The Practice of Algebraic Curves: A Second Course in Algebraic Geometry

Main subject categories: • Algebraic curves • Algebraic geometry • History of algebraic geometryThis textbook provides readers with a working knowledge of the modern theory of complex projective algebraic curves. Also known as compact Riemann surfaces, such curves shaped the development of algebraic geometry itself, making this theory essential background for anyone working in or using this discipline. Examples underpin the presentation throughout, illustrating techniques that range across classical geometric theory, modern commutative algebra, and moduli theory.The book begins with two chapters covering basic ideas, including maps to projective space, invertible sheaves, and the Riemann–Roch theorem. Subsequent chapters alternate between a detailed study of curves up to genus six and more advanced topics such as Jacobians, Hilbert schemes, moduli spaces of curves, Severi varieties, dualizing sheaves, and linkage of curves in 3-space. Three chapters treat the refinements of the Brill–Noether theorem, including applications and a complete proof of the basic result. Two chapters on free resolutions, rational normal scrolls, and canonical curves build context for Green’s conjecture. The book culminates in a study of Hilbert schemes of curves through examples. A historical appendix by Jeremy Gray captures the early development of the theory of algebraic curves. Exercises, illustrations, and open problems accompany the text throughout.The Practice of Algebraic Curves offers a masterclass in theory that has become essential in areas ranging from algebraic geometry itself to mathematical physics and other applications. Suitable for students and researchers alike, the text bridges the gap from a first course in algebraic geometry to advanced literature and active research.Readership • Graduate students considering working in the field of algebraic curves and researchers in a related field whose work has led them to questions about algebraic curves. Cover Title page Copyright Dedication Publisher's Notice Contents Preface Introduction Why you want to read this book Why we wrote this book What's with the title? What's in this book Exercises and hints Relation of this book to other texts Prerequisites, notation and conventions Commutative algebra Projective geometry Sheaves and cohomology Chapter 1. Linear series and morphisms to projective space 1.1. Divisors 1.2. Divisors and rational functions Generalizations Divisors of functions Invertible sheaves Invertible sheaves and line bundles 1.3. Linear series and maps to projective space 1.4. The geometry of linear series An upper bound on h0(L) Incomplete linear series Sums of linear series Which linear series define embeddings? Exercises Chapter 2. The Riemann–Roch theorem 2.1. How many sections? Riemann–Roch without duality 2.2. The most interesting linear series The adjunction formula Hurwitz's theorem 2.3. Riemann–Roch with duality Residues Arithmetic genus and geometric genus 2.4. The canonical morphism Geometric Riemann–Roch Linear series on a hyperelliptic curve 2.5. Clifford's theorem 2.6. Curves on surfaces The intersection pairing The Riemann–Roch theorem for smooth surfaces Blowups of smooth surfaces 2.7. Quadrics in P3 and the curves they contain The classification of quadrics Some classes of curves on quadrics 2.8. Exercises Chapter 3. Curves of genus 0 3.1. Rational normal curves 3.2. Other rational curves Smooth rational quartics Some open problems about rational curves 3.3. The Cohen–Macaulay property 3.4. Exercises Chapter 4. Smooth plane curves and curves of genus 1 4.1. Riemann, Clebsch, Brill and Noether 4.2. Smooth plane curves 4.2.1. Differentials on a smooth plane curve 4.2.2. Linear series on a smooth plane curve 4.2.3. The Cayley–Bacharach–Macaulay theorem 4.3. Curves of genus 1 and the group law of an elliptic curve 4.4. Low degree divisors on curves of genus 1 The dimension of families Double covers of P1 Plane cubics 4.5. Genus 1 quartics in P3 4.6. Genus 1 quintics in P4 4.7. Exercises Chapter 5. Jacobians 5.1. Symmetric products and the universal divisor Finite group quotients 5.2. The Picard varieties 5.3. Jacobians 5.4. Abel's theorem 5.5. The g+3 theorem 