An Introduction to Algebraic Geometry
A Computational Approach
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Beschreibung
Algebraic Geometry is a huge area of mathematics which went through several phases: Hilbert's fundamental paper from 1890, sheaves and cohomology introduced by Serre in the 1950s, Grothendieck's theory of schemes in the 1960s and so on. This book covers the basic material known before Serre's introduction of sheaves to the subject with an emphasis on computational methods. In particular, we will use Gröbner basis systematically.
The highlights are the Nullstellensatz, Gröbner basis, Hilbert's syzygy theorem and the Hilbert function, Bézout’s theorem, semi-continuity of the fiber dimension, Bertini's theorem, Cremona resolution of plane curves and parametrization of rational curves.
In the final chapter we discuss the proof of the Riemann-Roch theorem due to Brill and Noether, and give its basic applications.The algorithm to compute the Riemann-Roch space of a divisor on a curve, which has a plane model with only ordinary singularities, use adjoint systems. The proof of the completeness of adjoint systems becomes much more transparent if one use cohomology of coherent sheaves. Instead of giving the original proof of Max Noether, we explain in an appendix how this easily follows from standard facts on cohomology of coherent sheaves.
The book aims at undergraduate students. It could be a course book for a first Algebraic Geometry lecture, and hopefully motivates further studies.
Algebraic Geometry is a huge area of mathematics which went through several phases: Hilbert's fundamental paper from 1890, sheaves and cohomology introduced by Serre in the 1950s, Grothendieck's theory of schemes in the 1960s and so on. This book covers the basic material known before Serre's introduction of sheaves to the subject with an emphasis on computational methods. In particular, we will use Gröbner basis systematically.
The highlights are the Nullstellensatz, Gröbner basis, Hilbert's syzygy theorem and the Hilbert function, Bézout’s theorem, semi-continuity of the fiber dimension, Bertini's theorem, Cremona resolution of plane curves and parametrization of rational curves.
In the final chapter we discuss the proof of the Riemann-Roch theorem due to Brill and Noether, and give its basic applications.The algorithm to compute the Riemann-Roch space of a divisor on a curve, which has a plane model with only ordinary singularities, use adjoint systems. The proof of the completeness of adjoint systems becomes much more transparent if one use cohomology of coherent sheaves. Instead of giving the original proof of Max Noether, we explain in an appendix how this easily follows from standard facts on cohomology of coherent sheaves.
The book aims at undergraduate students. It could be a course book for a first Algebraic Geometry lecture, and hopefully motivates further studies.
Uses Groebner basis systematically in proofs of classical theorem Has a computational focus, featuring code in Macaulay2 for exercise Adds to classical Algebraic Geometry modern computational methods
Autor*in
Frank-Olaf Schreyer
Themen in »An Introduction to Algebraic Geometry«
Hilbert's basis theorem Nullstellensatz algorithms in algebraic geometry computation of Riemann-Roch spaces projective geometry semi-continuity of fiber dimensions Gröbner basis Primary decomposition Bertini's theorem Hilbert functions and polynomials Hurwitz formula Bezout's theorem parametrization of rational curves
Stimmen zu »An Introduction to Algebraic Geometry«
“The present book is an Introduction to Algebraic Geometry at a more or less graduate level. It makes the point on computational methods, and the purpose is that the text be useful for computer science students. ... The author has made a great work in making the book as self-contained as possible, and this results very well explained.” (José Javier Etayo, zbMATH 1573.14001, 2026)
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