UID:
almafu_9960117003402883
Format:
1 online resource (xiv, 616 pages) :
,
digital, PDF file(s).
ISBN:
1-316-67887-3
,
1-316-68006-1
,
1-139-06204-2
Content:
This book can form the basis of a second course in algebraic geometry. As motivation, it takes concrete questions from enumerative geometry and intersection theory, and provides intuition and technique, so that the student develops the ability to solve geometric problems. The authors explain key ideas, including rational equivalence, Chow rings, Schubert calculus and Chern classes, and readers will appreciate the abundant examples, many provided as exercises with solutions available online. Intersection is concerned with the enumeration of solutions of systems of polynomial equations in several variables. It has been an active area of mathematics since the work of Leibniz. Chasles' nineteenth-century calculation that there are 3264 smooth conic plane curves tangent to five given general conics was an important landmark, and was the inspiration behind the title of this book. Such computations were motivation for Poincaré's development of topology, and for many subsequent theories, so that intersection theory is now a central topic of modern mathematics.
Note:
Title from publisher's bibliographic system (viewed on 08 Mar 2016).
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Cover -- Half-title -- Title page -- Copyright information -- Table of contents -- Preface -- Epigraph -- Chapter 0 Introduction -- Why you want to read this book -- Why we wrote this book -- What's with the title? -- What is in this book -- Overture -- Grassmannians -- Chern classes -- Applications, I: Using the tools -- Parameter spaces -- Applications, II: Further developments -- Advanced topics -- Appendices -- Relation of this book to Intersection theory -- Keynote problems -- Exercises -- Prerequisites, notation and conventions -- What you need to know before starting -- Language -- Basic results on dimension and smoothness -- Chapter 1 Introducing the Chow ring -- 1.1 The goal of intersection theory -- 1.2 The Chow ring -- 1.2.1 Cycles -- 1.2.2 Rational equivalence and the Chow group -- 1.2.3 Transversality and the Chow ring -- 1.2.4 The moving lemma -- 1.3 Some techniques for computing the Chow ring -- 1.3.1 The fundamental class -- 1.3.2 Rational equivalence via divisors -- 1.3.3 Affine space -- 1.3.4 Mayer-Vietoris and excision -- 1.3.5 Affine stratifications -- 1.3.6 Functoriality -- 1.3.7 Dimensional transversality and multiplicities -- 1.3.8 The multiplicity of a scheme at a point -- 1.4 The first Chern class of a line bundle -- 1.4.1 The canonical class -- 1.4.2 The adjunction formula -- 1.4.3 Canonical classes of hypersurfaces and complete intersections -- 1.5 Exercises -- Chapter 2 First examples -- 2.1 The Chow rings of Pn and some related varieties -- 2.1.1 B[eacute]zout's theorem -- 2.1.2 Degrees of Veronese varieties -- 2.1.3 Degree of the dual of a hypersurface -- 2.1.4 Products of projective spaces -- 2.1.5 Degrees of Segre varieties -- 2.1.6 The class of the diagonal -- 2.1.7 The class of a graph -- 2.1.8 Nested pairs of divisors on [mathbb(P)sup(1)] -- 2.1.9 The blow-up of [mathbb(P)sup(n)] at a point.