5.6. The schemes Wrd(C) 5.7. Examples in low genus Genus 1 Genus 2 Genus 3 5.8. Martens' theorem 5.9. Exercises Chapter 6. Hyperelliptic curves and curves of genus 2 and 3 6.1. Hyperelliptic curves The equation of a hyperelliptic curve Differentials on a hyperelliptic curve 6.2. Branched covers with specified branching Branched covers of P1 6.3. Curves of genus 2 Maps of C to P1 Maps of C to P2 Embeddings in P3 The dimension of the family of genus 2 curves 6.4. Curves of genus 3 Other representations of a curve of genus 3 6.5. Theta characteristics Counting theta characteristics (proof of Theorem 6.8) 6.6. Exercises Chapter 7. Fine moduli spaces 7.1. What is a moduli problem? 7.2. What is a solution to a moduli problem? 7.3. Hilbert schemes 7.3.1. The tangent space to the Hilbert scheme 7.3.2. Parametrizing twisted cubics 7.3.3. Construction of the Hilbert scheme in general 7.3.4. Grassmannians 7.3.5. Equations defining the Hilbert scheme 7.4. Bounding the number of maps between curves 7.5. Exercises Chapter 8. Moduli of curves 8.1. Curves of genus 1 M1 is a coarse moduli space The good news Compactifying M1 8.2. Higher genus Stable, semistable, unstable 8.3. Stable curves How we deal with the fact that 2muto 6.8pt-2mu M-0mu0mug is not fine 8.4. Can one write down a general curve of genus g? 8.5. Hurwitz spaces The dimension of Mg Irreducibility of Mg 8.6. The Severi variety Local geometry of the Severi variety 8.7. Exercises Chapter 9. Curves of genus 4 and 5 9.1. Curves of genus 4 The canonical model Maps to projective space 9.2. Curves of genus 5 9.3. Canonical curves of genus 5 First case: the intersection of the quadrics has dimension 1 Second case: the intersection of the quadrics is a surface 9.4. Exercises Chapter 10. Hyperplane sectionsof a curve 10.1. Linearly general position 10.2. Castelnuovo's theorem Proof of Castelnuovo's bound Consequences and special cases 10.3. Other applications of linearly general position Existence of good projections The case of equality in Martens' theorem The g+2 theorem 10.4. Exercises Chapter 11. Monodromy of hyperplane sections 11.1. Uniform position and monodromy The monodromy group of a generically finite morphism Uniform position 11.2. Flexes and bitangents are isolated Not every tangent line is tangent at a flex Not every tangent is bitangent 11.3. Proof of the uniform position lemma Uniform position for higher-dimensional varieties 11.4. Applications of uniform position Irreducibility of fiber powers Numerical uniform position Sums of linear series Nodes of plane curves 11.5. Exercises Chapter 12. Brill–Noether theory and applications to genus 6 12.1. What linear series exist? 12.2. Brill–Noether theory 12.2.1. A Brill–Noether inequality 12.2.2. Refinements of the Brill–Noether theorem 12.3. Linear series on curves of genus 6 12.3.1. General curves of genus 6 12.3.2. Del Pezzo surfaces 12.3.3. The canonical image of a general curve of genus 6 12.4. Classification of curves of genus 6 |D| has a basepoint C is not trigonal and the image of phi(D) is two-to-one onto a plane curve of degree 3. 12.5. Exercises Chapter 13. Inflection points 13.1. Inflection points, Plücker formulas and Weierstrass points Definitions The Plücker formula Flexes of plane curves Weierstrass points Another characterization of Weierstrass points 13.2. Finiteness of the automorphism group 13.3. Curves with automorphisms are special 13.4. Inflections of linear series on P1 Schubert cycles Special Schubert cycles and Pieri's formula Conclusion 13.5. Exercises Chapter 14. Proof of the Brill–Noether theorem 14.1. Castelnuovo's approach Upper bound on the codimension of Wrd(C) 14.2. Specializing to a g-cuspidal curve Constructing curves with cusps Smoothing a cuspidal curve 14.3. The family of Picard varieties The Picard variety of a cuspidal curve The relative Picard variety Limits of invertible sheaves 14.4. Putting it all together Nonexistence Existence 14.5. Brill–Noether with inflection 14.6. Exercises Chapter 15. Using a singular plane model 15.1. Nodal plane curves 15.1.1. Differentials on a nodal plane curve 15.1.2. Linear series on a nodal plane curve 15.2. Arbitrary plane curves The conductor ideal and linear series on the normalization Differentials 15.3. Exercises Chapter 16. Linkage and the canonical sheaves of singular curves 16.1. Introduction 16.2. Linkage of twisted cubics 16.3. Linkage of smooth curves in P3 16.4. Linkage of purely 1-dimensional schemes in P3 16.5. Degree and genus of linked curves Dualizing sheaves for singular curves 16.6. The construction of dualizing sheaves 16.7. The linkage equivalence relation 16.8. Comparing the canonical sheaf with that of the normalization 16.9. A general Riemann–Roch theorem 16.10. Exercises Ropes and ribbons General adjunction Chapter 17. Scrolls and the curves they contain 17.1. Some classical geometry 17.2. 