UID:
almafu_9959338382902883
Format:
1 online resource (237 pages) :
,
illustrations
Edition:
1st ed. 2019.
ISBN:
9783030290160
,
3030290166
Series Statement:
Universitext,
Content:
The book presents the central facts of the local, projective and intrinsic theories of complex algebraic plane curves, with complete proofs and starting from low-level prerequisites. It includes Puiseux series, branches, intersection multiplicity, Bézout theorem, rational functions, Riemann-Roch theorem and rational maps. It is aimed at graduate and advanced undergraduate students, and also at anyone interested in algebraic curves or in an introduction to algebraic geometry via curves.
Note:
Intro -- Preface -- General Conventions -- Contents -- Chapter 1 -- Hypersurfaces, Elementary Facts -- 1.1 Projective Hypersurfaces -- 1.2 Affine Hypersurfaces -- 1.3 Singular Points -- 1.4 Rational Functions on Projective Spaces -- 1.5 Linear Families and Linear Conditions -- 1.6 Analytic Curves and Analytic Maps -- 1.7 Exercises -- Chapter 2 -- Local Properties of Plane Curves -- 2.1 Power Series -- 2.2 The Newton-Puiseux Algorithm -- 2.3 Further y-roots. Convergence -- 2.4 Branches of a Plane Curve -- 2.5 Branches and Irreducible Components -- 2.6 Intersection Multiplicity -- 2.7 Exercises -- Chapter 3 -- Projective Properties of Plane Curves -- 3.1 Sylvester's Resultant -- 3.2 Plane Curves and their Points -- 3.3 Bézout's Theorem -- 3.4 Pencils of Curves -- 3.5 Singular Points, Tangent Lines and Polar Curves -- 3.6 On Some Rings and Ideals -- 3.7 Rational Functions on an Irreducible Plane Curve -- 3.8 M. Noether's Fundamental Theorem -- 3.9 Exercises -- Chapter 4 -- The Intrinsic Geometry on a Curve -- 4.1 Rational and Birational Maps -- 4.2 Reducing Singularities -- 4.3 Birational Equivalence and Intrinsic Geometry -- 4.4 Divisors and Linear Series -- 4.5 Complete Linear Series, Riemann's Problem -- 4.6 The Genus of a Curve, Riemann's Inequality -- 4.7 Differentials on a Curve -- 4.8 The Canonical Series -- 4.9 The Riemann-Roch Theorem -- 4.10 Rational Maps Between Curves -- 4.11 Rational Maps Associated to Linear Series -- 4.12 Plane Rational Images of Curves -- 4.13 Exercises -- Bibliography -- Index.
Additional Edition:
ISBN 9783030290153
Additional Edition:
ISBN 3030290158
Language:
English
DOI:
10.1007/978-3-030-29016-0
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