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Symbol index

(123, 94), 118(166), 522(l, r), 152(a0, . . . , ar), 5112k , 402A f , 36Ak , 34Bg, 193C(d), 212CN , 361D( f ), 563Dψ, 2Dr(φ), 162E∨, 1Eg, 193F(C), 537G(3,AP3( f ))σ, 263Gr(Pn), 21G168, 273G216, 121HA f (t), 50

I1,N , 331IZ , 38KX , 75Ln(q), 269Nθ, 244Ω f , 51Pak (X), 5Q(V)±, 204Qg, 195RC , 252Rϑ, 216Rg, 239S (C), 533S d(E), 1S d(E), 1S a,n, 347S a1 ,...,ak ,n, 560

V22, 264W(EN ), 334Wr

g−1, 199

X∨, 29X0(n), 85Δ, 533Δk , 533Γ(3), 121Γ(ϑ), 218Γ∗(F ), 164ΓT , 212Γ f , 285Λp, 515Ω(A0), 511Ω(A0, A1, . . . , Ar), 511Θ, 199Θk(A, B), 98δx, 171γk, 516D( f ), 563Gk , 514P(1, 2, 2, 3, 3), 111P(E), 320V(E), 320EN , 334Fn, 322kN , 332H2, 522H3(3), 493L[n], 166Mar

3 , 250Mev

3 , 250Mev

g , 197TCd , 197TCX/S , 196A6, 102C, 252, 517C(C), 252

620

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Symbol index 621

CW , 528Cω, 518a, 202cx, 171BlX(Z), 282BlX(a), 282Bl+X(Z), 282CR4, 478Nϑ, 245S, 119S3, 472SW , 528Sω, 518T, 119ZX , 517ZG, 510μn, 245σm(C), 570τi j, 198Matm,k, 160Matm,k(r), 161O(1,N), 331Prym(S /S ), 238SL(2,Fq), 121Symm(r), 181VSP( f , s)o, 47ϑT , 194ϑi, jkl, 246ak , 176d(ϑ), 218dk , 176eγ(z), 200ek , 176fd, 280g1

n, 85iW , 529iω, 519km+1, 571pi1 ...im , 509qϑ, 188rα, 334rm, 570tg, 206(n, d, k, s), 45[n], 591Arf(q), 191Cr(n), 342Jac(X), 114Kum(A), 538OG(2,Q), 552O(N), 334Pf(A), 108SO(n + 1), 553Sp(V), 191Tr(A), 97

dom( f ), 280pd, 165

A : B, 269adj(A), 7apk

f , 48AP( f ), 36

B.A, 269Bs(| V |), 281b(| V |), 281

Catk(d, n), 48Catk( f ), 50Cay(| L |), 24Cay(X), 21

D2(n), 15Dd(n), 19, 23∂ j, 2D(A; u, v), 154D(L), 23Δi1 ,...,ik , 97Dm,n, 65

E, 1e(X, x), 34ECa(X), 9

He(X), 17He( f ), 4He( f ), 13He(X), 13Hilbs(P(E)), 39Hq(E), 57HS(X), 19

i, 2

j, 115Jac(L), 23

〈, ′〉, 526L(Z), 54Λ, 45

μ(φ), 33multxX, 35μ(X, x), 33

N( f ), 25

Ω f , 51Ω∨f , 51






622 Symbol index

O(E, q), 57

Pn, 4P(E), 4PB(| L |), 25Pn, 84, 85PO(3), 73PO(n + 1), 63PS(s, d; n), 47

Ram(φ), 29

〈S 1, S 2〉, 559Sd , 1S (3), 120S (E), 1s(n, d), 43ΣPGL(E), 11SL(U), 3Sn,C , 86St(X), 19

st, 20St(| L |), 23

T, 44T (E), 1Tx(X), 7Tφ, 44Tφ, 44TCa(X), 8Θk , 97

U, 235

Vnd , 32

vd , 32

wrk( f ), 52

X(3), 117

Z(s), 87






Subject index

(−1)-curve, 248(−n)-curve, 289(166)-configuration, 522Ak-singularity, 34F-locus, 281N-lateral, 256α-plane, 512β-plane, 512EN -lattice, 361k-secant line, 215s-lateral, 38

Abel–Jacobi Theorem, 198abelian surfaces

moduli, 545abelian variety

principally polarized, 199Abo, H., 45absolute invariant, 115aCM

sheaf, 164, 166, 167, 177, 296, 385ε-symmetric sheaf, 168of rank 1, 179symmetric, 168

subscheme, 296, 297, 299ADE singularity, 376adjoint

orbitminimal, 553nilpotent, 553supminimal, 553

variety, 553adjugate matrix, 7Alberich-Carraminana, M., 334Alexander, J. E., 42Alexander, J.W., 345Allcock, D., 482almost general position, 357Altman, A., 170

anti-polar, 51conic, 277

anticanonicaldivisor, 387embedding, 264linear system, 387model

of a del Pezzo surface, 384ring, 385

Antonelli, G., 591apolar

homogeneous form, 35quadrics, 98ring, 36subscheme, 36

apolarityduality, 131First Main Theorem, 42map, 48of conics, 112

Apollonius of Perga, 104Apollonius problem, 104apparent boundary, 7Arbarello, E., xi, 188, 218, 222, 224Arf invariant, 191Aronhold invariant, 115, 119, 130

symbolic expression, 137Aronhold set, 249Aronhold, S., 144, 278arrangement of lines, 256Artebani, M., 144, 266Artin, M., 351, 424associated

curve, 570line, 589sets of points, 455

association involution, 478August, F., 451, 452, 505azygetic, 207

623






624 Subject index

set of seven bitangents, 228triad of Steiner complexes, 208set in a symplectic space, 204tetrad of bitangents, 229triad in a symplectic space, 204triad of bitangents, 228, 229

