ϑ), 218 CN , 361 Γ∗(F ), 164 D( f ), 563 ΓT , 212 Dψ, 2 Γ f , 285 φ Dr( ), 162 Λp, 515 ∨ E , 1 Ω(A0), 511 Eg, 193 Ω(A0, A1,...,Ar), 511 F(C), 537 Θ, 199 , G(3 AP3( f ))σ, 263 Θk(A, B), 98 Pn Gr( ), 21 δx, 171 G168, 273 γk, 516 G216, 121 D( f ), 563

HA f (t), 50 Gk, 514 I1,N , 331 P(1, 2, 2, 3, 3), 111 IZ , 38 P(E), 320 KX, 75 V(E), 320 Ln(q), 269 EN , 334 Nθ, 244 Fn, 322 Ω f , 51 kN , 332 H Pak (X), 5 2, 522 Q(V)±, 204 H3(3), 493 [n] Qg, 195 L , 166 Mar RC , 252 3 , 250 ϑ Mev R , 216 3 , 250 Mev Rg, 239 g , 197 S (C), 533 TCd, 197 d S (E), 1 TCX/S , 196 S d(E), 1 A6, 102 S a,n, 347 C, 252, 517

S a1,...,ak,n, 560 C(C), 252

620

Symbol index 621

CW , 528 dom( f ), 280 Cω, 518 pd, 165 a, 202 cx, 171 A : B, 269 BlX(Z), 282 adj(A), 7 k BlX(a), 282 ap , 48 + f BlX(Z), 282 AP( f ), 36 CR4, 478 Nϑ, 245 B.A, 269 S, 119 Bs(| V |), 281 S3, 472 b(| V |), 281 SW , 528 Sω, 518 Catk(d, n), 48 T, 119 Catk( f ), 50 ZX, 517 Cay(| L |), 24 ZG, 510 Cay(X), 21 μn, 245 σm(C), 570 D2(n), 15 τij, 198 Dd(n), 19, 23 Matm,k, 160 ∂ j, 2 Matm,k(r), 161 D(A; u, v), 154 , O(1 N), 331 D(L), 23 Prym(S˜ /S ), 238 Δi ,...,i , 97 SL(2, F ), 121 1 k q Dm,n, 65 Symm(r), 181 f, s o VSP( ) , 47 E, 1 ϑ T , 194 e(X, x), 34 ϑi, jkl, 246 ECa(X), 9 ak, 176 d(ϑ), 218 He(X), 17 d , 176 k He( f ), 4 eγ(z), 200 He( f ), 13 e , 176 k He(X), 13 f , 280 d Hilbs(P(E)), 39 g1, 85 n H (E), 57 i , 529 q W HS(X), 19 iω, 519 k + , 571 m 1 i, 2 pi1...im , 509 qϑ, 188 j, 115 rα, 334 Jac(L), 23 rm, 570 t , 206 g , (n, d, k, s), 45 , 526 [n], 591 L(Z), 54 Λ Arf(q), 191 , 45 Cr(n), 342 Jac(X), 114 μ(φ), 33 Kum(A), 538 multxX, 35 OG(2, Q), 552 μ(X, x), 33 O(N), 334 Pf(A), 108 N( f ), 25 SO(n + 1), 553 Sp(V), 191 Ω f , 51 Ω∨ Tr(A), 97 f , 51

622 Symbol index

O(E, q), 57 st, 20 St(| L |), 23 Pn, 4 P(E), 4 T, 44 PB(| L |), 25 T(E), 1 T Pn, 84, 85 x(X), 7 PO(3), 73 Tφ, 44 PO(n + 1), 63 Tφ, 44 PS(s, d; n), 47 TCa(X), 8 Θk, 97 Ram(φ), 29 U, 235

S 1, S 2 , 559 Vn Sd, 1 d, 32 S (3), 120 vd, 32 S (E), 1 s(n, d), 43 wrk( f ), 52 ΣPGL(E), 11 SL(U), 3 X(3), 117 Sn,C , 86 St(X), 19 Z(s), 87

Subject index

(−1)-curve, 248 anti-polar, 51 (−n)-curve, 289 conic, 277 (166)-conﬁguration, 522 anticanonical Ak-singularity, 34 divisor, 387 F-locus, 281 embedding, 264 N-lateral, 256 linear system, 387 α-plane, 512 model β-plane, 512 of a del Pezzo surface, 384 EN -lattice, 361 ring, 385 k-secant line, 215 Antonelli, G., 591 s-lateral, 38 apolar homogeneous form, 35 Abel–Jacobi Theorem, 198 quadrics, 98 abelian surfaces ring, 36 moduli, 545 subscheme, 36 abelian variety apolarity principally polarized, 199 duality, 131 Abo, H., 45 First Main Theorem, 42 absolute invariant, 115 map, 48 aCM of conics, 112 sheaf, 164, 166, 167, 177, 296, 385 Apollonius of Perga, 104 -symmetric sheaf, 168 Apollonius problem, 104 of rank 1, 179 apparent boundary, 7 symmetric, 168 Arbarello, E., xi, 188, 218, 222, 224 subscheme, 296, 297, 299 Arf invariant, 191 ADE singularity, 376 Aronhold invariant, 115, 119, 130 adjoint symbolic expression, 137 orbit Aronhold set, 249 minimal, 553 Aronhold, S., 144, 278 nilpotent, 553 arrangement of lines, 256 supminimal, 553 Artebani, M., 144, 266 variety, 553 Artin, M., 351, 424 adjugate matrix, 7 associated Alberich-Carraminana,˜ M., 334 curve, 570 Alexander, J. E., 42 line, 589 Alexander, J.W., 345 sets of points, 455 Allcock, D., 482 association involution, 478 almost general position, 357 August, F., 451, 452, 505 Altman, A., 170 azygetic, 207

