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Classical Algebraic Geometry: A Modern View by Igor V. Dolgachev

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Classical Algebraic Geometry: a modern view written by IGOR V. DOLGACHEV . This is an other book of mathematics cover the following topics.

  • Polarity page
    Polar hypersurfaces (The polar pairing, First polars, Polar quadrics, The Hessian hypersurface, Parabolic points, The Steinerian hypersurface, The Jacobian hypersurface), The dual hypersurface (The polar map, Dual varieties, Plucker formulas), Polar s-hedra (Apolar schemes, Sums of powers, Generalized polar s-hedra, Secant varieties and sums of powers, The Waring problems), Dual homogeneous forms (Catalecticant matrices, Dual homogeneous forms, The Waring rank of a homogeneous form, Mukai’s skew-symmetric form, Harmonic polynomials, First examples (Binary forms, Quadrics), Exercises, Historical Notes

  • Conics and quadric surfaces
    Self-polar triangles (Veronese quartic surfaces, Polar lines, The variety of self-polar triangles, Conjugate triangles, Poncelet relation Darboux’s Theorem, Poncelet curves and vector bundles, Complex circles, Quadric surfaces (Polar properties of quadrics, Invariants of a pair of quadrics, Invariants of a pair of conics, The Salmon conic), Exercises, Historical Notes

  • Plane cubics
    Equations (Elliptic curves, The Hesse equation, The Hesse pencil, The Hesse group), Polars of a plane cubic (The Hessian of a cubic hypersurface, The Hessian of a plane cubic, The dual curve, Polar s-gons), Projective generation of cubic curves (Projective generation, Projective generation of a plane cubic), Invariant theory of plane cubics (Mixed concomitants, Clebsch’s transfer principle, Invariants of plane cubics), Exercises, Historical Notes

  • Determinantal equations
    Plane curves (The problem, Plane curves, The symmetric case, Contact curves, First examples, The moduli space), Determinantal equations for hypersurfaces (Determinantal varieties, Arithmetically Cohen-Macaulay sheaves, Symmetric and skew-symmetric aCM sheaves, Singular plane curves, Linear determinantal representations of surfaces, Symmetroid surfaces), Exercises, Historical Notes

  • Theta characteristics
    Odd and even theta characteristics (First definitions and examples, Quadratic forms over a field of characteristic), Hyperelliptic curves (Equations of hyperelliptic curves, 2-torsion points on a hyperelliptic curve, Theta characteristics on a hyperelliptic curve, Families of curves with odd or even theta characteristic), Theta functions (Jacobian variety, Theta functions, Hyperelliptic curves again), Odd theta characteristics (Syzygetic triads, Steiner complexes, Fundamental sets), Scorza correspondence (Correspondences on an algebraic curve, Scorza correspondence, Scorza quartic hypersurfaces, Contact hyperplanes of canonical curves), Exercises, Historical Notes

  • Plane Quartics
    Bitangents (28 bitangents, Aronhold sets, Riemann’s equations for bitangents), Determinant equations of a plane quartic (Quadratic determinantal representations, Symmetric quadratic determinants), Even theta characteristics (Contact cubics, Cayley octads, Seven points in the plane, The Clebsch covariant quartic, Clebsch and Luroth quartics, A Fano model of VSP(f, 6)), Invariant theory of plane quartics, Automorphisms of plane quartic curves (Automorphisms of finite order, Automorphism groups, The Klein quartic), Exercises, Historical Notes

  • Cremona transformations
    Homaloidal linear systems (Linear systems and their base schemes, Resolution of a rational map, The graph of a Cremona transformation, F-locus and P-locus, Computation of the multidegree), First examples (Quadro-quadratic transformations, Bilinear Cremona transformations, de Jonquieres transformations), Planar Cremona transformations (Exceptional configurations, The bubble space of a surface, Nets of isologues and fixed points, Quadratic transformations, Symmetric Cremona transformations, de Jonquieres transformations and hyperelliptic curves), Elementary transformations(Minimal rational ruled surfaces, Elementary transformations, Birational automorphisms of P 1 × P 1, Noether’s Factorization Theorem ( Characteristic matrices, The Weyl groups, Noether-Fano inequality, Noether’s Factorization Theorem, Exercises, Historical Notes

  • Del Pezzo surfaces
    First properties, Surfaces of degree d in Pd, Rational double points, A blow-up model of a del Pezzo surface), The EN -lattice (Quadratic lattices, The EN -lattice, Roots, Fundamental weights, Gosset polytopes, (−1)-curves on del Pezzo surfaces, Effective roots, Cremona isometries), Anticanonical models (Anticanonical linear systems, Anticanonical model), Del Pezzo surfaces of degree ≥ 6 (Del Pezzo surfaces of degree 7, 8, 9, Del Pezzo surfaces of degree 6), Del Pezzo surfaces of degree 5 (Lines and singularities, Equations, OADP varieties, Automorphism group), Quartic del Pezzo surfaces (Equations, Cyclid quartics, Lines and singularities, Automorphisms), Del Pezzo surfaces of degree 2 (Singularities, Geiser involution, Automorphisms of del Pezzo surfaces of degree 2), Del Pezzo surfaces of degree 1 (Singularities, Bertini involution, Rational elliptic surfaces, Automorphisms of del Pezzo surfaces of degree 1) , Exercises, Historical Notes

  • Cubic surfaces
    Lines on a nonsingular cubic surface (More about the E6-lattice, Lines and tritangent planes, Schur’s quadrics, Eckardt points), Singularities (Non-normal cubic surfaces, Lines and singularities), Determinantal equations (Cayley-Salmon equation, Hilbert-Burch Theorem, Cubic symmetroids), Representations as sums of cubes (Sylvester’s pentahedron, The Hessian surface, Cremona’s hexahedral equations, The Segre cubic primal, Moduli spaces of cubic surfaces), Automorphisms of cubic surfaces (Cyclic groups of automorphisms, Maximal subgroups of W(E6), Groups of automorphisms, The Clebsch diagonal cubic), Exercises, Historical Notes

  • Geometry of Lines
    Grassmannians of lines (Generalities about Grassmannians, Schubert varieties, Secant varieties of Grassmannians of lines), Linear line complexes (Linear line complexes and apolarity, Six lines, Linear systems of linear line complexes), Quadratic line complexes (Generalities, Intersection of two quadrics, Kummer surfaces, Harmonic complex, The tangential line complex, Tetrahedral line complex), Ruled surfaces (Scrolls, Cayley-Zeuthen formulas, Developable ruled surfaces, Quartic ruled surfaces in P3, Ruled surfaces in P3 and the tetrahedral line, complex, Exercises, Historical Notes

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