Bohning, C., 227Babbage’s conjecture, 184Babbage, D.W., 184, 392Baker, H., 111, 215, 225, 405, 422, 425, 440,

503, 504Bardelli, F., 250Barth’s condition, 46Barth, W., 46, 85, 90, 112, 279, 324Bateman, H., 277, 279, 313Battaglini line complex, 547, 588Battaglini, G., 590Bauer, Th., 85, 90, 112Beauville, A., 168, 185, 186, 197, 392, 553Beklemishev, N., 479Beltrametti, M., xiiBertini

involution, 317, 413Theorem

on elliptic pencils, 344, 346on irreducibility, 75, 305, 443on singularities, 20, 177, 299, 305, 382, 562

Bertini, E., 345, 346, 424bezoutiant, 184bielliptic curve, 241bifid map, 193binary form

quadratic invariant, 138Binet, J., 591binode, 424birational map, 280Birkenhake, Ch., 216biscribed triangle, 244

of the Klein curve, 274bitangent

defined by Aronhold set, 228honest, 35hyperplane, 204matrix, 277their number, 215

bitangential curve, 215, 225Blache, R., 178blowing down structure, 355Bobillier, E., 66, 112bordered determinant, 99, 140, 154, 278Bordiga scroll, 300Bordiga surface, 300, 423, 440Bordiga, G., 300, 423, 440Borel, A., 366

Bos, H., 112Bottema, O., 584Bourbaki, N., 365, 425, 506, 591bracket-function, 467Bragadin, G., xiiBrambilla, M., 42, 45branch divisor, 29Brianchon’s Theorem, 80Brianchon, Ch., 80Briancon, J., 131Brill, A., 214, 225Bring curve, 498, 500Brioschi covariant, 140Brioschi, F., 140Bronowski, J., 67Bruno, A., 296bubble cycle, 308

admissible order, 308fundamental, 308

bubble space, 307height function, 307proper points, 307

Burch, L., 453Burhardt quartic threefold, 186Burns, D., 500

C. van Oss, 276Calabi–Yau variety, 25, 528Campbell, J.E., 67canonical

classof a Fano variety, 534of a normal surface, 178of a projective bundle, 321of a ruled surface, 322, 561of blow-up, 332of Grassmann variety, 75

equationof plane cubic, 117

map, 221Caporali quartic, 277Caporali, E., 277Caporaso, L., 204, 228Carletti, E., xiiCarlini, E., 45Cartan cubic, 435Cartan matrix, 362

irreducible, 363Cartan, E., 435Carter, R., 483Casnati, G., 168Castelnuovo, G., 345, 591Castelnuovo–Richmond quartic, 478, 481, 524,

545catalecticant, 266






Subject index 625

determinant, 50hypersurface, 50matrix, 48

Catanese, F., 168, 177, 183, 186, 187, 238, 239Cayley

cubic surface, 449dianode surface, 250family of cubic surfaces, 504octad, 245quartic symmetroid, 28

Cayley, A., 17, 28, 29, 35, 36, 83, 100, 112, 140,141, 144, 181, 214, 225, 250, 278, 345,433, 504, 549, 590, 591

Cayley–Brill formula, 214Cayley–Salmon equation, 503Cayley–Zeuthen formulas, 564Cayleyan

contravariant, 140curve, 21, 65, 142

of plane cubic, 127variety, 21

of a linear system, 24center variety, 528Chandler, K., 42characteristic matrix, 330Chasles

covariant quadric, 94, 112Principle of Correspondence, 225Theorem

on conjugate triangles, 77on linear line complex, 525, 590on polar tetrahedra, 93

Chasles, M., 93, 111, 112, 144, 224, 590–592Chipalkatti, J., 45Chisini, O., 65, 66Chow

form, 517, 548, 568, 590group, 290ring, 512

Ciani, E., 276, 279Ciliberto, C., 14, 67, 393circle

complex, 89real, 91

circulant matrix, 48class, 4, 30

of a space curve, 570, 573of immersion, 564

Clebschdiagonal cubic surface, 439, 497quartic curve, 255

nondegenerate, 256weakly nondegenerate, 256

Theorem, 334transfer principle, 138, 474, 481

Clebsch, A., 138, 140, 142, 144, 278, 424, 479,505

Clemens, C.H., 224Clifford, W., 345Coble, A., 112, 211, 224, 251, 278, 425, 481,

482, 506, 513Cohen, T., 266Cohen–Macaulay

module, 164sheaf, 169variety, 161

collineation, 10Colombo, E., 482complete

ideal, 283pentalateral, 256quadrangle, 264

complex equationof a quadric, 99

complex reflection, 122group, 103, 122, 276

compound matrix, 98adjugate, 98

conductor formula, 171conductor ideal, 170Cone Theorem, 371congruence of lines, 513

class, 513order, 513

conicapolar, 112conjugate triangles of, 76invariants of a pair, 100mutually apolar, 103Poncelet n-related, 82Poncelet related, 81self-polar triangles of, 73variety of pairs, 85

conic bundle, 328conjugate

conics, 112linear forms, 51linear subspaces, 92triangle, 76

contactcurves, 155

conics, 240cubics, 244of degree d − 1, 240

hyperplaneof a canonical curve, 189

manifold, 553contravariant, 22, 136

Cayleyan, 140Hermite, 141






626 Subject index

of a pair of conics, 104of a pair of quadrics, 112of a plane quartic, 266of a quartic curve, 265of a ternary cubic, 139on quartic ternary forms, 139Pippian, 141

Cook, R, 186Coolidge, J., 111, 112, 311, 313, 344coresidual point, 136, 251Cornalba, M., xi, 188, 197, 218, 222, 224correlation, 10

composition, 11conjugate points, 10dual, 11polarity, 11

correspondence, 212direct lateral, 215inverse, 213of type (4, 4), 224Scorza, 216symmetric, 213united point, 213valence, 213with valence, 224

