DOCSLIB.ORG

Classical Algebraic Geometry: a Modern View Igor V

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

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 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-01765-8 - Classical Algebraic Geometry: A Modern View Igor V. Dolgachev Index More information 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 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-01765-8 - Classical Algebraic Geometry: A Modern View Igor V. Dolgachev Index More information 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 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-01765-8 - Classical Algebraic Geometry: A Modern View Igor V. Dolgachev Index More information 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 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-01765-8 - Classical Algebraic Geometry: A Modern View Igor V. Dolgachev Index More information 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 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-01765-8 - Classical Algebraic Geometry: A Modern View Igor V. Dolgachev Index More information 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 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-01765-8 - Classical Algebraic Geometry: A Modern View Igor V.
Recommended publications
  • The Geometry of Syzygies

    The Geometry of Syzygies

    The Geometry of Syzygies A second course in Commutative Algebra and Algebraic Geometry David Eisenbud University of California, Berkeley with the collaboration of Freddy Bonnin, Clement´ Caubel and Hel´ ene` Maugendre For a current version of this manuscript-in-progress, see www.msri.org/people/staff/de/ready.pdf Copyright David Eisenbud, 2002 ii Contents 0 Preface: Algebra and Geometry xi 0A What are syzygies? . xii 0B The Geometric Content of Syzygies . xiii 0C What does it mean to solve linear equations? . xiv 0D Experiment and Computation . xvi 0E What’s In This Book? . xvii 0F Prerequisites . xix 0G How did this book come about? . xix 0H Other Books . 1 0I Thanks . 1 0J Notation . 1 1 Free resolutions and Hilbert functions 3 1A Hilbert’s contributions . 3 1A.1 The generation of invariants . 3 1A.2 The study of syzygies . 5 1A.3 The Hilbert function becomes polynomial . 7 iii iv CONTENTS 1B Minimal free resolutions . 8 1B.1 Describing resolutions: Betti diagrams . 11 1B.2 Properties of the graded Betti numbers . 12 1B.3 The information in the Hilbert function . 13 1C Exercises . 14 2 First Examples of Free Resolutions 19 2A Monomial ideals and simplicial complexes . 19 2A.1 Syzygies of monomial ideals . 23 2A.2 Examples . 25 2A.3 Bounds on Betti numbers and proof of Hilbert’s Syzygy Theorem . 26 2B Geometry from syzygies: seven points in P3 .......... 29 2B.1 The Hilbert polynomial and function. 29 2B.2 . and other information in the resolution . 31 2C Exercises . 34 3 Points in P2 39 3A The ideal of a finite set of points .
  • INTRODUCTION to ALGEBRAIC GEOMETRY, CLASS 25 Contents 1

    INTRODUCTION to ALGEBRAIC GEOMETRY, CLASS 25 Contents 1

    INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 25 RAVI VAKIL Contents 1. The genus of a nonsingular projective curve 1 2. The Riemann-Roch Theorem with Applications but No Proof 2 2.1. A criterion for closed immersions 3 3. Recap of course 6 PS10 back today; PS11 due today. PS12 due Monday December 13. 1. The genus of a nonsingular projective curve The definition I’m going to give you isn’t the one people would typically start with. I prefer to introduce this one here, because it is more easily computable. Definition. The tentative genus of a nonsingular projective curve C is given by 1 − deg ΩC =2g 2. Fact (from Riemann-Roch, later). g is always a nonnegative integer, i.e. deg K = −2, 0, 2,.... Complex picture: Riemann-surface with g “holes”. Examples. Hence P1 has genus 0, smooth plane cubics have genus 1, etc. Exercise: Hyperelliptic curves. Suppose f(x0,x1) is a polynomial of homo- geneous degree n where n is even. Let C0 be the affine plane curve given by 2 y = f(1,x1), with the generically 2-to-1 cover C0 → U0.LetC1be the affine 2 plane curve given by z = f(x0, 1), with the generically 2-to-1 cover C1 → U1. Check that C0 and C1 are nonsingular. Show that you can glue together C0 and C1 (and the double covers) so as to give a double cover C → P1. (For computational convenience, you may assume that neither [0; 1] nor [1; 0] are zeros of f.) What goes wrong if n is odd? Show that the tentative genus of C is n/2 − 1.(Thisisa special case of the Riemann-Hurwitz formula.) This provides examples of curves of any genus.
  • [Math.AG] 1 Nov 2005 Where Uoopimgop Hscntuto Ensahlmrhcperio Holomorphic a Defines Construction This Group

