Kunz, Ernst
Translated by Belshoff, R.G.
Based on the original German edition, "Ebene algebraische Kurven", Der Regensburg Trichter, 23, Universität Regensburg, ISBN 3-88246-167-5, © 1991 Ernst Kunz; English translation by Richard G. Belshoff.
2005, XII, 293 p. 52 illus.
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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 referred to, as needed.
Kunz's proven conception of teaching topics in commutative algebra together with their applications to algebraic geometry makes this book significantly different from others on plane algebraic curves. The exposition focuses on the purely algebraic aspects of plane curve theory, leaving the topological and analytical viewpoints in the background, with only casual references to these subjects and suggestions for further reading.
Most important to this text:
* Emphasizes and utilizes the theory of filtered algebras, their graduated rings and Rees algebras, to deduce basic facts about the intersection theory of plane curves
* Presents residue theory in the affine plane and its applications to intersection theory
* Methods of proof for the Riemann–Roch theorem conform to the presentation of curve theory, formulated in the language of filtrations and associated graded rings
* Examples, exercises, figures and suggestions for further study round out this fairly self-contained textbook
From a review of the German edition:
"[T]he reader is invited to learn some topics from commutative ring theory by mainly studying their illustrations and applications in plane curve theory. This methodical approach is certainly very enlightening and efficient for both teachers and students… The whole text is a real masterpiece of clarity, rigor, comprehension, methodical skill, algebraic and geometric motivation…highly enlightening, motivating and entertaining at the same time… One simply cannot do better in writing such a textbook."
—Zentralblatt MATH
Content Level » Graduate
Keywords » Algebraic curve - Belshoff - Kunz - algebra - computer algebra - ksa - ring theory
Related subjects » Birkhäuser Mathematics
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