Corti, A., 346Cossec, F., 28, 183, 187, 402, 459covariant, 4, 22, 136

Brioschi, 140Clebsch, 252, 263Hermite, 140Hessian, 255of a binary cubic, 61of a pair of conics, 104, 107, 108of a pair of quadrics, 112of a plane quartic, 265of a ternary cubic, 139of binary quartic, 61quadric, 94Scorza, 252

Coxeter, H., 66, 333, 369, 425, 507Coxeter–Dynkin diagram, 333, 363

extended, 366Crauder, B., 293Cremona

group, 326, 342, 346hexahedral equations, 464inequalities, 286

Cremona transformation, 284Bertini involution, 317Clebsch Theorem, 334cubo-cubic, 299de Jonquieres, 303de Jonquieres involution, 301determinantal, 345

F-locus, 287fixed points, 310fundamental point, 287, 308Geiser involution, 316given by pfaffians, 517indeterminacy point, 287multidegree, 286of degree 5, 378, 436ordered resolution, 329P-locus, 287quadro-cubic in P4, 293quadro-quadratic, 295regularizable, 342symmetric, 315

Cremona, L., 67, 112, 143, 144, 186, 225, 286,319, 344, 424, 505, 513, 585, 591

cross ratio, 105, 110, 467, 553cubic hypersurface, 65, 88, 124, 125, 472, 573

catalecticant, 49determinantal, 42, 49, 387fourfold, 392in P6, 388pfaffian, 517symmetroid, 402variety of lines, 589

cubic plane curveabsolute invariant, 115canonical equation, 117covariants and contravariants, 145dual curve, 128equianharmonic, 116harmonic, 116Hesse equation, 117Hesse pencil, 119its Cayleyan curve, 127its Hessian curve, 125Legendre equations, 116Weierstrass equation, 115

cubic scrollin P3, 444in P4, 444

cubic surface, 884-nodal, 551as a base of a Palatini scroll, 531Cayley–Salmon equation, 450Cayley surface, 449Cremona’s hexahedral equations, 465cyclic, 441, 462dual surface, 504Eckardt point, 440lines on it, 432moduli, 528moduli space, 479non-normal, 444projective generation, 452






Subject index 627

Sylvester nondegenerate, 260, 462symmetroid, 456tritangent plane, 412

cubic symmetroid, 456cuspidal edge, 575cyclide

curve, 400degenerate surface, 400Dupont surface, 422quartic surface, 399

D’Almeida, J., 569Dale, M., 41Darboux curve, 46, 256

its equation, 257Darboux’s Theorem, 83Darboux, G., 45, 67, 83, 112, 279, 423Dardanelli, E., 463, 479, 480de Jonquieres involution, 301de Jonquieres, E., 225, 344de Siebenthal, J., 366Debarre, O., 384defect, 515defective, 41

k-defective, 515variety, 515

degeneracy locus, 530, 533, 566degenerate

homogeneous form, 50multilinear form, 26

del Pezzo surface, 349, 353, 423Cremona isometries, 377degree, 355effective cone, 371its secant variety, 388lines on it, 383marked, 356nef cone, 372of degree 1, 411, 413, 414, 422, 500of degree 2, 242, 248, 250, 405, 410, 424,

441, 443of degree 4, 396, 400, 403, 422, 423, 459,

534, 535of degree 5, 389–392, 395, 422, 529, 589of degree 6, 288, 299, 386–388, 422, 451, 551of degree 7, 386of degree 8, 386, 422, 444

del Pezzo, P., 423Deligne, P., 306, 308Deligne–Hoskin formula, 308Demazure, M., 326, 425Dersch, O., 225Desargues’ Theorem, 77, 79

in space, 91Desargues, G., 77, 79, 91

desmictetrahedra, 96

determinantalhypersurface, 146equation, 240formula, 511hypersurface, 148, 151, 240representation

equivalence, 146of singular plane curves, 169of surfaces, 177quartic surfaces, 185

variety, 74resolution of singularities, 65

developable surface, 17, 580, 587, 589of a space curve, 570quartic, 587

Dickson, L., 144, 186, 506difference map, 216directrix, 558discrepancy divisor, 178discriminant, 19

hypersurface, 30cubic, 69its degree, 19its dual hypersurface, 32of linear system, 23of quadrics, 15tangent space, 32

of a binary cubic, 60of a binary form, 10of a binary quartic, 61of a general polynomial, 9symbolic expression, 138

divisor classbig, 353nef, 353

Dixmier, J., 265Dixon, A., 47, 186, 503Dolgachev, I., 3, 29, 35, 47, 51, 64, 67, 112, 115,

144, 148, 192, 219, 225, 239, 251, 258,262, 274, 276, 334, 346, 395, 402, 439,440, 455, 459, 467, 471, 478, 479, 482,506, 528, 589

double-pointclass, 563formula, 563set, 563

double-six, 427azygetic duad, 428azygetic triad, 429Steiner complex of triads, 429syzygetic duad, 428syzygetic triad, 429

Du Val singularity, 376






628 Subject index

Du Val, P., 345, 376, 406, 412, 424, 506dual

homogeneous form, 48dual variety, 29

degree, 33of a hypersurface, 29of a plane cubic, 142of a Veronese surface, 69of Grassmann variety, 519of Segre cubic primal, 478projective space, 4Reflexivity Theorem, 29

duality map, 30dualizing sheaf, 170Dupin, Ch., 423Durege, H., 144Dyck, W., 279Dynkin curve, 373Dynkin, E., 333, 366

