Titlebook: Introduction to Plane Algebraic Curves; Ernst Kunz Textbook 2005 Birkh?user Boston 2005 Algebraic curve.Belshoff.Kunz.algebra.computer alg

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nt departure from other works on plane algebraic curves in w.This work treats an introduction to commutative ring theory and algebraic plane curves, requiring of the student only a basic knowledge of algebra, with all of the algebraic facts collected into several appendices that can be easily referr

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Regular and Singular Points of Algebraic Curves. Tangentside whether a point is simple or singular with the help of the local ring at the point.The facts from Appendix E on Noetherian rings and discrete valuation rings will play a role in this chapter.Toward the end, some theorems from Appendix F on integral ring extensions wil l al so be needed.

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Applications of Residue Theory to Curvesue of differentials on a smooth curve. See also Griffiths-Harris [.], Chapter V. The theorems presented here have far-reaching higher-dimensional generalizations ([.],[.], [.],[.], [.]). In his thesis [.] Gerhard Quarg has discovered further global geometric applications of algebraic residue theory. [.] contains an outline of part of this thesis.

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The Riemann-Roch Theoremtheorem, one for the curve itself and one for its Riemann surface (its function field). The theorem leads to an important birational invariant of irreducible curves, namely the genus of the associated function held. An excellent presentation of the corresponding complex-analytic theory is given by Forst er [.].

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Residue Calculusion as here. What we are talking about is sometimes called Grothendieck residue theory. It was originally introduced in [.], Chapter 3, §9, in great generality. For different approaches, see also [.] and [.]. Chapters 11 and 12 will not be used in Chapter 13 and later. The reader may go directly from here to the Riemann-Roch theorem.

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The Coordinate Ring of an Algebraic Curve and the Intersections of Two CurvesFrom now on, we assume that the reader is familiar with the material in Appendices A and B. Above all,we will use the methods contained in Appendix B repeatedly. We will also apply the elementary Lemmas D.5 and I.4.

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