Classical Algebraic Geometry: a modern view IGOR V. DOLGACHEV Preface The main purpose of the present treatise is to give an account of some of the topics in algebraic geometry which while having occupied the minds of many mathematicians in previous generations have fallen out of fashion in modern times. Often in the history of mathematics new ideas and techniques make the work of previous generations of researchers obsolete, especially this applies to the foundations of the subject and the fundamental general theoretical facts used heavily in research. Even the greatest achievements of the past genera- tions which can be found for example in the work of F. Severi on algebraic cycles or in the work of O. Zariski’s in the theory of algebraic surfaces have been greatly generalized and clarified so that they now remain only of histor- ical interest. In contrast, the fact that a nonsingular cubic surface has 27 lines or that a plane quartic has 28 bitangents is something that cannot be improved upon and continues to fascinate modern geometers. One of the goals of this present work is then to save from oblivion the work of many mathematicians who discovered these classic tenets and many other beautiful results. In writing this book the greatest challenge the author has faced was distilling the material down to what should be covered. The number of concrete facts, examples of special varieties and beautiful geometric constructions that have accumulated during the classical period of development of algebraic geometry is enormous and what the reader is going to find in the book is really only the tip of the iceberg; a work that is like a taste sampler of classical algebraic geometry. It avoids most of the material found in other modern books on the subject, such as, for example, [10] where one can find many of the classical results on algebraic curves. Instead, it tries to assemble or, in other words, to create a compendium of material that either cannot befound, is too dispersed to be found easily, or is simply not treated adequately by contemporary research papers. On the other hand, while most of the material treated in the book exists in classical treatises in algebraic geometry, their somewhat archaic terminology iv Preface and what is by now completely forgotten background knowledge makes these books useful to but a handful of experts in the classical literature. Lastly, one must admit that the personal taste of the author also has much sway in the choice of material. The reader should be warned that the book is by no means an introduction to algebraic geometry. Although some of the exposition can be followed with only a minimum background in algebraic geometry, for example, based on Shafarevich’s book [528], it often relies on current cohomological techniques, such as those found in Hartshorne’s book [281]. The idea was to reconstruct a result by using modern techniques but not necessarily its original proof. For one, the ingenious geometric constructions in those proofs were often beyond the authors abilities to follow them completely. Understandably, the price of this was often to replace a beautiful geometric argument with a dull cohomo- logical one. For those looking for a less demanding sample of some of the topics covered in the book, the recent beautiful book [39] may be of great use. No attempt has been made to give a complete bibliography. To give an idea of such an enormous task one could mention that the report on the status of topics in algebraic geometry submitted to the National Research Council in Washington in 1928 [533] contains more than 500 items of bibliography by 130 different authors only in the subject of planar Cremona transformations (covered in one of the chapters of the present book.) Another example is the bibliography on cubic surfaces compiled by J. E. Hill [294] in 1896 which alone contains 205 titles. Meyer’s article [383] cites around 130 papers pub- lished 1896-1928. The title search in MathSciNet reveals more than 200 papers refereed since 1940, many of them published only in the past 20 years. How sad it is when one