Eckardt point, 440, 480–482defining an involution, 441

Eckardt, F., 506Edge variety, 392Edge, W., 274, 392, 395, 404, 422, 423, 581,

585, 592effective cone, 371Ehrenborg, R., 42Ein, L., 293Eisenbud, D., 36, 37, 219, 385, 452, 455elementary transformation, 324

of vector bundles, 324Ellingsrud, G., 297, 299Elliott, E., 145elliptic normal curve, 114Elte, E., 425, 507Emch, A., 91Enriques diagram, 308Enriques surface, 463, 589Enriques, F., 65, 66, 308, 326, 463envelope, 4enveloping cone, 9equianharmonic

plane cubic, 131quadruple, 105

Euler exact sequence, 321dual, 321

Euler formula, 6exceptional

curve, 351divisor, 282section, 320type, 335vector, 367

extremal ray, 371

Fano variety, 64, 109, 263, 264, 355, 501, 528,531, 534

degree, 263genus, 263index, 263of genus 12, 264, 502toric, 288

Fano, G., 264Farkas, G., 221fat point, 40Fay, J., 222Fermat hypersurface

cubic curve, 125cubic surface, 442, 480, 497plane cubic, 129, 130, 143, 253plane quartic, 252, 269, 272, 276

Ferrers, N., 112fiber

of a sheaf, 162Fielder, W., 479, 506Finkelnberg, H., 477First Fundamental Theorem, 3, 509Fischer, G., 16, 35Fitting ideal, 162Flatto, L., 90, 112Formanek, E., 159Frahm, W., 67Freitag, E., 482Fresnel’s wave surface, 551Fresnel, A., 551Fricke, R., 274Frobenius, G., 278Fulton, W., 35, 104, 134, 163, 290, 394, 477,

511, 512, 530, 533, 563, 565, 566, 591functional determinant, 13fundamental

cycle, 351point, 281set, 210

normal, 211weight, 366

Gopel, A., 224Gallarati, D., xiiGantmacher, F., 397Gauss curvature, 16Gauss map, 30, 218, 222, 516Geiser involution, 316, 408, 409Geiser, C., 345, 424Gelfand, I., 19, 29, 35, 144, 517, 568general position, 357genus 4 curve, 189, 214, 224, 237, 241, 277,

414, 438, 498, 573geometric basis, 355geometric marking, 355






Subject index 629

Geramita, A., 43, 67Gerbardi, F., 85, 102Gergonne, J., 66Giambelli, G., 187Giorgini, G., 590GIT-quotient, 81, 101, 111, 228, 266, 395, 467,

479, 506Gizatullin, M., 45, 245Glass, J., 278Godeaux, L., 346Gonzalez-Aguilera, V., 269Gonzales-Sprinberg, G., 345Goodman, R., 57Gordan, P., 14, 102, 112, 144, 275Gorenstein

curve, 174Fano variety, 264local Artinian ring, 36normal surface, 178ring, 171singularity, 352

Gosset, T., 369, 425, 507Grace, J., 108, 112, 145Grassmann bundle, 510Grassmann variety, 21, 508

canonical sheaf, 510cohomology ring, 511degree, 512its dimension, 510of lines, 508Plucker embedding, 508Plucker equations, 509secant variety of, 513, 515tangent sheaf, 510tangent space, 516universal quotient bundle, 509universal subbundle, 509

Grassmann, H., 143, 505, 590Greuel, G.-M., 174, 176Griffiths, Ph., xi, 28, 35, 112, 134, 188, 198,

215, 218, 222, 224, 358, 570Gross, B., 191Grothendieck A., 166Grushevsky, S., 221Guardia, J., 234Gundelfinger quartic, 245Gundelfinger, S., 144, 245

Hacking, P., 482Halphen pencil, 129, 344, 346Halphen, G., 144, 346Hankel matrix, 48Hankel, H., 48harmonic

binary quartic, 116

conic-locus, 113conjugate, 72, 105, 110cubic curve, 116line complex, 547polar line, 118polynomial

as a pfaffian, 60of degree 2, 98

quadruple, 105tensor, 56

harmonizant, 67Harris, J., xi, 7, 28, 35, 42, 74, 112, 117, 134,

175, 181, 188, 198, 215, 218, 222, 224,347, 358, 394, 559, 570, 572

Harris. J., 191Hartshorne, R., xii, 8, 114, 165, 172, 213, 270,

283, 290, 303, 305, 320, 322, 324, 350,358, 382, 510, 558, 560, 562, 569, 583, 584

Hassett, B., 266Hawkins, T., 591Heisenberg group, 493, 495, 522, 543, 547Henderson, A., 504Hermite

contravariant, 141covariant, 141curve, 143

Hermite, Ch., 141Hesse

arrangement of lines, 118dual, 119

canonical equationof plane cubic, 117

formof a plane cubic curve, 114

group, 121pencil, 119quadrilateral, 110Theorem, 110

Hesse, O., 67, 110, 112, 117, 118, 144, 154, 278Hesse–Salmon configuration, 142Hessian

determinant, 4hypersurface, 13matrix, 13of a binary quartic, 61surface, 65

hexad, 230Hilbert modular surface, 500Hilbert scheme

of aCM subschemes, 297of lines, 264of projective plane, 81, 109, 130, 131of projective space, 47punctual, 39

Hilbert, D., 47, 67, 453






630 Subject index

Hilbert–Birch Theorem, 504Hill, J. E., xiiHirschowitz, A., 42Hirzebruch, F., 500Hitchin, N., 57, 60, 112, 502Hodge Index Theorem, 358Hodge type inequality, 286Hodge, W., 397, 509, 511, 591homaloid, 284, 345homaloidal

netcharacteristic, 306

polynomial, 148type, 335

homology, 440harmonic, 440its center, 440

Hoskin, M., 306, 308Hosoh, T., 506Hudson, H., 342, 344Hudson, R.W., 590Humbert curve of genus 5, 404, 405Humbert, G., 404Hunt, B., 479Hurwitz formula, 29Hurwitz’s Theorem, 270Hurwitz, A., 225, 270Hutchinson, J., 506hyperelliptic curve, 224