623

624 Subject index

set of seven bitangents, 228 Bos, H., 112 triad of Steiner complexes, 208 Bottema, O., 584 set in a symplectic space, 204 Bourbaki, N., 365, 425, 506, 591 tetrad of bitangents, 229 bracket-function, 467 triad in a symplectic space, 204 Bragadin, G., xii triad of bitangents, 228, 229 Brambilla, M., 42, 45 branch divisor, 29 Bohning,¨ C., 227 Brianchon’s Theorem, 80 Babbage’s conjecture, 184 Brianchon, Ch., 80 Babbage, D.W., 184, 392 Briancon, J., 131 Baker, H., 111, 215, 225, 405, 422, 425, 440, Brill, A., 214, 225 503, 504 Bring curve, 498, 500 Bardelli, F., 250 Brioschi covariant, 140 Barth’s condition, 46 Brioschi, F., 140 Barth, W., 46, 85, 90, 112, 279, 324 Bronowski, J., 67 Bateman, H., 277, 279, 313 Bruno, A., 296 Battaglini line complex, 547, 588 bubble cycle, 308 Battaglini, G., 590 admissible order, 308 Bauer, Th., 85, 90, 112 fundamental, 308 Beauville, A., 168, 185, 186, 197, 392, 553 bubble space, 307 Beklemishev, N., 479 height function, 307 Beltrametti, M., xii proper points, 307 Bertini Burch, L., 453 involution, 317, 413 Burhardt quartic threefold, 186 Theorem Burns, D., 500 on elliptic pencils, 344, 346 on irreducibility, 75, 305, 443 C. van Oss, 276 on singularities, 20, 177, 299, 305, 382, 562 Calabi–Yau variety, 25, 528 Bertini, E., 345, 346, 424 Campbell, J.E., 67 bezoutiant, 184 canonical bielliptic curve, 241 class biﬁd map, 193 of a Fano variety, 534 binary form of a normal surface, 178 quadratic invariant, 138 of a projective bundle, 321 Binet, J., 591 of a ruled surface, 322, 561 binode, 424 of blow-up, 332 birational map, 280 of Grassmann variety, 75 Birkenhake, Ch., 216 equation biscribed triangle, 244 of plane cubic, 117 of the Klein curve, 274 map, 221 bitangent Caporali quartic, 277 deﬁned by Aronhold set, 228 Caporali, E., 277 honest, 35 Caporaso, L., 204, 228 hyperplane, 204 Carletti, E., xii matrix, 277 Carlini, E., 45 their number, 215 Cartan cubic, 435 bitangential curve, 215, 225 Cartan matrix, 362 Blache, R., 178 irreducible, 363 blowing down structure, 355 Cartan, E., 435 Bobillier, E., 66, 112 Carter, R., 483 bordered determinant, 99, 140, 154, 278 Casnati, G., 168 Bordiga scroll, 300 Castelnuovo, G., 345, 591 Bordiga surface, 300, 423, 440 Castelnuovo–Richmond quartic, 478, 481, 524, Bordiga, G., 300, 423, 440 545 Borel, A., 366 catalecticant, 266

Subject index 625

determinant, 50 Clebsch, A., 138, 140, 142, 144, 278, 424, 479, hypersurface, 50 505 matrix, 48 Clemens, C.H., 224 Catanese, F., 168, 177, 183, 186, 187, 238, 239 Cliﬀord, W., 345 Cayley Coble, A., 112, 211, 224, 251, 278, 425, 481, cubic surface, 449 482, 506, 513 dianode surface, 250 Cohen, T., 266 family of cubic surfaces, 504 Cohen–Macaulay octad, 245 module, 164 quartic symmetroid, 28 sheaf, 169 Cayley, A., 17, 28, 29, 35, 36, 83, 100, 112, 140, variety, 161 141, 144, 181, 214, 225, 250, 278, 345, collineation, 10 433, 504, 549, 590, 591 Colombo, E., 482 Cayley–Brill formula, 214 complete Cayley–Salmon equation, 503 ideal, 283 Cayley–Zeuthen formulas, 564 pentalateral, 256 Cayleyan quadrangle, 264 contravariant, 140 complex equation curve, 21, 65, 142 of a quadric, 99 of plane cubic, 127 complex reﬂection, 122 variety, 21 group, 103, 122, 276 of a linear system, 24 compound matrix, 98 center variety, 528 adjugate, 98 Chandler, K., 42 conductor formula, 171 characteristic matrix, 330 conductor ideal, 170 Chasles Cone Theorem, 371 covariant quadric, 94, 112 congruence of lines, 513 Principle of Correspondence, 225 class, 513 Theorem order, 513 on conjugate triangles, 77 conic on linear line complex, 525, 590 apolar, 112 on polar tetrahedra, 93 conjugate triangles of, 76 Chasles, M., 93, 111, 112, 144, 224, 590–592 invariants of a pair, 100 Chipalkatti, J., 45 mutually apolar, 103 Chisini, O., 65, 66 Poncelet n-related, 82 Chow Poncelet related, 81 form, 517, 548, 568, 590 self-polar triangles of, 73 group, 290 variety of pairs, 85 ring, 512 conic bundle, 328 Ciani, E., 276, 279 conjugate Ciliberto, C., 14, 67, 393 conics, 112 circle linear forms, 51 complex, 89 linear subspaces, 92 real, 91 triangle, 76 circulant matrix, 48 contact class, 4, 30 curves, 155 of a space curve, 570, 573 conics, 240 of immersion, 564 cubics, 244 Clebsch of degree d − 1, 240 diagonal cubic surface, 439, 497 hyperplane quartic curve, 255 of a canonical curve, 189 nondegenerate, 256 manifold, 553 weakly nondegenerate, 256 contravariant, 22, 136 Theorem, 334 Cayleyan, 140 transfer principle, 138, 474, 481 Hermite, 141