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2.1.10 Intersection multiplicities via blow-ups -- 2.2 Loci of singular plane cubics -- 2.2.1 Reducible cubics -- 2.2.2 Triangles -- 2.2.3 Asterisks -- 2.3 The circles of Apollonius -- 2.3.1 What is a circle? -- 2.3.2 Circles tangent to a given circle -- 2.3.3 Conclusion of the argument -- 2.4 Curves on surfaces -- 2.4.1 The genus formula -- 2.4.2 The self-intersection of a curve on a surface -- 2.4.3 Linked curves in [mathbb(P)sup(3)] -- 2.4.4 The blow-up of a surface -- 2.4.5 Canonical class of a blow-up -- 2.4.6 The genus formula with singularities -- 2.5 Intersections on singular varieties -- 2.6 Exercises -- Chapter 3 Introduction to Grassmannians and lines in [mathbb(P)sup(3)] -- 3.1 Enumerative formulas -- 3.1.1 What are enumerative problems, and how do we solve them? -- 3.1.2 The content of an enumerative formula -- 3.2 Introduction to Grassmannians -- 3.2.1 The Pl[ddot(u)]cker embedding -- 3.2.2 Covering by affine spaces -- local coordinates -- 3.2.3 Universal sub and quotient bundles -- 3.2.4 The tangent bundle of the Grassmannian -- 3.2.5 The differential of a morphism to the Grassmannian -- 3.2.6 Tangent spaces via the universal property -- 3.3 The Chow ring of [mathbb(G)](1, 3) -- 3.3.1 Schubert cycles in [mathbb(G)](1, 3) -- 3.3.2 Ring structure -- 3.4 Lines and curves in [mathbb(P)sup(3)] -- 3.4.1 How many lines meet four general lines? -- 3.4.2 Lines meeting a curve of degree d -- 3.4.3 Chords to a space curve -- 3.5 Specialization -- 3.5.1 Schubert calculus by static specialization -- 3.5.2 Dynamic projection -- 3.5.3 Lines meeting a curve by specialization -- 3.5.4 Chords via specialization: multiplicity problems -- 3.5.5 Common chords to twisted cubics via specialization -- 3.6 Lines and surfaces in [mathbb(P)sup(3)] -- 3.6.1 Lines lying on a quadric -- 3.6.2 Tangent lines to a surface -- 3.7 Exercises.
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Chapter 4 Grassmannians in general -- 4.1 Schubert cells and Schubert cycles -- 4.1.1 Schubert classes and Chern classes -- 4.1.2 The affine stratification by Schubert cells -- 4.1.3 Equations of the Schubert cycles -- 4.2 Intersection products -- 4.2.1 Transverse flags -- 4.2.2 Intersections in complementary dimension -- 4.2.3 Varieties swept out by linear spaces -- 4.2.4 Pieri's formula -- 4.3 Grassmannians of lines -- 4.4 Dynamic specialization -- 4.5 Young diagrams -- 4.5.1 Pieri's formula for the other special Schubert classes -- 4.6 Linear spaces on quadrics -- 4.7 Giambelli's formula -- 4.8 Generalizations -- 4.8.1 Flag manifolds -- 4.8.2 Lagrangian Grassmannians and beyond -- 4.9 Exercises -- Chapter 5 Chern classes -- 5.1 Introduction: Chern classes and the lines on a cubic surface -- 5.2 Characterizing Chern classes -- 5.3 Constructing Chern classes -- 5.4 The splitting principle -- 5.5 Using Whitney's formula with the splitting principle -- 5.5.1 Tensor products with line bundles -- 5.5.2 Tensor product of two bundles -- 5.6 Tautological bundles -- 5.6.1 Projective spaces -- 5.6.2 Grassmannians -- 5.7 Chern classes of varieties -- 5.7.1 Tangent bundles of projective spaces -- 5.7.2 Tangent bundles to hypersurfaces -- 5.7.3 The topological Euler characteristic -- 5.7.4 First Chern class of the Grassmannian -- 5.8 Generators and relations for A(G(k, n)) -- 5.9 Steps in the proofs of Theorem 5.3 -- 5.9.1 Whitney's formula for globally generated bundles -- 5.10 Exercises -- Chapter 6 Lines on hypersurfaces -- 6.1 What to expect -- 6.1.1 Definition of the Fano scheme -- 6.2 Fano schemes and Chern classes -- 6.2.1 Counting lines on cubics -- 6.3 Definition and existence of Hilbert schemes -- 6.3.1 A universal property of the Grassmannian -- 6.3.2 A universal property of the Fano scheme -- 6.3.3 The Hilbert scheme and its universal property.