2-torsion divisor classes, 193, 210and Kummer surface, 539de Jonqueres transformations, 318equation, 192its Jacobian variety, 534of genus 3, 185, 224plane model, 319, 343theta characteristics, 194, 212Weil pairing, 194

hyperosculating point, 570multiplicity, 572their number, 572

hyperplane, 4hypersurface, 4

monoidal, 301submonoidal, 301

Iano-Fletcher, A., 192Iarrobino, A., 36, 40, 47, 50, 67, 309icosahedron

fundamental set, 499icosahedral set, 501

Igusa quartic, 545Iliev, A., 47incidence variety, 510indeterminacy point, 280

infinitely near point, 307inflection

bitangent, 226, 266point, 17

order, 18tangent, 12, 65, 276

honest, 35triangle, 118

integral closure, 282intersection matrix, 352invariant, 136

absolute, 116Aronhold, 137bracket-function, 467First Fundamental Theorem, 3Joubert, 471of 6 lines in P3, 527of a pair of binary forms, 106of binary forms, 10of binary quartic, 117of binary quartics, 49, 61, 104of complex reflection group, 122of cubic surface, 479of Hesse group, 123of plane quartics, 265of ternary cubic, 136of two symmetric matrices, 99of Valentiner group, 103relative, 123symbolic expression, 137tact, 100Toeplitz, 45weight, 137

inversion transformation, 314Inversive group, 326Iskovskikh, V., 264, 346isologue, 309

center, 309net, 310

isotropic subspace, 190Izadi, E., 221, 239

j-invariant, 115Jozefiak, T., 181Jacobi, C., 278Jacobian

curve, 410determinant, 13hypersurface, 23ideal, 33variety, 197

intermediate, 534Jessop, C., 405, 423, 590Jeurissen, R., 276join






Subject index 631

of projective subbundles, 558of scrolls, 559

Jordan, C., 425, 506Joubert functions, 471Joubert, P., 471, 506Jung, H., 171Jung–Milnor formula, 171

K3 surface, 28, 56, 264, 395, 404, 405, 463,537, 538, 543, 550

Kane, R., 365, 406Kanev, V., 47, 67, 225, 258, 262, 274, 276Kantor, S., 345, 425, 506, 591Kapranov, M., 19, 29, 35, 112, 144, 439, 440,

517, 568, 589Katsylo, P., 227, 239, 250Katz, S., 293Keel, S., 482Kers, C., 112Keum, J., 506, 546Kirkman points, 111Kirkman, J., 111, 112Kleiman, S., 35, 170, 510Klein

coordinates, 522quadric, 512quartic curve, 252, 270, 276, 589

automorphisms, 273its bitangents, 277its hessian, 275

sextic, 499singularities, 376

Klein, F., 270, 274, 375, 376, 400, 499, 500,505, 512, 590

Kleppe, H., 74Knorrer, H., 176Kneebone, G., 591Kodaira

fibers of elliptic fibration, 415, 550Kodaira, K., 415, 550Kollar, J., 371, 372, 375, 425Kondo, S., 266, 482, 546Kravitsky, N., 184, 186Krazer, J., 224Kummer cones, 423Kummer surface, 536, 538

and 5-cuspidal sextic, 404and Segre cubic, 589as the Hessian surface, 506associate hyperelliptic curve, 536, 538associated to a quadratic line complex, 538automorphisms, 547double plane model, 542octic model, 539of an abelian surface, 538

quartic equation, 543self-duality, 546Tetrahedroid, 549wave surface, 552

Kummer variety, 239, 538jacobian, 538

Kummer, E., 423, 590

Luroth quartic, 68, 257determinantal representation, 260pentalateral theta characteristic, 262

Luroth, J., 255, 257, 279Le Dung Trang, 33La Hire, Ph., 113Laguerre net, 248, 312Laksov, D., 74Lange, H., 216Laplace operator, 57Lascoux, A., 181lattice, 331, 358

I1,N , 331, 360EN , 334, 361discriminant, 358discriminant group, 358embedding, 360even, 360isometry, 360nondegenerate, 358orthogonal group, 360primitive embedding, 360signature, 358sublattice, 358

finite index, 358primitive, 358

unimodular, 358Lazarsfeld, R., 282, 286, 294, 354, 563Le Potier, J., 263LeBarz, P., 216Lefschetz’s fixed-point formula, 483Lefschetz, S., 483Legendre equation

of a plane cubic, 116Legendre, A.-M., 116Lehavi, D., 228, 243Libgober, A., 175Lie, S., 518, 590Lifsic, M., 186Lindemann, F., 140, 142, 144, 424line complex, 517

apolar, 520cubic

Montesano, 589degree, 517linear, 517

center, 518






632 Subject index

monoidal, 589quadratic, 99, 156, 531

Battaglini, 547harmonic, 547Kummer surface, 538lines on it, 537of tangent lines to a quadric, 156tangential, 552tetrahedral, 553, 586

rank, 518singular line, 533singular variety, 533special, 517

line-equationof a quadric, 99

linear systembase ideal, 281base locus, 281base scheme, 281base-point-free, 281homaloidal, 284

linearly d-independent, 39lines

conjugate, 77in a quadratic line complex, 537on a cubic threefold, 589on a weak del Pezzo surface, 383six linearly dependent, 526two transversals to four, 525

Lipman, J., 409Lo Giudice, G. , 239London, F., 45, 68, 144Looijenga, E., 266, 425, 482Loria, G., 590Lossen, C., 14Lurie, J., 435