626 Subject index

of a pair of conics, 104 F-locus, 287 of a pair of quadrics, 112 fixed points, 310 of a plane quartic, 266 fundamental point, 287, 308 of a quartic curve, 265 Geiser involution, 316 of a ternary cubic, 139 given by pfaffians, 517 on quartic ternary forms, 139 indeterminacy point, 287 Pippian, 141 multidegree, 286 Cook, R, 186 of degree 5, 378, 436 Coolidge, J., 111, 112, 311, 313, 344 ordered resolution, 329 coresidual point, 136, 251 P-locus, 287 Cornalba, M., xi, 188, 197, 218, 222, 224 quadro-cubic in P4, 293 correlation, 10 quadro-quadratic, 295 composition, 11 regularizable, 342 conjugate points, 10 symmetric, 315 dual, 11 Cremona, L., 67, 112, 143, 144, 186, 225, 286, polarity, 11 319, 344, 424, 505, 513, 585, 591 correspondence, 212 cross ratio, 105, 110, 467, 553 direct lateral, 215 cubic hypersurface, 65, 88, 124, 125, 472, 573 inverse, 213 catalecticant, 49 of type (4, 4), 224 determinantal, 42, 49, 387 Scorza, 216 fourfold, 392 symmetric, 213 in P6, 388 united point, 213 pfaffian, 517 valence, 213 symmetroid, 402 with valence, 224 variety of lines, 589 Corti, A., 346 cubic plane curve Cossec, F., 28, 183, 187, 402, 459 absolute invariant, 115 covariant, 4, 22, 136 canonical equation, 117 Brioschi, 140 covariants and contravariants, 145 Clebsch, 252, 263 dual curve, 128 Hermite, 140 equianharmonic, 116 Hessian, 255 harmonic, 116 of a binary cubic, 61 Hesse equation, 117 of a pair of conics, 104, 107, 108 Hesse pencil, 119 of a pair of quadrics, 112 its Cayleyan curve, 127 of a plane quartic, 265 its Hessian curve, 125 of a ternary cubic, 139 Legendre equations, 116 of binary quartic, 61 Weierstrass equation, 115 quadric, 94 cubic scroll Scorza, 252 in P3, 444 Coxeter, H., 66, 333, 369, 425, 507 in P4, 444 Coxeter–Dynkin diagram, 333, 363 cubic surface, 88 extended, 366 4-nodal, 551 Crauder, B., 293 as a base of a Palatini scroll, 531 Cremona Cayley–Salmon equation, 450 group, 326, 342, 346 Cayley surface, 449 hexahedral equations, 464 Cremona’s hexahedral equations, 465 inequalities, 286 cyclic, 441, 462 Cremona transformation, 284 dual surface, 504 Bertini involution, 317 Eckardt point, 440 Clebsch Theorem, 334 lines on it, 432 cubo-cubic, 299 moduli, 528 de Jonquieres,` 303 moduli space, 479 de Jonquieres` involution, 301 non-normal, 444 determinantal, 345 projective generation, 452

Subject index 627

Sylvester nondegenerate, 260, 462 desmic symmetroid, 456 tetrahedra, 96 tritangent plane, 412 determinantal cubic symmetroid, 456 hypersurface, 146 cuspidal edge, 575 equation, 240 cyclide formula, 511 curve, 400 hypersurface, 148, 151, 240 degenerate surface, 400 representation Dupont surface, 422 equivalence, 146 quartic surface, 399 of singular plane curves, 169 of surfaces, 177 D’Almeida, J., 569 quartic surfaces, 185 Dale, M., 41 variety, 74 Darboux curve, 46, 256 resolution of singularities, 65 its equation, 257 developable surface, 17, 580, 587, 589 Darboux’s Theorem, 83 of a space curve, 570 Darboux, G., 45, 67, 83, 112, 279, 423 quartic, 587 Dardanelli, E., 463, 479, 480 Dickson, L., 144, 186, 506 de Jonquieres` involution, 301 diﬀerence map, 216 de Jonquieres,` E., 225, 344 directrix, 558 de Siebenthal, J., 366 discrepancy divisor, 178 Debarre, O., 384 discriminant, 19 defect, 515 hypersurface, 30 defective, 41 cubic, 69 k-defective, 515 its degree, 19 variety, 515 its dual hypersurface, 32 degeneracy locus, 530, 533, 566 of linear system, 23 degenerate of quadrics, 15 homogeneous form, 50 tangent space, 32 multilinear form, 26 of a binary cubic, 60 del Pezzo surface, 349, 353, 423 of a binary form, 10 Cremona isometries, 377 of a binary quartic, 61 degree, 355 of a general polynomial, 9 eﬀective cone, 371 symbolic expression, 138 its secant variety, 388 divisor class lines on it, 383 big, 353 marked, 356 nef, 353 nef cone, 372 Dixmier, J., 265 of degree 1, 411, 413, 414, 422, 500 Dixon, A., 47, 186, 503 of degree 2, 242, 248, 250, 405, 410, 424, Dolgachev, I., 3, 29, 35, 47, 51, 64, 67, 112, 115, 441, 443 144, 148, 192, 219, 225, 239, 251, 258, of degree 4, 396, 400, 403, 422, 423, 459, 262, 274, 276, 334, 346, 395, 402, 439, 534, 535 440, 455, 459, 467, 471, 478, 479, 482, of degree 5, 389–392, 395, 422, 529, 589 506, 528, 589 of degree 6, 288, 299, 386–388, 422, 451, 551 double-point of degree 7, 386 class, 563 of degree 8, 386, 422, 444 formula, 563 del Pezzo, P., 423 set, 563 Deligne, P., 306, 308 double-six, 427 Deligne–Hoskin formula, 308 azygetic duad, 428 Demazure, M., 326, 425 azygetic triad, 429 Dersch, O., 225 Steiner complex of triads, 429 Desargues’ Theorem, 77, 79 syzygetic duad, 428 in space, 91 syzygetic triad, 429 Desargues, G., 77, 79, 91 Du Val singularity, 376