considers the impossibility of saving from oblivion so many names of researchers of the past who have contributed so much to our subject. A word about exercises: some of them are easy and follow from the defi- nitions, some of them are hard and are meant to provide additional facts not covered in the main text. In this case we indicate the sources for the statements and solutions. I am very grateful to many people for their comments and corrections to many previous versions of the manuscript. I am especially thankful to Sergey Tikhomirov whose help in the mathematical editing of the book was essential for getting rid of many mistakes in the previous versions. For all the errors still found in the book the author bears sole responsibility. Contents 1 Polarity page 1 1.1 Polar hypersurfaces 1 1.1.1 The polar pairing 1 1.1.2 First polars 7 1.1.3 Polar quadrics 13 1.1.4 The Hessian hypersurface 15 1.1.5 Parabolic points 18 1.1.6 The Steinerian hypersurface 21 1.1.7 The Jacobian hypersurface 25 1.2 The dual hypersurface 32 1.2.1 The polar map 32 1.2.2 Dual varieties 33 1.2.3 Pl ¨ ucker formulas 37 1.3 Polar s-hedra 40 1.3.1 Apolar schemes 40 1.3.2 Sums of powers 42 1.3.3 Generalized polar s-hedra 44 1.3.4 Secant varieties and sums of powers 45 1.3.5 The Waring problems 52 1.4 Dual homogeneous forms 54 1.4.1 Catalecticant matrices 54 1.4.2 Dual homogeneous forms 57 1.4.3 The Waring rank of a homogeneous form 58 1.4.4 Mukai’s skew-symmetric form 59 1.4.5 Harmonic polynomials 62 1.5 First examples 67 1.5.1 Binary forms 67 vi Contents 1.5.2 Quadrics 70 Exercises 72 Historical Notes 74 2 Conics and quadric surfaces 77 2.1 Self-polar triangles 77 2.1.1 Veronese quartic surfaces 77 2.1.2 Polar lines 79 2.1.3 The variety of self-polar triangles 81 2.1.4 Conjugate triangles 85 2.2 Poncelet relation 91 2.2.1 Darboux’s Theorem 91 2.2.2 Poncelet curves and vector bundles 96 2.2.3 Complex circles 99 2.3 Quadric surfaces 102 2.3.1 Polar properties of quadrics 102 2.3.2 Invariants of a pair of quadrics 108 2.3.3 Invariants of a pair of conics 112 2.3.4 The Salmon conic 117 Exercises 121 Historical Notes 125 3 Plane cubics 127 3.1 Equations 127 3.1.1 Elliptic curves 127 3.1.2 The Hesse equation 131 3.1.3 The Hesse pencil 133 3.1.4 The Hesse group 134 3.2 Polars of a plane cubic 138 3.2.1 The Hessian of a cubic hypersurface 138 3.2.2 The Hessian of a plane cubic 139 3.2.3 The dual curve 143 3.2.4 Polar s-gons 144 3.3 Projective generation of cubic curves 149 3.3.1 Projective generation 149 3.3.2 Projective generation of a plane cubic 151 3.4 Invariant theory of plane cubics 152 3.4.1 Mixed concomitants 152 3.4.2 Clebsch’s transfer principle 153 3.4.3 Invariants of plane cubics 155 Exercises 157 Contents vii Historical Notes 160 4 Determinantal equations 162 4.1 Plane curves 162 4.1.1 The problem 162 4.1.2 Plane curves 163 4.1.3 The symmetric case 168 4.1.4 Contact curves 170 4.1.5 First examples 174 4.1.6 The moduli space 176 4.2 Determinantal equations for hypersurfaces 178 4.2.1 Determinantal varieties 178 4.2.2 Arithmetically Cohen-Macaulay sheaves 182 4.2.3 Symmetric and skew-symmetric aCM sheaves 187 4.2.4 Singular plane curves 189 4.2.5 Linear determinantal representations of surfaces 197 4.2.6 Symmetroid surfaces 201 Exercises 204 Historical Notes 207 5 Theta characteristics 209 5.1 Odd and even theta characteristics 209 5.1.1 First definitions and examples 209 5.1.2 Quadratic forms over a field of characteristic 2 210 5.2 Hyperelliptic curves 213 5.2.1 Equations of hyperelliptic curves 213 5.2.2 2-torsion points on a hyperelliptic curve 214 5.2.3 Theta characteristics on a hyperelliptic curve 216 5.2.4 Families of curves with odd or even theta characteristic 218 5.3 Theta functions 219 5.3.1 Jacobian variety 219 5.3.2 Theta functions 222 5.3.3 Hyperelliptic curves again 224 5.4 Odd theta characteristics 226 5.4.1 Syzygetic triads 226 5.4.2 Steiner complexes 229 5.4.3 Fundamental sets 233 5.5 Scorza correspondence 236 5.5.1 Correspondences on an algebraic curve 236 5.5.2 Scorza correspondence 240 viii Contents 5.5.3 Scorza quartic hypersurfaces 243 5.5.4 Contact hyperplanes