Mobius, A., 590Muller, H., 591Merindol, J., 425Macaulay, F.S., 36, 187MacLaurin, C., 111, 344Magnus, L., 345, 504Manin, Yu., 346, 425, 483, 507Marcus, A., 186marking, 356

geometric, 356Massoti Biggiogero, G., 66Mathews, R., 96Mella, M., 47, 393Melliez, F., 276, 277, 502Meyer, W., xii, 504, 513Michel, J., 112Miles, E.P., 58Milne, W., 141

Milnor number, 33Milnor, J., 171minimal degree varieties, 347minimal rational ruled surface, 320minus vector, 492mixed combinant, 136mixed concomitant, 136modular

family, 120surface, 500

moduli spaceRg, 239Mev

3 , 250Mev

g , 197of 6 points, 477, 480of 7 points, 250of abelian surfaces, 545of bielliptic curves, 277of cubic surfaces, 440, 479, 482, 491, 528of determinantal representations, 159of elliptic curves, 121of nets of quadrics, 262of plane quartics, 266of quadratic line complexes, 549of reflexive sheaves, 170

Monge’s differential equation, 518Mori, S., 371, 372, 375Morley, F., 262, 279, 326Morley, F.V., 326Morrison, I., 572Moutard, M., 423Muir, T., 65Mukai skew-form, 54Mukai, S., 47, 75, 112, 244, 264, 265, 396, 501multidegree

of a rational map, 285multiplicity

of a singular point, 351Mumford, D., 172, 178, 188, 189, 196, 200, 213,

224, 239, 352

Nagata, M., 345Naruki, I., 482, 491, 500net, 45

of conics, 109, 110, 143, 241, 243, 245, 386of cubics, 250, 317of quadrics, 45, 46, 246, 247, 260, 261, 265,

274, 460, 462, 513, 584Newton, I., 144Nikulin, V., 360node, 34

on a surfaceeven set, 183weakly even set, 183

Noether formula, 566






Subject index 633

Noether’s Reduction Theorem, 339Noether, M., 14, 345nondegenerate

Clebsch quartic, 256homogeneous form, 50subvariety, 347

normallinearly, 349projectively, 177, 385scroll, 557subvariety, 349surface

canonical class, 178intersection theory, 178

normal system, 210null polarity, 11null-circle, 90null-plane, 520null-point, 520null-system, 11, 210, 520

OADP subvariety, 392, 423Okonek, C., 300, 531Oort, F., 112ordinary singularities, 563orthogonal group, 57, 63, 331, 334Ortland, D., 219, 251, 395, 455, 471, 478oscnode, 407osculating

developable surface, 570hyperplane, 142plane, 570sheaf, 570

Ottaviani, G., 42, 43, 45, 130, 263, 279, 300,313, 591

pairs of conicsGIT-quotient, 101invariants, 100

Palatiniruled surface, 531scroll, 530

Palatini, F., 47, 67, 530, 591Pan, I., 302, 342, 345parabolic hypersurface, 17parabolic point, 17partial normalization, 171Pascal line, 80Pascal’s Theorem, 79Pascal, B., 79Pascal, E., 66, 279, 504, 590Pash, M., 590Pedoe, D., 397, 509, 511, 591pencil, 45

of conics, 83, 131, 135, 144, 237, 276, 390,585

of cubics, 126, 129, 159of quadrics, 100, 246, 250, 396, 397, 403,

404, 423, 424, 534, 550, 556, 587of quartics, 252

Perazzo primal, 504Perazzo, U., 504period matrix, 198perspectivity, 77

center, 77line of , 77of simplexes, 91

perspectrix, 77Persson, U., 415Petersen graph, 390, 422Pfaff differential equation, 518pfaffian, 74, 108, 130, 514, 516, 529pfaffian hypersurface, 515

and Palatini scroll, 530cubic, 517cubic fourfold, 392cubic in P14, 517

Pfister, G., 174Picard scheme, 198

relative, 159Piene, R., 570, 573Pieri’s formula, 512pinch point, 563Pinkham, H., 351, 375, 425Piontkowski, J., 172, 176Pippian contravariant, 141pippiana, 67Plucker

coordinates, 509formula

for hypersurfaces, 33for plane curves, 35for space curves, 571

formulas, 33lines, 111

Plucker, J., 66, 111, 225, 278, 504, 590Plucker–Teissier formula, 33, 60plane quartic curve

even theta characteristic, 246simple singularities, 406

Plaumann, D., 277plus vector, 492Poisson bracket, 53polar

s-gon, 38s-hedron, 38

generalized, 39nondegenerate, 38

base locus, 25






634 Subject index

bilinear form, 5, 189duality, 520hypersurface, 5

first, 7second, 124

line, 71, 520linear subspaces, 92map, 28net of quadrics, 45pairing, 2pentagon, 255pentalateral, 255quadrangle, 129

generalized nondegenerate, 129quadric, 12subspace, 528

polarity, 11polarization map

partial, 2total, 1

Polarraum, 111pole, 71, 74Polo-Blanco, I., 584poloconic, 140, 141polygon, 82

side, 82vertex, 82

polytoperegular, 369semi-regular, 369

Poncelet related curve, 86Poncelet, J.-V., 111, 278, 344Popescu, S., 219, 455porism, 91Postulation formula, 284Pragacz, P., 181prime-form, 222principal curve

total, 330principal parts, 569Principle of Correspondence, 224pro-Hessian surface, 17projective bundle, 320

canonical class, 321projective coordinates, 4projective generation, 163

determinantal varieties, 163of a cubic curve, 144of a cubic surface, 187, 452of plane curves, 310of rational normal curve, 134Steiner’s construction, 93

projective space, 4prospector, 77Prym canonical map, 223

Prym map, 239Prym, F., 223

quadratic formeven, 191odd, 191

quadratic transformation, 314quadric

complex equation, 99invariants of a pair, 96line-equation, 99polar pentahedron, 95polar tetrahedra, 93