628 Subject index

Du Val, P., 345, 376, 406, 412, 424, 506 Fano variety, 64, 109, 263, 264, 355, 501, 528, dual 531, 534 homogeneous form, 48 degree, 263 dual variety, 29 genus, 263 degree, 33 index, 263 of a hypersurface, 29 of genus 12, 264, 502 of a plane cubic, 142 toric, 288 of a Veronese surface, 69 Fano, G., 264 of Grassmann variety, 519 Farkas, G., 221 of Segre cubic primal, 478 fat point, 40 projective space, 4 Fay, J., 222 Reflexivity Theorem, 29 Fermat hypersurface duality map, 30 cubic curve, 125 dualizing sheaf, 170 cubic surface, 442, 480, 497 Dupin, Ch., 423 plane cubic, 129, 130, 143, 253 Durege,` H., 144 plane quartic, 252, 269, 272, 276 Dyck, W., 279 Ferrers, N., 112 Dynkin curve, 373 fiber Dynkin, E., 333, 366 of a sheaf, 162 Fielder, W., 479, 506 Eckardt point, 440, 480–482 Finkelnberg, H., 477 defining an involution, 441 First Fundamental Theorem, 3, 509 Eckardt, F., 506 Fischer, G., 16, 35 Edge variety, 392 Fitting ideal, 162 Edge, W., 274, 392, 395, 404, 422, 423, 581, Flatto, L., 90, 112 585, 592 Formanek, E., 159 effective cone, 371 Frahm, W., 67 Ehrenborg, R., 42 Freitag, E., 482 Ein, L., 293 Fresnel’s wave surface, 551 Eisenbud, D., 36, 37, 219, 385, 452, 455 Fresnel, A., 551 elementary transformation, 324 Fricke, R., 274 of vector bundles, 324 Frobenius, G., 278 Ellingsrud, G., 297, 299 Fulton, W., 35, 104, 134, 163, 290, 394, 477, Elliott, E., 145 511, 512, 530, 533, 563, 565, 566, 591 elliptic normal curve, 114 functional determinant, 13 Elte, E., 425, 507 fundamental Emch, A., 91 cycle, 351 Enriques diagram, 308 point, 281 Enriques surface, 463, 589 set, 210 Enriques, F., 65, 66, 308, 326, 463 normal, 211 envelope, 4 weight, 366 enveloping cone, 9 equianharmonic Gopel,¨ A., 224 plane cubic, 131 Gallarati, D., xii quadruple, 105 Gantmacher, F., 397 Euler exact sequence, 321 Gauss curvature, 16 dual, 321 Gauss map, 30, 218, 222, 516 Euler formula, 6 Geiser involution, 316, 408, 409 exceptional Geiser, C., 345, 424 curve, 351 Gelfand, I., 19, 29, 35, 144, 517, 568 divisor, 282 general position, 357 section, 320 genus 4 curve, 189, 214, 224, 237, 241, 277, type, 335 414, 438, 498, 573 vector, 367 geometric basis, 355 extremal ray, 371 geometric marking, 355

Subject index 629

Geramita, A., 43, 67 conic-locus, 113 Gerbardi, F., 85, 102 conjugate, 72, 105, 110 Gergonne, J., 66 cubic curve, 116 Giambelli, G., 187 line complex, 547 Giorgini, G., 590 polar line, 118 GIT-quotient, 81, 101, 111, 228, 266, 395, 467, polynomial 479, 506 as a pfaffian, 60 Gizatullin, M., 45, 245 of degree 2, 98 Glass, J., 278 quadruple, 105 Godeaux, L., 346 tensor, 56 Gonzalez-Aguilera,` V., 269 harmonizant, 67 Gonzales-Sprinberg, G., 345 Harris, J., xi, 7, 28, 35, 42, 74, 112, 117, 134, Goodman, R., 57 175, 181, 188, 198, 215, 218, 222, 224, Gordan, P., 14, 102, 112, 144, 275 347, 358, 394, 559, 570, 572 Gorenstein Harris. J., 191 curve, 174 Hartshorne, R., xii, 8, 114, 165, 172, 213, 270, Fano variety, 264 283, 290, 303, 305, 320, 322, 324, 350, local Artinian ring, 36 358, 382, 510, 558, 560, 562, 569, 583, 584 normal surface, 178 Hassett, B., 266 ring, 171 Hawkins, T., 591 singularity, 352 Heisenberg group, 493, 495, 522, 543, 547 Gosset, T., 369, 425, 507 Henderson, A., 504 Grace, J., 108, 112, 145 Hermite Grassmann bundle, 510 contravariant, 141 Grassmann variety, 21, 508 covariant, 141 canonical sheaf, 510 curve, 143 cohomology ring, 511 Hermite, Ch., 141 degree, 512 Hesse its dimension, 510 arrangement of lines, 118 of lines, 508 dual, 119 Plucker¨ embedding, 508 canonical equation Plucker¨ equations, 509 of plane cubic, 117 secant variety of, 513, 515 form tangent sheaf, 510 of a plane cubic curve, 114 tangent space, 516 group, 121 universal quotient bundle, 509 pencil, 119 universal subbundle, 509 quadrilateral, 110 Grassmann, H., 143, 505, 590 Theorem, 110 Greuel, G.-M., 174, 176 Hesse, O., 67, 110, 112, 117, 118, 144, 154, 278 Griffiths, Ph., xi, 28, 35, 112, 134, 188, 198, Hesse–Salmon configuration, 142 215, 218, 222, 224, 358, 570 Hessian Gross, B., 191 determinant, 4 Grothendieck A., 166 hypersurface, 13 Grushevsky, S., 221 matrix, 13 Guardia,` J., 234 of a binary quartic, 61 Gundelfinger quartic, 245 surface, 65 Gundelfinger, S., 144, 245 hexad, 230 Hilbert modular surface, 500 Hacking, P., 482 Hilbert scheme Halphen pencil, 129, 344, 346 of aCM subschemes, 297 Halphen, G., 144, 346 of lines, 264 Hankel matrix, 48 of projective plane, 81, 109, 130, 131 Hankel, H., 48 of projective space, 47 harmonic punctual, 39 binary quartic, 116 Hilbert, D., 47, 67, 453