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6.3.4 Sketch of the construction of the Hilbert scheme -- 6.4 Tangent spaces to Fano and Hilbert schemes -- 6.4.1 Normal bundles and the smoothness of the Fano scheme -- 6.4.2 First-order deformations as tangents to the Hilbert scheme -- 6.4.3 Normal bundles of k-planes on hypersurfaces -- 6.4.4 The case of lines -- 6.5 Lines on quintic threefolds and beyond -- 6.6 The universal Fano scheme and the geometry of families of lines -- 6.6.1 Lines on the quartic surfaces in a pencil -- 6.7 Lines on a cubic with a double point -- 6.8 The Debarre-de Jong Conjecture -- 6.8.1 Further open problems -- 6.9 Exercises -- Chapter 7 Singular elements of linear series -- 7.1 Singular hypersurfaces and the universal singularity -- 7.2 Bundles of principal parts -- 7.3 Singular elements of a pencil -- 7.3.1 From pencils to degeneracy loci -- 7.3.2 The Chern class of a bundle of principal parts -- 7.3.3 Triple points of plane curves -- 7.3.4 Cones -- 7.4 Singular elements of linear series in general -- 7.4.1 Number of singular elements of a pencil -- 7.4.2 Pencils of curves on a surface -- 7.4.3 The second fundamental form -- 7.5 Inflection points of curves in [mathbb(P)sup(r)] -- 7.5.1 Vanishing sequences and osculating planes -- 7.5.2 Total inflection: the Pl[ddot(u)]cker formula -- 7.5.3 The situation in higher dimension -- 7.6 Nets of plane curves -- 7.6.1 Class of the universal singular point -- 7.6.2 The discriminant of a net of plane curves -- 7.7 The topological Hurwitz formula -- 7.7.1 Pencils of curves on a surface, revisited -- 7.7.2 Multiplicities of the discriminant hypersurface -- 7.7.3 Tangent cones of the discriminant hypersurface -- 7.8 Exercises -- Chapter 8 Compactifying parameter spaces -- 8.1 Approaches to the five conic problem -- 8.2 Complete conics -- 8.2.1 Informal description -- 8.2.2 Rigorous description.
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8.2.3 Solution to the five conic problem -- 8.2.4 Chow ring of the space of complete conics -- 8.3 Complete quadrics -- 8.4 Parameter spaces of curves -- 8.4.1 Hilbert schemes -- 8.4.2 Report card for the Hilbert scheme -- 8.4.3 The Kontsevich space -- 8.4.4 Report card for the Kontsevich space -- 8.5 How the Kontsevich space is used: rational plane curves -- 8.6 Exercises -- Chapter 9 Projective bundles and their Chow rings -- 9.1 Projective bundles and the tautological divisor class -- 9.1.1 Example: rational normal scrolls -- 9.2 Maps to a projective bundle -- 9.3 Chow ring of a projective bundle -- 9.3.1 The universal k-plane over [mathbb(G)](k, n) -- 9.3.2 The blow-up of [mathbb(P)sup(n)] along a linear space -- 9.3.3 Nested pairs of divisors on [mathbb(P)sup(1)] revisited -- 9.4 Projectivization of a subbundle -- 9.4.1 Ruled surfaces -- 9.4.2 Self-intersection of the zero section -- 9.5 Brauer-Severi varieties -- 9.6 Chow ring of a Grassmannian bundle -- 9.7 Conics in [mathbb(P)sup(3)] meeting eight lines -- 9.7.1 The parameter space as projective bundle -- 9.7.2 The class delta of the cycle of conics meeting a line -- 9.7.3 The degree of [[delta]sup(8)] -- 9.7.4 The parameter space as Hilbert scheme -- 9.7.5 Tangent spaces to incidence cycles -- 9.7.6 Proof of transversality -- 9.8 Exercises -- Chapter 10 Segre classes and varieties of linear spaces -- 10.1 Segre classes -- 10.2 Varieties swept out by linear spaces -- 10.3 Secant varieties -- 10.3.1 Symmetric powers -- 10.3.2 Secant varieties in general -- 10.4 Secant varieties of rational normal curves -- 10.4.1 Secants to rational normal curves -- 10.4.2 Degrees of the secant varieties -- 10.4.3 Expression of a form as a sum of powers -- 10.5 Special secant planes -- 10.5.1 The class of the locus of secant planes -- 10.5.2 Secants to curves of positive genus.
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10.6 Dual varieties and conormal varieties.
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English
Additional Edition:
ISBN 1-107-60272-6
Additional Edition:
ISBN 1-107-01708-4
Language:
English
URL:
Volltext
(URL des Erstveröffentlichers)
URL:
Volltext
(lizenzpflichtig)
URL:
https://doi.org/10.1017/CBO9781139062046
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