    [Math.AG] 1 Nov 2005 Where Uoopimgop Hscntuto Ensahlmrhcperio Holomorphic a Defines Construction This Group

    A COMPACTIFICATION OF M3 VIA K3 SURFACES MICHELA ARTEBANI Abstract. S. Kond¯odefined a birational period map from the moduli space of genus three curves to a moduli space of degree four polarized K3 surfaces. In this paper we extend the period map to a surjective morphism on a suitable compactification of M3 and describe its geometry. Introduction 14 Let V =| OP2 (4) |=∼ P be the space of plane quartics and V0 be the open subvariety of smooth curves. The degree four cyclic cover of the plane branched along a curve C ∈ V0 is a K3 surface equipped with an order four non-symplectic automorphism group. This construction defines a holomorphic period map: P0 : V0 −→ M, where V0 is the geometric quotient of V0 by the action of P GL(3) and M is a moduli space of polarized K3 surfaces. In [15] S. Kond¯oshows that P0 gives an isomorphism between V0 and the com- plement of two irreducible divisors Dn, Dh in M. Moreover, he proves that the generic points in Dn and Dh correspond to plane quartics with a node and to smooth hyperelliptic genus three curves respectively. The moduli space M is an arithmetic quotient of a six dimensional complex ball, hence a natural compactification is given by the Baily-Borel compactification M∗ (see [1]). On the other hand, geometric invariant theory provides a compact pro- jective variety V containing V0 as a dense subset, given by the categorical quotient of the semistable locus in V for the natural action of P GL(3). In this paper we prove that the map P0 can be extended to a holomorphic surjective map P : V −→M∗ on the blowing-up V of V in the point v0 corresponding to the orbit of double arXiv:math/0511031v1 [math.AG] 1 Nov 2005 e conics.
  • Shadow of a Cubic Surface

    Shadow of a Cubic Surface

    Faculteit B`etawetenschappen Shadow of a cubic surface Bachelor Thesis Rein Janssen Groesbeek Wiskunde en Natuurkunde Supervisors: Dr. Martijn Kool Departement Wiskunde Dr. Thomas Grimm ITF June 2020 Abstract 3 For a smooth cubic surface S in P we can cast a shadow from a point P 2 S that does not lie on one of the 27 lines of S onto a hyperplane H. The closure of this shadow is a smooth quartic curve. Conversely, from every smooth quartic curve we can reconstruct a smooth cubic surface whose closure of the shadow is this quartic curve. We will also present an algorithm to reconstruct the cubic surface from the bitangents of a quartic curve. The 27 lines of S together with the tangent space TP S at P are in correspondence with the 28 bitangents or hyperflexes of the smooth quartic shadow curve. Then a short discussion on F-theory is given to relate this geometry to physics. Acknowledgements I would like to thank Martijn Kool for suggesting the topic of the shadow of a cubic surface to me and for the discussions on this topic. Also I would like to thank Thomas Grimm for the suggestions on the applications in physics of these cubic surfaces. Finally I would like to thank the developers of Singular, Sagemath and PovRay for making their software available for free. i Contents 1 Introduction 1 2 The shadow of a smooth cubic surface 1 2.1 Projection of the first polar . .1 2.2 Reconstructing a cubic from the shadow . .5 3 The 27 lines and the 28 bitangents 9 3.1 Theorem of the apparent boundary .
  • Hyperbolic Monopoles and Rational Normal Curves