630 Subject index

Hilbert–Birch Theorem, 504 infinitely near point, 307 Hill, J. E., xii inflection Hirschowitz, A., 42 bitangent, 226, 266 Hirzebruch, F., 500 point, 17 Hitchin, N., 57, 60, 112, 502 order, 18 Hodge Index Theorem, 358 tangent, 12, 65, 276 Hodge type inequality, 286 honest, 35 Hodge, W., 397, 509, 511, 591 triangle, 118 homaloid, 284, 345 integral closure, 282 homaloidal intersection matrix, 352 net invariant, 136 characteristic, 306 absolute, 116 polynomial, 148 Aronhold, 137 type, 335 bracket-function, 467 homology, 440 First Fundamental Theorem, 3 harmonic, 440 Joubert, 471 its center, 440 of 6 lines in P3, 527 Hoskin, M., 306, 308 of a pair of binary forms, 106 Hosoh, T., 506 of binary forms, 10 Hudson, H., 342, 344 of binary quartic, 117 Hudson, R.W., 590 of binary quartics, 49, 61, 104 Humbert curve of genus 5, 404, 405 of complex reflection group, 122 Humbert, G., 404 of cubic surface, 479 Hunt, B., 479 of Hesse group, 123 Hurwitz formula, 29 of plane quartics, 265 Hurwitz’s Theorem, 270 of ternary cubic, 136 Hurwitz, A., 225, 270 of two symmetric matrices, 99 Hutchinson, J., 506 of Valentiner group, 103 hyperelliptic curve, 224 relative, 123 2-torsion divisor classes, 193, 210 symbolic expression, 137 and Kummer surface, 539 tact, 100 de Jonqueres` transformations, 318 Toeplitz, 45 equation, 192 weight, 137 its Jacobian variety, 534 inversion transformation, 314 of genus 3, 185, 224 Inversive group, 326 plane model, 319, 343 Iskovskikh, V., 264, 346 theta characteristics, 194, 212 isologue, 309 Weil pairing, 194 center, 309 hyperosculating point, 570 net, 310 multiplicity, 572 isotropic subspace, 190 their number, 572 Izadi, E., 221, 239 hyperplane, 4 hypersurface, 4 j-invariant, 115 monoidal, 301 Jozefiak,´ T., 181 submonoidal, 301 Jacobi, C., 278 Jacobian Iano-Fletcher, A., 192 curve, 410 Iarrobino, A., 36, 40, 47, 50, 67, 309 determinant, 13 icosahedron hypersurface, 23 fundamental set, 499 ideal, 33 icosahedral set, 501 variety, 197 Igusa quartic, 545 intermediate, 534 Iliev, A., 47 Jessop, C., 405, 423, 590 incidence variety, 510 Jeurissen, R., 276 indeterminacy point, 280 join

Subject index 631

of projective subbundles, 558 quartic equation, 543 of scrolls, 559 self-duality, 546 Jordan, C., 425, 506 Tetrahedroid, 549 Joubert functions, 471 wave surface, 552 Joubert, P., 471, 506 Kummer variety, 239, 538 Jung, H., 171 jacobian, 538 Jung–Milnor formula, 171 Kummer, E., 423, 590

K3 surface, 28, 56, 264, 395, 404, 405, 463, Luroth¨ quartic, 68, 257 537, 538, 543, 550 determinantal representation, 260 Kane, R., 365, 406 pentalateral theta characteristic, 262 Kanev, V., 47, 67, 225, 258, 262, 274, 276 Luroth,¨ J., 255, 257, 279 Kantor, S., 345, 425, 506, 591 LeDˆ ung˜ Trang,´ 33 Kapranov, M., 19, 29, 35, 112, 144, 439, 440, La Hire, Ph., 113 517, 568, 589 Laguerre net, 248, 312 Katsylo, P., 227, 239, 250 Laksov, D., 74 Katz, S., 293 Lange, H., 216 Keel, S., 482 Laplace operator, 57 Kers, C., 112 Lascoux, A., 181 Keum, J., 506, 546 lattice, 331, 358 Kirkman points, 111 I1,N , 331, 360 Kirkman, J., 111, 112 EN , 334, 361 Kleiman, S., 35, 170, 510 discriminant, 358 Klein discriminant group, 358 coordinates, 522 embedding, 360 quadric, 512 even, 360 quartic curve, 252, 270, 276, 589 isometry, 360 automorphisms, 273 nondegenerate, 358 its bitangents, 277 orthogonal group, 360 its hessian, 275 primitive embedding, 360 sextic, 499 signature, 358 singularities, 376 sublattice, 358 Klein, F., 270, 274, 375, 376, 400, 499, 500, finite index, 358 505, 512, 590 primitive, 358 Kleppe, H., 74 unimodular, 358 Knorrer,¨ H., 176 Lazarsfeld, R., 282, 286, 294, 354, 563 Kneebone, G., 591 Le Potier, J., 263 Kodaira LeBarz, P., 216 fibers of elliptic fibration, 415, 550 Lefschetz’s fixed-point formula, 483 Kodaira, K., 415, 550 Lefschetz, S., 483 Kollar,` J., 371, 372, 375, 425 Legendre equation Kondo,¯ S., 266, 482, 546 of a plane cubic, 116 Kravitsky, N., 184, 186 Legendre, A.-M., 116 Krazer, J., 224 Lehavi, D., 228, 243 Kummer cones, 423 Libgober, A., 175 Kummer surface, 536, 538 Lie, S., 518, 590 and 5-cuspidal sextic, 404 Lifsic,˘ M., 186 and Segre cubic, 589 Lindemann, F., 140, 142, 144, 424 as the Hessian surface, 506 line complex, 517 associate hyperelliptic curve, 536, 538 apolar, 520 associated to a quadratic line complex, 538 cubic automorphisms, 547 Montesano, 589 double plane model, 542 degree, 517 octic model, 539 linear, 517 of an abelian surface, 538 center, 518

