Contents Preface ix Chapter 1. Interseetions of Hypersurfaees 1 1.1. Early history (Bezout, Poncelet) 1 1.2. Class of a curve (Plüeker) 2 1.3. Degree of a dual surface (Salmon) 2 1.4. The problem of five conics 4 1.5. Á dynamic formula (Severi, Lazarsfeld) 5 1.6. Algebraie multiplicity, resultants 6 Chapter 2. Multiplicity and Normal Cones 9 2.1. Geometrie multiplicity 9 2.2. Hubert polynomials 9 2.3. Á refinement of Bezout's theorem 10 2.4. Samuel's intersection multiplicity 11 2.5. Normal cones 12 2.6. Deformation to the normal cone 15 2.7. Intersection produets: a preview 16 Chapter 3. Divisors and Rational Equivalence 19 3.1. Homology and cohomology 19 3.2. Divisors 21 3.3. Rational equivalence 22 3.4. Intersecting with divisors 24 3.5. Applications 26 Chapter 4. Chern Classes and Segre Classes 29 4.1. Chern classes of vector bundles 29 4.2. Segre classes of cones and subvarieties 31 4.3. Intersection forumulas 33 Chapter 5. Gysin Maps and Intersection Rings 37 5.1. Gysin homomorphisms 37 5.2. The intersection ring of a nonsingular variety 39 5.3. Grassmannians and flag varieties 41 5.4. Enumerating tangents 43 Chapter 6. Degeneracy Loci 47
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