    Hyperbolic Monopoles and Rational Normal Curves

    HYPERBOLIC MONOPOLES AND RATIONAL NORMAL CURVES Nigel Hitchin (Oxford) Edinburgh April 20th 2009 204 Research Notes A NOTE ON THE TANGENTS OF A TWISTED CUBIC B Y M. F. ATIYAH Communicated by J. A. TODD Received 8 May 1951 1. Consider a rational normal cubic C3. In the Klein representation of the lines of $3 by points of a quadric Q in Ss, the tangents of C3 are represented by the points of a rational normal quartic O4. It is the object of this note to examine some of the consequences of this correspondence, in terms of the geometry associated with the two curves. 2. C4 lies on a Veronese surface V, which represents the congruence of chords of O3(l). Also C4 determines a 4-space 2 meeting D. in Qx, say; and since the surface of tangents of O3 is a developable, consecutive tangents intersect, and therefore the tangents to C4 lie on Q, and so on £lv Hence Qx, containing the sextic surface of tangents to C4, must be the quadric threefold / associated with C4, i.e. the quadric determining the same polarity as C4 (2). We note also that the tangents to C4 correspond in #3 to the plane pencils with vertices on O3, and lying in the corresponding osculating planes. 3. We shall prove that the surface U, which is the locus of points of intersection of pairs of osculating planes of C4, is the projection of the Veronese surface V from L, the pole of 2, on to 2. Let P denote a point of C3, and t, n the tangent line and osculating plane at P, and let T, T, w denote the same for the corresponding point of C4.
  • Combination of Cubic and Quartic Plane Curve

    Combination of Cubic and Quartic Plane Curve

    IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 2 (Mar. - Apr. 2013), PP 43-53 www.iosrjournals.org Combination of Cubic and Quartic Plane Curve C.Dayanithi Research Scholar, Cmj University, Megalaya Abstract The set of complex eigenvalues of unistochastic matrices of order three forms a deltoid. A cross-section of the set of unistochastic matrices of order three forms a deltoid. The set of possible traces of unitary matrices belonging to the group SU(3) forms a deltoid. The intersection of two deltoids parametrizes a family of Complex Hadamard matrices of order six. The set of all Simson lines of given triangle, form an envelope in the shape of a deltoid. This is known as the Steiner deltoid or Steiner's hypocycloid after Jakob Steiner who described the shape and symmetry of the curve in 1856. The envelope of the area bisectors of a triangle is a deltoid (in the broader sense defined above) with vertices at the midpoints of the medians. The sides of the deltoid are arcs of hyperbolas that are asymptotic to the triangle's sides. I. Introduction Various combinations of coefficients in the above equation give rise to various important families of curves as listed below. 1. Bicorn curve 2. Klein quartic 3. Bullet-nose curve 4. Lemniscate of Bernoulli 5. Cartesian oval 6. Lemniscate of Gerono 7. Cassini oval 8. Lüroth quartic 9. Deltoid curve 10. Spiric section 11. Hippopede 12. Toric section 13. Kampyle of Eudoxus 14. Trott curve II. Bicorn curve In geometry, the bicorn, also known as a cocked hat curve due to its resemblance to a bicorne, is a rational quartic curve defined by the equation It has two cusps and is symmetric about the y-axis.
  • Algebraic Curves and Surfaces

    Algebraic Curves and Surfaces

    Notes for Curves and Surfaces Instructor: Robert Freidman Henry Liu April 25, 2017 Abstract These are my live-texed notes for the Spring 2017 offering of MATH GR8293 Algebraic Curves & Surfaces . Let me know when you find errors or typos. I'm sure there are plenty. 1 Curves on a surface 1 1.1 Topological invariants . 1 1.2 Holomorphic invariants . 2 1.3 Divisors . 3 1.4 Algebraic intersection theory . 4 1.5 Arithmetic genus . 6 1.6 Riemann{Roch formula . 7 1.7 Hodge index theorem . 7 1.8 Ample and nef divisors . 8 1.9 Ample cone and its closure . 11 1.10 Closure of the ample cone . 13 1.11 Div and Num as functors . 15 2 Birational geometry 17 2.1 Blowing up and down . 17 2.2 Numerical invariants of X~ ...................................... 18 2.3 Embedded resolutions for curves on a surface . 19 2.4 Minimal models of surfaces . 23 2.5 More general contractions . 24 2.6 Rational singularities . 26 2.7 Fundamental cycles . 28 2.8 Surface singularities . 31 2.9 Gorenstein condition for normal surface singularities . 33 3 Examples of surfaces 36 3.1 Rational ruled surfaces . 36 3.2 More general ruled surfaces . 39 3.3 Numerical invariants . 41 3.4 The invariant e(V ).......................................... 42 3.5 Ample and nef cones . 44 3.6 del Pezzo surfaces . 44 3.7 Lines on a cubic and del Pezzos . 47 3.8 Characterization of del Pezzo surfaces . 50 3.9 K3 surfaces . 51 3.10 Period map . 54 a 3.11 Elliptic surfaces .
  • Tropical Curves