632 Subject index

monoidal, 589 Milnor number, 33 quadratic, 99, 156, 531 Milnor, J., 171 Battaglini, 547 minimal degree varieties, 347 harmonic, 547 minimal rational ruled surface, 320 Kummer surface, 538 minus vector, 492 lines on it, 537 mixed combinant, 136 of tangent lines to a quadric, 156 mixed concomitant, 136 tangential, 552 modular tetrahedral, 553, 586 family, 120 rank, 518 surface, 500 singular line, 533 moduli space singular variety, 533 Rg, 239 Mev special, 517 3 , 250 Mev line-equation g , 197 of a quadric, 99 of 6 points, 477, 480 linear system of 7 points, 250 base ideal, 281 of abelian surfaces, 545 base locus, 281 of bielliptic curves, 277 base scheme, 281 of cubic surfaces, 440, 479, 482, 491, 528 base-point-free, 281 of determinantal representations, 159 homaloidal, 284 of elliptic curves, 121 linearly d-independent, 39 of nets of quadrics, 262 lines of plane quartics, 266 conjugate, 77 of quadratic line complexes, 549 in a quadratic line complex, 537 of reﬂexive sheaves, 170 on a cubic threefold, 589 Monge’s diﬀerential equation, 518 on a weak del Pezzo surface, 383 Mori, S., 371, 372, 375 six linearly dependent, 526 Morley, F., 262, 279, 326 two transversals to four, 525 Morley, F.V., 326 Lipman, J., 409 Morrison, I., 572 Lo Giudice, G. , 239 Moutard, M., 423 London, F., 45, 68, 144 Muir, T., 65 Looijenga, E., 266, 425, 482 Mukai skew-form, 54 Loria, G., 590 Mukai, S., 47, 75, 112, 244, 264, 265, 396, 501 Lossen, C., 14 multidegree Lurie, J., 435 of a rational map, 285 multiplicity Mobius,¨ A., 590 of a singular point, 351 Muller,¨ H., 591 Mumford, D., 172, 178, 188, 189, 196, 200, 213, Merindol,´ J., 425 224, 239, 352 Macaulay, F.S., 36, 187 MacLaurin, C., 111, 344 Nagata, M., 345 Magnus, L., 345, 504 Naruki, I., 482, 491, 500 Manin, Yu., 346, 425, 483, 507 net, 45 Marcus, A., 186 of conics, 109, 110, 143, 241, 243, 245, 386 marking, 356 of cubics, 250, 317 geometric, 356 of quadrics, 45, 46, 246, 247, 260, 261, 265, Massoti Biggiogero, G., 66 274, 460, 462, 513, 584 Mathews, R., 96 Newton, I., 144 Mella, M., 47, 393 Nikulin, V., 360 Melliez, F., 276, 277, 502 node, 34 Meyer, W., xii, 504, 513 on a surface Michel, J., 112 even set, 183 Miles, E.P., 58 weakly even set, 183 Milne, W., 141 Noether formula, 566

Subject index 633

Noether’s Reduction Theorem, 339 of conics, 83, 131, 135, 144, 237, 276, 390, Noether, M., 14, 345 585 nondegenerate of cubics, 126, 129, 159 Clebsch quartic, 256 of quadrics, 100, 246, 250, 396, 397, 403, homogeneous form, 50 404, 423, 424, 534, 550, 556, 587 subvariety, 347 of quartics, 252 normal Perazzo primal, 504 linearly, 349 Perazzo, U., 504 projectively, 177, 385 period matrix, 198 scroll, 557 perspectivity, 77 subvariety, 349 center, 77 surface line of , 77 canonical class, 178 of simplexes, 91 intersection theory, 178 perspectrix, 77 normal system, 210 Persson, U., 415 null polarity, 11 Petersen graph, 390, 422 ff ff null-circle, 90 Pfa di erential equation, 518 ffi null-plane, 520 pfa an, 74, 108, 130, 514, 516, 529 ffi null-point, 520 pfa an hypersurface, 515 null-system, 11, 210, 520 and Palatini scroll, 530 cubic, 517 cubic fourfold, 392 OADP subvariety, 392, 423 cubic in P14, 517 Okonek, C., 300, 531 Pfister, G., 174 Oort, F., 112 Picard scheme, 198 ordinary singularities, 563 relative, 159 orthogonal group, 57, 63, 331, 334 Piene, R., 570, 573 Ortland, D., 219, 251, 395, 455, 471, 478 Pieri’s formula, 512 oscnode, 407 pinch point, 563 osculating Pinkham, H., 351, 375, 425 developable surface, 570 Piontkowski, J., 172, 176 hyperplane, 142 Pippian contravariant, 141 plane, 570 pippiana, 67 sheaf, 570 Plucker¨ Ottaviani, G., 42, 43, 45, 130, 263, 279, 300, coordinates, 509 313, 591 formula for hypersurfaces, 33 pairs of conics for plane curves, 35 GIT-quotient, 101 for space curves, 571 invariants, 100 formulas, 33 Palatini lines, 111 ruled surface, 531 Plucker,¨ J., 66, 111, 225, 278, 504, 590 scroll, 530 Plucker–Teissier¨ formula, 33, 60 Palatini, F., 47, 67, 530, 591 plane quartic curve Pan, I., 302, 342, 345 even theta characteristic, 246 parabolic hypersurface, 17 simple singularities, 406 parabolic point, 17 Plaumann, D., 277 partial normalization, 171 plus vector, 492 Pascal line, 80 Poisson bracket, 53 Pascal’s Theorem, 79 polar Pascal, B., 79 s-gon, 38 Pascal, E., 66, 279, 504, 590 s-hedron, 38 Pash, M., 590 generalized, 39 Pedoe, D., 397, 509, 511, 591 nondegenerate, 38 pencil, 45 base locus, 25

634 Subject index

bilinear form, 5, 189 Prym map, 239 duality, 520 Prym, F., 223 hypersurface, 5 first, 7 quadratic form second, 124 even, 191 line, 71, 520 odd, 191 linear subspaces, 92 quadratic transformation, 314 map, 28 quadric net of quadrics, 45 complex equation, 99 pairing, 2 invariants of a pair, 96 pentagon, 255 line-equation, 99 pentalateral, 255 polar pentahedron, 95 quadrangle, 129 polar tetrahedra, 93 generalized nondegenerate, 129 quadric bundle, 156, 531 quadric, 12 discriminant locus, 531 subspace, 528 quadrilateral, 88, 110 polarity, 11 quartic hypersurface polarization map Burhardt, 186 partial, 2 Castelnuovo–Richmond, 478, 481, 524, 545 total, 1 Igusa, 545 Polarraum, 111 Scorza, 189 pole, 71, 74 quartic plane curve Polo-Blanco, I., 584 Aronhold invariant, 258 poloconic, 140, 141 automorphisms, 266 polygon, 82 bitangents, 226 side, 82 Aronhod sets, 228 vertex, 82 azygetic triads, 228 polytope Steiner complexes, 226 regular, 369 syzygetic triads, 226 semi-regular, 369 Caporali, 277 Poncelet related curve, 86 Clebsch, 255 Poncelet, J.-V., 111, 278, 344 contravariants, 266 Popescu, S., 219, 455 covariants, 265 porism, 91 determinantal equation, 235 Postulation formula, 284 symmetric, 239 Pragacz, P., 181 invariants, 265 prime-form, 222 Klein, 270 principal curve Luroth,¨ 257 total, 330 the variety of sums of powers, 263 principal parts, 569 quartic surface Principle of Correspondence, 224 4-nodal, 402 pro-Hessian surface, 17 del Pezzo, 397 projective bundle, 320 desmic, 96 canonical class, 321 developable, 580 projective coordinates, 4 dual of Cayley cubic, 449 projective generation, 163 Gundelfinger, 245 determinantal varieties, 163 Kummer, 404 of a cubic curve, 144 ruled, 575 of a cubic surface, 187, 452 classification, 576 of plane curves, 310 equations, 586 of rational normal curve, 134 Segre, 397 Steiner’s construction, 93 Steiner, 449 projective space, 4 Tetrahedroid, 549 prospector, 77 wave surface, 551 Prym canonical map, 223 Quippian contravariant, 141