of canonical curves 246 Exercises 249 Historical Notes 249 6 Plane Quartics 251 6.1 Bitangents 251 6.1.1 28 bitangents 251 6.1.2 Aronhold sets 253 6.1.3 Riemann’s equations for bitangents 256 6.2 Determinant equations of a plane quartic 261 6.2.1 Quadratic determinantal representations 261 6.2.2 Symmetric quadratic determinants 265 6.3 Even theta characteristics 270 6.3.1 Contact cubics 270 6.3.2 Cayley octads 271 6.3.3 Seven points in the plane 275 6.3.4 The Clebsch covariant quartic 279 6.3.5 Clebsch and L ¨ uroth quartics 283 6.3.6 A Fano model of VSP(f, 6) 291 6.4 Invariant theory of plane quartics 294 6.5 Automorphisms of plane quartic curves 296 6.5.1 Automorphisms of finite order 296 6.5.2 Automorphism groups 299 6.5.3 The Klein quartic 302 Exercises 306 Historical Notes 308 7 Cremona transformations 311 7.1 Homaloidal linear systems 311 7.1.1 Linear systems and their base schemes 311 7.1.2 Resolution of a rational map 313 7.1.3 The graph of a Cremona transformation 316 7.1.4 F-locus and P-locus 318 7.1.5 Computation of the multidegree 323 7.2 First examples 327 7.2.1 Quadro-quadratic transformations 327 7.2.2 Bilinear Cremona transformations 329 7.2.3 de Jonqui ` eres transformations 334 7.3 Planar Cremona transformations 337 7.3.1 Exceptional configurations 337 Contents ix 7.3.2 The bubble space of a surface 341 7.3.3 Nets of isologues and fixed points 344 7.3.4 Quadratic transformations 349 7.3.5 Symmetric Cremona transformations 351 7.3.6 de Jonqui ` eres transformations and hyperellip- tic curves 353 7.4 Elementary transformations 356 7.4.1 Minimal rational ruled surfaces 356 7.4.2 Elementary transformations 359 7.4.3 Birational automorphisms of P 1 × P 1 361 7.5 Noether’s Factorization Theorem 366 7.5.1 Characteristic matrices 366 7.5.2 The Weyl groups 372 7.5.3 Noether-Fano inequality 376 7.5.4 Noether’s Factorization Theorem 378 Exercises 381 Historical Notes 383 8 Del Pezzo surfaces 386 8.1 First properties 386 8.1.1 Surfaces of degree d in P d 386 8.1.2 Rational double points 390 8.1.3 A blow-up model of a del Pezzo surface 392 8.2 The E N -lattice 398 8.2.1 Quadratic lattices 398 8.2.2 The E N -lattice 401 8.2.3 Roots 403 8.2.4 Fundamental weights 408 8.2.5 Gosset polytopes 410 8.2.6 (−1)-curves on del Pezzo surfaces 412 8.2.7 Effective roots 415 8.2.8 Cremona isometries 418 8.3 Anticanonical models 422 8.3.1 Anticanonical linear systems 422 8.3.2 Anticanonical model 427 8.4 Del Pezzo surfaces of degree ≥ 6 429 8.4.1 Del Pezzo surfaces of degree 7, 8, 9 429 8.4.2 Del Pezzo surfaces of degree 6 430 8.5 Del Pezzo surfaces of degree 5 433 8.5.1 Lines and singularities 433 x Contents 8.5.2 Equations 434 8.5.3 OADP varieties 436 8.5.4 Automorphism group 437 8.6 Quartic del Pezzo surfaces 441 8.6.1 Equations 441 8.6.2 Cyclid quartics 444 8.6.3 Lines and singularities 446 8.6.4 Automorphisms 448 8.7 Del Pezzo surfaces of degree 2 451 8.7.1 Singularities 451 8.7.2 Geiser involution 454 8.7.3 Automorphisms of del Pezzo surfaces of degree 2 457 8.8 Del Pezzo surfaces of degree 1 458 8.8.1 Singularities 458 8.8.2 Bertini involution 460 8.8.3 Rational elliptic surfaces 462 8.8.4 Automorphisms of del Pezzo surfaces of degree 1 463 Exercises 470 Historical Notes 471 9 Cubic surfaces 475 9.1 Lines on a nonsingular cubic surface 475 9.1.1 More about the E 6 -lattice 475 9.1.2 Lines and tritangent planes 482 9.1.3 Schur’s quadrics 486 9.1.4 Eckardt points 491 9.2 Singularities 494 9.2.1 Non-normal cubic surfaces 494 9.2.2 Lines and singularities 495 9.3 Determinantal equations 501 9.3.1 Cayley-Salmon equation 501 9.3.2 Hilbert-Burch Theorem 504 9.3.3 Cubic symmetroids 509 9.4 Representations as sums of cubes 512 9.4.1 Sylvester’s pentahedron 512 9.4.2 The Hessian surface 515 9.4.3 Cremona’s hexahedral equations 517 9.4.4 The Segre cubic primal 520 [...]