    Tropical Curves

    Tropical Curves Nathan Pflueger 24 February 2011 Abstract A tropical curve is a graph with specified edge lengths, some of which may be infinite. Various facts and attributes about algebraic curves have analogs for tropical curves. In this article, we focus on divisors and linear series, and prove the Riemann-Roch formula for divisors on tropical curves. We describe two ways in which algebraic curves may be transformed into tropical curves: by aboemas and by specialization on arithmetic surfaces. We discuss how the study of linear series on tropical curves can be used to obtain results about linear series on algebraic curves, and summarize several recent applications. Contents 1 Introduction 2 2 From curves to graphs 3 2.1 Amoebas of plane curves . .3 2.2 Curves over the field of Puiseux series . .4 2.3 Specialization . .5 3 Metric graphs and tropical curves 6 4 Divisors and linear equivalence on tropical curves 9 4.1 The Riemann-Roch criterion . 12 4.2 Tropical Riemann-Roch . 14 5 Tropical plane curves 17 5.1 Tropical algebra and tropical projective space . 17 5.2 Tropical curves in R2 ................................... 18 5.3 Calculation of the genus . 23 5.4 Stable intersection and the tropical B´ezouttheorem . 24 5.5 Classical B´ezoutfrom tropical B´ezout . 29 5.6 Enumerative geometry of tropical plane curves . 31 6 Tropical curves via specialization 32 6.1 The specialization map and specialization lemma . 32 6.2 The canonical divisor of a graph is canonical . 34 6.3 A tropical proof of the Brill-Noether theorem . 34 1 1 Introduction The origins of tropical geometry lie in the study of tropical algebra, whose basic object is the set R [ {−∞} equipped with the operations x ⊕ y = max(x; y) and x ⊗ y = x + y.
  • Classical Algebraic Geometry

    Classical Algebraic Geometry

    CLASSICAL ALGEBRAIC GEOMETRY Daniel Plaumann Universität Konstanz Summer A brief inaccurate history of algebraic geometry - Projective geometry. Emergence of ’analytic’geometry with cartesian coordinates, as opposed to ’synthetic’(axiomatic) geometry in the style of Euclid. (Celebrities: Plücker, Hesse, Cayley) - Complex analytic geometry. Powerful new tools for the study of geo- metric problems over C.(Celebrities: Abel, Jacobi, Riemann) - Classical school. Perfected the use of existing tools without any ’dog- matic’approach. (Celebrities: Castelnuovo, Segre, Severi, M. Noether) - Algebraization. Development of modern algebraic foundations (’com- mutative ring theory’) for algebraic geometry. (Celebrities: Hilbert, E. Noether, Zariski) from Modern algebraic geometry. All-encompassing abstract frameworks (schemes, stacks), greatly widening the scope of algebraic geometry. (Celebrities: Weil, Serre, Grothendieck, Deligne, Mumford) from Computational algebraic geometry Symbolic computation and dis- crete methods, many new applications. (Celebrities: Buchberger) Literature Primary source [Ha] J. Harris, Algebraic Geometry: A first course. Springer GTM () Classical algebraic geometry [BCGB] M. C. Beltrametti, E. Carletti, D. Gallarati, G. Monti Bragadin. Lectures on Curves, Sur- faces and Projective Varieties. A classical view of algebraic geometry. EMS Textbooks (translated from Italian) () [Do] I. Dolgachev. Classical Algebraic Geometry. A modern view. Cambridge UP () Algorithmic algebraic geometry [CLO] D. Cox, J. Little, D.
  • Cubic Curves and Totally Geodesic Subvarieties of Moduli Space