quadric bundle, 156, 531discriminant locus, 531

quadrilateral, 88, 110quartic hypersurface

Burhardt, 186Castelnuovo–Richmond, 478, 481, 524, 545Igusa, 545Scorza, 189

quartic plane curveAronhold invariant, 258automorphisms, 266bitangents, 226

Aronhod sets, 228azygetic triads, 228Steiner complexes, 226syzygetic triads, 226

Caporali, 277Clebsch, 255contravariants, 266covariants, 265determinantal equation, 235

symmetric, 239invariants, 265Klein, 270Luroth, 257the variety of sums of powers, 263

quartic surface4-nodal, 402del Pezzo, 397desmic, 96developable, 580dual of Cayley cubic, 449Gundelfinger, 245Kummer, 404ruled, 575

classification, 576equations, 586

Segre, 397Steiner, 449Tetrahedroid, 549wave surface, 551

Quippian contravariant, 141






Subject index 635

Ramanujam’s Vanishing Theorem, 354Ramanujam, C., 354Ranestad, K., 47, 263, 276, 277, 502rank

of a curve, 570rational elliptic surface, 344, 414, 415, 423rational map, 280

inverse transform under, 281its resolution, 282

rational normal curveassociated to a net of quadrics, 264equations, 135of degree 4, 581, 589secant variety, 49, 88secants of, 513

rational plane curvedeterminantal equation, 184

Raven, D., 112real sphere, 400reciprocity theorem, 6reflection, 334, 365reflexive sheaf, 165Reflexivity Theorem, 29Rego, C., 170regular linear system, 26Reichstein, Z., 144Reid, M., 351, 353, 504relative Picard scheme, 196Reskine, C., 297resolution

minimal, 330Reye

congruence, 28line, 24line complex, 591variety, 27

Reye, T., 67, 111, 112, 506, 513, 590Richmond, H., 47, 67Riemann

constant, 200equation of bitangents, 230

Riemann, B., 189, 200, 223, 224, 278Riemann–Kempf Theorem, 199Riemann–Mumford relation, 189Ritzenthaler, C., 243Rodenberg, C., 463Rodriguez, R., 269Room, T., 260, 440root, 362

basis, 362canonical, 363irreducible, 362

effective, 372function, 223lattice, 363

nodal, 372positive, negative, 364sublattice, 366

Rosanes, J., 67, 112, 345Rosenberg, J., 506Rosenhein, J., 224Rota, G.-C., 42Roth, L., 299Roth, P., 278Rowe, D., 591ruled surface, 17, 560

of degree 4, 585contact curve, 564elliptic, 575elliptic of degree 6, 531exceptional section, 561genus, 562minimal, 561minimal surface Fn, 322normalized vector bundle, 561of degree 3, 443, 575of degree 8, 300Palatini, 531

ruled variety, 557Russo, F., 393

Saavedra Rivano, N., 224Salmon

conic, 106, 107envelope conic, 107invariant, 266

Salmon, G., 67, 111–113, 115, 136, 144, 215,225, 247, 252, 266, 278, 479, 504, 506, 591

Salvatti Manni, R., 221Sankaran, G., 239Sarkisov program, 346Sarkisov, V., 346satellite conic

of a plane cubic, 143Scheyer, F.-O., 47Schlafli equation, 142Schlafli’s Theorem, 503Schlafli, L., 144, 505Schlesinger, O., 143, 144Schoenberg, I., 90Schoute, P., 506Schroter, H., 145Schreyer, F.-O., 263, 264Schubert

class, 511cycle, 511variety, 511

special, 511Schubert, H., 591Schur quadric, 437, 498, 527






636 Subject index

Schur sextic, 438, 499Schur, F., 436Schwarz, H., 592Schwarzenberger vector bundle, 86, 588Schwarzenberger, R. L. E., 86, 112Scorza

correspondence, 216, 221, 224, 568general pair, 219

covariant, 252map, 265quartic hypersurface, 221

Scorza, G., 216, 219, 225, 279scroll, 349, 444, 530, 557, 577

r-directrix, 558Bordiga, 300cubic in P3, 444cubic in P4, 444degree, 559generator, 557join, 559normal, 557Palatini, 530rational normal, 347

of dimension 2, 347, 575tangential, 577, 579, 580

secant variety, 41defective, 44of a del Pezzo surface, 388of a rational normal curve, 42, 65, 588of Grassmann variety, 513of rational normal curve, 49of Segre–Veronese variety, 44of Veronese variety, 41, 42

Segreclass, 290cubic primal, 472, 529, 543, 544, 549, 589quartic surface, 397symbol, 397, 424variety, 65, 422, 514, 559

Segre, B., 505, 591Segre, C., 225, 345, 400, 423, 505, 588, 590Segre–Veronese embedding, 43Segre–Veronese variety, 40

secant variety of, 45self-associated sets, 219self-conjugate

hexad, 111pentad, 95polyhedron, 91tetrad in the plane, 110tetrahedron, 92

semi-stable points, 467Semple, J., 299, 591Sernesi, E., 204, 228, 279, 313Severi F., xi

