Algebraic Curves, the Brill and Noether Way

verfasst von: Prof. Eduardo Casas-Alvero

Verlag: Springer International Publishing

Buchreihe : Universitext

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten

Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Hypersurfaces, Elementary Facts

Abstract

The reader is referred to any book on projective geometry (for instance [4]) for the most basic facts about projective spaces and homogeneous coordinates.

Eduardo Casas-Alvero

Chapter 2. Local Properties of Plane Curves

Abstract

Let f ∈ C[x, y]. Our first aim in this chapter is to find some sort of power series s = s(x), in the variable x, such that f(x, s(x)) is identically zero in x; in other words s is a root of f viewed as a polynomial in y with coefficients in C[x].

Eduardo Casas-Alvero

Chapter 3. Projective Properties of Plane Curves

Abstract

This section is devoted to introducing and proving the main properties of the resultant, in the form due to Sylvester. Roughly speaking the resultant of two polynomials is a polynomial expression in the coefficients of the polynomials that is zero if and only if either the polynomials have a common root or they both have degree strictly less than prescribed.

Eduardo Casas-Alvero

Chapter 4. The Intrinsic Geometry on a Curve

Abstract

The study of an irreducible curve modulo the action of birational maps is called the intrinsic geometry on the curve. It may of course also be called birationally invariant geometry.

Eduardo Casas-Alvero

Backmatter

Titel: Algebraic Curves, the Brill and Noether Way
verfasst von: Prof. Eduardo Casas-Alvero
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-29016-0
Print ISBN: 978-3-030-29015-3
DOI: https://doi.org/10.1007/978-3-030-29016-0