1-generic matrices and the equations of scrolls 17.3. Scrolls as images of projective bundles 17.4. Curves on a 2-dimensional scroll Finding a scroll containing a given curve Finding curves on a given scroll 17.5. Exercises Chapter 18. Free resolutions and canonical curves 18.1. Free resolutions The classification of 1-generic 2 x f matrices How to look at a resolution When is a finite free complex a resolution? 18.2. Depth and the Cohen–Macaulay property The Gorenstein property 18.3. The Eagon–Northcott complex The case rankG = 1: the Koszul complex The case rankF = rankG + 1: the Hilbert–Burch complex The Hilbert–Burch theorem The general case of the Eagon–Northcott complex 18.4. Green's conjecture 18.5. Exercises Chapter 19. Hilbert Schemes 19.1. Degree 3 The other component of H0,3,3 19.2. Extraneous components 19.3. Degree 4 Genus 0 Genus 1 19.4. Degree 5 Genus 2 19.5. Degree 6 Genus 4 Genus 3 19.6. Degree 7 19.7. The expected dimension of Hg,r,d 19.8. Some open problems Brill–Noether in low codimension Maximally special curves Rigid curves? 19.9. Degree 8, genus 9 19.10. Degree 9, genus 10 19.11. Estimating the dimension of the restricted Hilbert schemes using the Brill–Noether theorem 19.12. Exercises Appendix: A historical essay on some topics in algebraic geometry to 30ptby Jeremy Gray 1. Greek mathematicians and conic sections 2. The first appearance of complex numbers 3. Conic sections from the 17th to the 19th centuries 4. Curves of higher degree from the 17th to the early 19th century 5. The birth of projective space 6. Riemann's theory of algebraic curves and its reception 7. First ideas about the resolution of singular points 8. The work of Brill and Noether 9. Historical bibliography Hints to marked exercises Bibliography Index Back Cover This textbook provides readers with a working knowledge of the modern theory of complex projective algebraic curves. Also known as compact Riemann surfaces, such curves shaped the development of algebraic geometry itself, making this theory essential background for anyone working in or using this discipline. Examples underpin the presentation throughout, illustrating techniques that range across classical geometric theory, modern commutative algebra, and moduli theory. The book begins with two chapters covering basic ideas, including maps to projective space, invertible sheaves, and the Riemann–Roch theorem. Subsequent chapters alternate between a detailed study of curves up to genus six and more advanced topics such as Jacobians, Hilbert schemes, moduli spaces of curves, Severi varieties, dualizing sheaves, and linkage of curves in 3-space. Three chapters treat the refinements of the Brill–Noether theorem, including applications and a complete proof of the basic result. Two chapters on free resolutions, rational normal scrolls, and canonical curves build context for Green’s conjecture. The book culminates in a study of Hilbert schemes of curves through examples. A historical appendix by Jeremy Gray captures the early development of the theory of algebraic curves. Exercises, illustrations, and open problems accompany the text throughout. The Practice of Algebraic Curves offers a masterclass in theory that has become essential in areas ranging from algebraic geometry itself to mathematical physics and other applications. Suitable for students and researchers alike, the text bridges the gap from a first course in algebraic geometry to advanced literature and active research. Readership Graduate students considering working in the field of algebraic curves and researchers in a related field whose work has led them to questions about algebraic curves.