Severi–Zak variety, 294, 515sextactic point, 142Shafarevich, I. R., xiiShephard, G., 122Shepherd-Barron, N., 293Shioda, T., 265simplex, 91

edge, 91face, 91facet, 91mutually polar, 91vertex, 91

singular line, 537singular point

of Kummer variety, 538singular subspace, 190singular variety

of a quadratic line complex of lines, 531of line complexes of lines, 528

singularityAk , 34ADE, 175binode, 424complete intersection, 33cusp

ordinary, 34rhamphoid, 407

Du Val, 376Gorenstein, 352multiplicity, 351node, 34ordinary isolated, 31rational, 351rational double point, 351rational Gorenstein, 352simple, 175, 376small resolution, 477tacnode, 407

sixer, 426Smith, R., 239socle, 36Sommerville, 99Sommerville, D., 113Sousley, C., 481space curve

m-rank, 570associated, 570dual, 570ordinary point, 570Plucker formula, 571quartic curves

species, 117stationary point, 570

Springer, T., 122, 416stable points, 467






Subject index 637

standard quadratic transformation, 287degenerate, 288

standard symplectic basis, 190standard tableaux, 471star, 163star-duality, 555Steenbrink, J., 276Steiner

complexin a symplectic space, 206

complexesazygetic triad, 208syzygetic, 207syzygetic triad, 208

points, 111polygon, 143projective generation, 93, 134quartic surface, 70

its dual, 449Steiner, J., 66, 70, 90, 93, 111, 112, 134, 143,

144, 278, 424, 505, 506Steinerian hypersurface, 19, 23

as a covariant, 22its degree, 20of a linear system, 23

Steinerian map, 20Stipins, J., 96Stuart, T., 47Sturm, R., 111, 345, 505, 592Sturmfels, B., 277sums of powers

variety VSP( f , s), 46, 560-dimensional, 47explicit description, 47Fano model, 263of a binary form, 62of a conic, 64of a plane curve, 264of a quadric, 64of a set of forms, 45

Waring problem, 47Sylvester

equation of a cubic surface, 261, 462nondegenerate, 260

pentahedron, 463Sylvester, J., 47, 67, 136, 345, 424, 504symbolic method, 3, 137symmetric algebra, 1symmetric power, 1symmetroid surface, 181

cubic, 238quintic, 185

symplectic group, 191syntheme, 465syzygetic

pencil, 144Steiner complexes, 207tetrad in a symplectic space, 204tetrad of bitangents, 229tetrad of theta characteristics, 206tetrads in Steiner complex, 224triad in a symplectic space, 204triad of Steiner complexes, 208triad of theta characteristics, 205

Szpiro, L., 297

tact-invariant, 100Takagi, H., 219tangent cone, 8tangent space

embedded, 7tangential variety

of a space curve, 570of a Veronese surface, 69of an elliptic normal curve, 573of rational normal curve, 572

tautological exact sequence, 509Taylor, D., 141Teissier, B., 33Terracini’s Lemma, 41, 42Terracini, A., 41, 45, 67tetrahedral line complex, 586Tetrahedroid quartic surface, 549Tevelev, E., 29, 482theta

characteristic, 151, 169, 188effective, 151even, odd, 151Scorza invariant of, 218syzygetic triad, 204their number, 189vanishing, 189, 224

divisoreven, odd, 199symmetric, 199

factor, 200function

Riemann, 200with characteristic, 200

theta function, 200Thom–Porteous formula, 533Thomas, A., 186Thomsen, H., 266Tikhomirov, A., 263Tikhomirov, S., xiiTimms, G., 401, 425Todd, J., 112, 122, 185, 506Toeplitz invariant, 45, 462Toeplitz map, 44Toeplitz, E., 45, 68, 462






638 Subject index

Togliatti, E., 422Top, J., 584toric variety, 326

Fano, 288of type An, 299surface, 387

torsal generator, 567, 589torsion-free sheaf

global invariant, 172local invariant, 174local type, 172

total, 465transversal lines, 452, 526Trautman, G., 112triangle, 71

circumscribed, 76inscribed, 76polar, 71polar degenerate, 73self-conjugate, 76side, 71vertex, 71

trianglesperspective, 77

trigonal construction, 238trisecant

plane, 73, 588ruled surface, 300

trisecant line, 299, 529, 536, 560tritangent plane, 433

conjugate triads, 431trope, 396, 545Tu, L., 74, 181Turnbull, H.W., 60, 99, 112Tyurin, A., 186, 263, 324, 455

Umemura, H., 501universal

quotient bundle, 63subbundle, 63

unode, 424Urabe, T, 425

V( f ), 4Valentiner group, 102Valentiner, H., 102Valles, J., 109, 112, 422Van de Ven, A., 294van den Bergh, M., 159van den Dries, B., 51van der Geer, G., 278, 478van der Put, M., 584van der Vries, J., 589van Geemen, B., 260, 463, 479, 480, 482Varley, R., 405

vector bundle, 320Veronese map, 32, 40Veronese surface, 41, 109, 588

dual variety of, 69of degree 4, 69projected, 70, 529

Veronese variety, 32, 40embedded tangent space, 65its dual hypersurface, 32secant variety of, 41, 42, 67, 240tangent space, 42

Verra, A., 239, 243, 296, 541Vinnikov, V., 186Vinzant, C., 277von Staudt’s Theorem, 554von Staudt, G., 112, 554, 590Voss, A., 590

Wall, C.T.C., 186Wallach, N., 57Waring problem, 46Waring rank, 52

exceptional cases, 52web, 45Weber, H., 223, 224, 235, 275, 278, 506Weierstrass

equationof a plane cubic, 115of elliptic surface, 415

formof a plane cubic, 114

point, 573Weierstrass, K., 144, 424weighted projective space, 192

adjunction formula, 192weights

minisculetritangent trio, 430

Weildivisor, 165pairing, 188Theorem, 534

Weil, A., 534Weiler, A., 424, 590Weyl chamber, 365

face, 365Weyl group, 334, 365Weyl, H., 506White surface, 440, 504White, F.S, 440White, H.W., 145Williams, E., 58Wiman pencil, 395, 423Wiman plane sextic, 395, 501Wiman, A., 506






Subject index 639

Winger, R., 501Wirtinger plane sextic, 237Wirtinger, W., 238Wong, B., 585, 592Wronskian, 588

Young, A., 112, 145Young, W., 591

Yuzvinsky, S., 96

Zak, F., 29, 41, 294Zariski, O., xi, 283, 308Zelevinsky, A., 19, 29, 35, 144, 517, 568Zeuthen, H., 591Zindler, K., 590Zuccone, F., 219






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