    Cubic Curves and Totally Geodesic Subvarieties of Moduli Space

    Cubic curves and totally geodesic subvarieties of moduli space Curtis T. McMullen, Ronen E. Mukamel and Alex Wright 16 March 2016 Abstract In this paper we present the first example of a primitive, totally geodesic subvariety F ⊂ Mg;n with dim(F ) > 1. The variety we consider is a surface F ⊂ M1;3 defined using the projective geometry of plane cubic curves. We also obtain a new series of Teichm¨ullercurves in M4, and new SL2(R){invariant varieties in the moduli spaces of quadratic differentials and holomorphic 1-forms. Contents 1 Introduction . 1 2 Cubic curves . 6 3 The flex locus . 11 4 The gothic locus . 14 5 F is totally geodesic . 20 6 F is primitive . 24 7 The Kobayashi metric on F ................... 26 8 Teichm¨ullercurves in M4 .................... 28 9 Explicit polygonal constructions . 30 Research supported in part by the NSF and the CMI (A.W.). Revised 15 Dec 2016. Typeset 2018-07-27 17:28. 1 Introduction Let Mg;n denote the moduli space of compact Riemann surfaces of genus g with n marked points. A complex geodesic is a holomorphic immersion f : H !Mg;n that is a local isometry for the Kobayashi metrics on its domain and range. It is known that Mg;n contains a complex geodesic through every point and in every possible direction. We say a subvariety V ⊂ Mg;n is totally geodesic if every complex geodesic tangent to V is contained in V . It is primitive if it does not arise from a simpler moduli space via a covering construction.
  • Chapter 2 Affine Algebraic Geometry

    Chapter 2 Affine Algebraic Geometry

    Chapter 2 Affine Algebraic Geometry 2.1 The Algebraic-Geometric Dictionary The correspondence between algebra and geometry is closest in affine algebraic geom- etry, where the basic objects are solutions to systems of polynomial equations. For many applications, it suffices to work over the real R, or the complex numbers C. Since important applications such as coding theory or symbolic computation require finite fields, Fq , or the rational numbers, Q, we shall develop algebraic geometry over an arbitrary field, F, and keep in mind the important cases of R and C. For algebraically closed fields, there is an exact and easily motivated correspondence be- tween algebraic and geometric concepts. When the field is not algebraically closed, this correspondence weakens considerably. When that occurs, we will use the case of algebraically closed fields as our guide and base our definitions on algebra. Similarly, the strongest and most elegant results in algebraic geometry hold only for algebraically closed fields. We will invoke the hypothesis that F is algebraically closed to obtain these results, and then discuss what holds for arbitrary fields, par- ticularly the real numbers. Since many important varieties have structures which are independent of the field of definition, we feel this approach is justified—and it keeps our presentation elementary and motivated. Lastly, for the most part it will suffice to let F be R or C; not only are these the most important cases, but they are also the sources of our geometric intuitions. n Let A denote affine n-space over F. This is the set of all n-tuples (t1,...,tn) of elements of F.
  • Geometry of Algebraic Curves

    Geometry of Algebraic Curves

    Geometry of Algebraic Curves Fall 2011 Course taught by Joe Harris Notes by Atanas Atanasov One Oxford Street, Cambridge, MA 02138 E-mail address: [email protected] Contents Lecture 1. September 2, 2011 6 Lecture 2. September 7, 2011 10 2.1. Riemann surfaces associated to a polynomial 10 2.2. The degree of KX and Riemann-Hurwitz 13 2.3. Maps into projective space 15 2.4. An amusing fact 16 Lecture 3. September 9, 2011 17 3.1. Embedding Riemann surfaces in projective space 17 3.2. Geometric Riemann-Roch 17 3.3. Adjunction 18 Lecture 4. September 12, 2011 21 4.1. A change of viewpoint 21 4.2. The Brill-Noether problem 21 Lecture 5. September 16, 2011 25 5.1. Remark on a homework problem 25 5.2. Abel's Theorem 25 5.3. Examples and applications 27 Lecture 6. September 21, 2011 30 6.1. The canonical divisor on a smooth plane curve 30 6.2. More general divisors on smooth plane curves 31 6.3. The canonical divisor on a nodal plane curve 32 6.4. More general divisors on nodal plane curves 33 Lecture 7. September 23, 2011 35 7.1. More on divisors 35 7.2. Riemann-Roch, finally 36 7.3. Fun applications 37 7.4. Sheaf cohomology 37 Lecture 8. September 28, 2011 40 8.1. Examples of low genus 40 8.2. Hyperelliptic curves 40 8.3. Low genus examples 42 Lecture 9. September 30, 2011 44 9.1. Automorphisms of genus 0 an 1 curves 44 9.2.