Subject index 635

Ramanujam’s Vanishing Theorem, 354 nodal, 372 Ramanujam, C., 354 positive, negative, 364 Ranestad, K., 47, 263, 276, 277, 502 sublattice, 366 rank Rosanes, J., 67, 112, 345 of a curve, 570 Rosenberg, J., 506 rational elliptic surface, 344, 414, 415, 423 Rosenhein, J., 224 rational map, 280 Rota, G.-C., 42 inverse transform under, 281 Roth, L., 299 its resolution, 282 Roth, P., 278 rational normal curve Rowe, D., 591 associated to a net of quadrics, 264 ruled surface, 17, 560 equations, 135 of degree 4, 585 of degree 4, 581, 589 contact curve, 564 secant variety, 49, 88 elliptic, 575 secants of, 513 elliptic of degree 6, 531 rational plane curve exceptional section, 561 determinantal equation, 184 genus, 562 Raven, D., 112 minimal, 561 real sphere, 400 minimal surface Fn, 322 reciprocity theorem, 6 normalized vector bundle, 561 reflection, 334, 365 of degree 3, 443, 575 reflexive sheaf, 165 of degree 8, 300 Reflexivity Theorem, 29 Palatini, 531 Rego, C., 170 ruled variety, 557 regular linear system, 26 Russo, F., 393 Reichstein, Z., 144 Reid, M., 351, 353, 504 Saavedra Rivano, N., 224 relative Picard scheme, 196 Salmon Reskine, C., 297 conic, 106, 107 resolution envelope conic, 107 minimal, 330 invariant, 266 Reye Salmon, G., 67, 111–113, 115, 136, 144, 215, congruence, 28 225, 247, 252, 266, 278, 479, 504, 506, 591 line, 24 Salvatti Manni, R., 221 line complex, 591 Sankaran, G., 239 variety, 27 Sarkisov program, 346 Reye, T., 67, 111, 112, 506, 513, 590 Sarkisov, V., 346 Richmond, H., 47, 67 satellite conic Riemann of a plane cubic, 143 constant, 200 Scheyer, F.-O., 47 equation of bitangents, 230 Schlafli¨ equation, 142 Riemann, B., 189, 200, 223, 224, 278 Schlafli’s¨ Theorem, 503 Riemann–Kempf Theorem, 199 Schlafli,¨ L., 144, 505 Riemann–Mumford relation, 189 Schlesinger, O., 143, 144 Ritzenthaler, C., 243 Schoenberg, I., 90 Rodenberg, C., 463 Schoute, P., 506 Rodriguez, R., 269 Schroter,¨ H., 145 Room, T., 260, 440 Schreyer, F.-O., 263, 264 root, 362 Schubert basis, 362 class, 511 canonical, 363 cycle, 511 irreducible, 362 variety, 511 effective, 372 special, 511 function, 223 Schubert, H., 591 lattice, 363 Schur quadric, 437, 498, 527

636 Subject index

Schur sextic, 438, 499 Severi–Zak variety, 294, 515 Schur, F., 436 sextactic point, 142 Schwarz, H., 592 Shafarevich, I. R., xii Schwarzenberger vector bundle, 86, 588 Shephard, G., 122 Schwarzenberger, R. L. E., 86, 112 Shepherd-Barron, N., 293 Scorza Shioda, T., 265 correspondence, 216, 221, 224, 568 simplex, 91 general pair, 219 edge, 91 covariant, 252 face, 91 map, 265 facet, 91 quartic hypersurface, 221 mutually polar, 91 Scorza, G., 216, 219, 225, 279 vertex, 91 scroll, 349, 444, 530, 557, 577 singular line, 537 r-directrix, 558 singular point Bordiga, 300 of Kummer variety, 538 cubic in P3, 444 singular subspace, 190 cubic in P4, 444 singular variety degree, 559 of a quadratic line complex of lines, 531 generator, 557 of line complexes of lines, 528 join, 559 singularity normal, 557 Ak, 34 Palatini, 530 ADE, 175 rational normal, 347 binode, 424 of dimension 2, 347, 575 complete intersection, 33 tangential, 577, 579, 580 cusp secant variety, 41 ordinary, 34 defective, 44 rhamphoid, 407 of a del Pezzo surface, 388 Du Val, 376 of a rational normal curve, 42, 65, 588 Gorenstein, 352 of Grassmann variety, 513 multiplicity, 351 of rational normal curve, 49 node, 34 of Segre–Veronese variety, 44 ordinary isolated, 31 of Veronese variety, 41, 42 rational, 351 Segre rational double point, 351 class, 290 rational Gorenstein, 352 cubic primal, 472, 529, 543, 544, 549, 589 simple, 175, 376 quartic surface, 397 small resolution, 477 symbol, 397, 424 tacnode, 407 variety, 65, 422, 514, 559 sixer, 426 Segre, B., 505, 591 Smith, R., 239 Segre, C., 225, 345, 400, 423, 505, 588, 590 socle, 36 Segre–Veronese embedding, 43 Sommerville, 99 Segre–Veronese variety, 40 Sommerville, D., 113 secant variety of, 45 Sousley, C., 481 self-associated sets, 219 space curve self-conjugate m-rank, 570 hexad, 111 associated, 570 pentad, 95 dual, 570 polyhedron, 91 ordinary point, 570 tetrad in the plane, 110 Plucker¨ formula, 571 tetrahedron, 92 quartic curves semi-stable points, 467 species, 117 Semple, J., 299, 591 stationary point, 570 Sernesi, E., 204, 228, 279, 313 Springer, T., 122, 416 Severi F., xi stable points, 467