... a singular quadric in P1 The converse is also true For example, a nonsingular quadric has no parabolic points, and all nonsingular points of a singular quadric are parabolic A generalization of a quadratic cone is a developable surface It is a special kind of a ruled surface which characterized by the condition that the tangent plane does not change along a ruling We will discuss these surfaces later... the Grassmannian of r-dimensional subspaces in Pn In the sequel we will also use the notation G(r + 1, E) = Gr (|E|) for the variety of linear r + 1-dimensional subspaces of a linear space E The map is not defined at the intersection of the diagonal with HS(X) We know that HS (a, a) = 0 means that Pad−1 (X) = 0, and the latter means that a is a singular point of X Thus the map is a regular map for a nonsingular... each r ≤ deg X − m, Par (X) has a singular point at a of multiplicity m and the tangent cone of Par (X) at a coincides with the tangent cone TCa (X) of X at a For any point b = a, the r-th polar Pbr (X) has multiplicity ≥ m − r at a and its tangent cone at a is equal to the r-th polar of TCa (X) with respect to b Proof Let us prove the first assertion Without loss of generality, we may assume that a. .. or a row of the adjugate matrix adj(He(f )) evaluated at the point a Thus, St(X) coincides with the image of the Hessian hypersurface under the rational map st : He(X) St(X), a → Sing(Pad−2 (X)), given by polynomials of degree n(d − 2) We call it the Steinerian map Of course, it is not defined when all polar quadrics are of corank > 1 Also, if the first polar hypersurface Pa (X) has an isolated singular... (1.18), the polar quadric Q is also singular at a and therefore it must be a cone over its image under the projection from a The union of inflection tangents is equal to Q Example 1.1.14 Assume a is a nonsingular point of an irreducible surface X in P3 A tangent hyperplane Ta (X) cuts out in X a curve C with a singular point a If a is an ordinary double point of C, there are two inflection tangents corresponding... adj (A) is the cofactor matrix (classically called the adjugate matrix of A, but not the adjoint matrix as it is often called in modern text-books) 1.1.2 First polars Let us consider some special cases Let X = V (f ) be a hypersurface of degree d Obviously, any 0-th polar of X is equal to X and, by (1.12), the d-th polar 8 Polarity Pad (X) is empty if a ∈ X and equals Pn if a ∈ X Now take k = 1, d − 1... this means that one of the lines is an inflection tangent A point a of a plane curve X such that there exists an inflection tangent at a is called an inflection point of X If n > 2, the inflection tangent lines at a point a ∈ X ∩He(X) sweep a cone over a singular quadric in Pn−2 (or the whole Pn−2 if the point is singular) Such a point is called a parabolic point of X The closure of the set of parabolic... 3 It is also the discriminant surface of a binary cubic, i.e the surface parameterizing binary cubics a0 u3 + 3a1 u2 v + 3a2 uv 2 + a3 v 3 with a multiple root The pro-Hessian of any quartic developable surface is the surface itself [84] 20 Polarity Assume now that X is a curve Let us see when it has infinitely many inflection points Certainly, this happens when X contains a line component; each of its... quadric Q, the map x → Px (Q) defines a projective isomorphism from the projective space to the dual projective space This is a special case of a correlation According to classical terminology, a projective automorphism of Pn is called a collineation An isomorphism from |E| to its dual space P(E) is called a correlation A correlation c : |E| → P(E) is given by an invertible linear map φ : E → E ∨ defined... implies that b, and hence , belongs to the tangent plane Ta (X) For s = 2, this condition implies that b ∈ Pad−2 (X) Since is tangent to X at a, and Pad−2 (X) is tangent to X at a, this is equivalent to that belongs to Pad−2 (X) It follows from (1.19) that a is a singular point of X of multiplicity ≥ s + 1 if and only if Pad−k (X) = Pn for k ≤ s In particular, the quadric polar Pad−2 (X) = Pn if and only . function A → adj (A) , where adj (A) is the cofactor matrix (classically called the adjugate matrix of A, but not the adjoint matrix as it is often called in modern. really only the tip of the iceberg; a work that is like a taste sampler of classical algebraic geometry. It avoids most of the material found in other modern