Subject index 637

standard quadratic transformation, 287 pencil, 144 degenerate, 288 Steiner complexes, 207 standard symplectic basis, 190 tetrad in a symplectic space, 204 standard tableaux, 471 tetrad of bitangents, 229 star, 163 tetrad of theta characteristics, 206 star-duality, 555 tetrads in Steiner complex, 224 Steenbrink, J., 276 triad in a symplectic space, 204 Steiner triad of Steiner complexes, 208 complex triad of theta characteristics, 205 in a symplectic space, 206 Szpiro, L., 297 complexes azygetic triad, 208 tact-invariant, 100 syzygetic, 207 Takagi, H., 219 syzygetic triad, 208 tangent cone, 8 points, 111 tangent space polygon, 143 embedded, 7 projective generation, 93, 134 tangential variety quartic surface, 70 of a space curve, 570 its dual, 449 of a Veronese surface, 69 Steiner, J., 66, 70, 90, 93, 111, 112, 134, 143, of an elliptic normal curve, 573 144, 278, 424, 505, 506 of rational normal curve, 572 Steinerian hypersurface, 19, 23 tautological exact sequence, 509 as a covariant, 22 Taylor, D., 141 its degree, 20 Teissier, B., 33 of a linear system, 23 Terracini’s Lemma, 41, 42 Steinerian map, 20 Terracini, A., 41, 45, 67 Stipins, J., 96 tetrahedral line complex, 586 Stuart, T., 47 Tetrahedroid quartic surface, 549 Sturm, R., 111, 345, 505, 592 Tevelev, E., 29, 482 Sturmfels, B., 277 theta sums of powers characteristic, 151, 169, 188 variety VSP( f, s), 46, 56 eﬀective, 151 0-dimensional, 47 even, odd, 151 explicit description, 47 Scorza invariant of, 218 Fano model, 263 syzygetic triad, 204 of a binary form, 62 their number, 189 of a conic, 64 vanishing, 189, 224 of a plane curve, 264 divisor of a quadric, 64 even, odd, 199 of a set of forms, 45 symmetric, 199 Waring problem, 47 factor, 200 Sylvester function equation of a cubic surface, 261, 462 Riemann, 200 nondegenerate, 260 with characteristic, 200 pentahedron, 463 theta function, 200 Sylvester, J., 47, 67, 136, 345, 424, 504 Thom–Porteous formula, 533 symbolic method, 3, 137 Thomas, A., 186 symmetric algebra, 1 Thomsen, H., 266 symmetric power, 1 Tikhomirov, A., 263 symmetroid surface, 181 Tikhomirov, S., xii cubic, 238 Timms, G., 401, 425 quintic, 185 Todd, J., 112, 122, 185, 506 symplectic group, 191 Toeplitz invariant, 45, 462 syntheme, 465 Toeplitz map, 44 syzygetic Toeplitz, E., 45, 68, 462

638 Subject index

Togliatti, E., 422 vector bundle, 320 Top, J., 584 Veronese map, 32, 40 toric variety, 326 Veronese surface, 41, 109, 588 Fano, 288 dual variety of, 69 of type An, 299 of degree 4, 69 surface, 387 projected, 70, 529 torsal generator, 567, 589 Veronese variety, 32, 40 torsion-free sheaf embedded tangent space, 65 global invariant, 172 its dual hypersurface, 32 local invariant, 174 secant variety of, 41, 42, 67, 240 local type, 172 tangent space, 42 total, 465 Verra, A., 239, 243, 296, 541 transversal lines, 452, 526 Vinnikov, V., 186 Trautman, G., 112 Vinzant, C., 277 triangle, 71 von Staudt’s Theorem, 554 circumscribed, 76 von Staudt, G., 112, 554, 590 inscribed, 76 Voss, A., 590 polar, 71 polar degenerate, 73 Wall, C.T.C., 186 self-conjugate, 76 Wallach, N., 57 side, 71 Waring problem, 46 vertex, 71 Waring rank, 52 triangles exceptional cases, 52 perspective, 77 web, 45 trigonal construction, 238 Weber, H., 223, 224, 235, 275, 278, 506 trisecant Weierstrass plane, 73, 588 equation ruled surface, 300 of a plane cubic, 115 trisecant line, 299, 529, 536, 560 of elliptic surface, 415 tritangent plane, 433 form conjugate triads, 431 of a plane cubic, 114 trope, 396, 545 point, 573 Tu, L., 74, 181 Weierstrass, K., 144, 424 Turnbull, H.W., 60, 99, 112 weighted projective space, 192 Tyurin, A., 186, 263, 324, 455 adjunction formula, 192 weights Umemura, H., 501 miniscule universal tritangent trio, 430 quotient bundle, 63 Weil subbundle, 63 divisor, 165 unode, 424 pairing, 188 Urabe, T, 425 Theorem, 534 Weil, A., 534 V( f ), 4 Weiler, A., 424, 590 Valentiner group, 102 Weyl chamber, 365 Valentiner, H., 102 face, 365 Valles, J., 109, 112, 422 Weyl group, 334, 365 Van de Ven, A., 294 Weyl, H., 506 van den Bergh, M., 159 White surface, 440, 504 van den Dries, B., 51 White, F.S, 440 van der Geer, G., 278, 478 White, H.W., 145 van der Put, M., 584 Williams, E., 58 van der Vries, J., 589 Wiman pencil, 395, 423 van Geemen, B., 260, 463, 479, 480, 482 Wiman plane sextic, 395, 501 Varley, R., 405 Wiman, A., 506

Subject index 639

Winger, R., 501 Yuzvinsky, S., 96 Wirtinger plane sextic, 237 Wirtinger, W., 238 Zak, F., 29, 41, 294 Wong, B., 585, 592 Zariski, O., xi, 283, 308 Wronskian, 588 Zelevinsky, A., 19, 29, 35, 144, 517, 568 Zeuthen, H., 591 Young, A., 112, 145 Zindler, K., 590 Young, W., 591 Zuccone, F., 219