An Introduction to Algebraic Geometry

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An Introduction to Algebraic Geometry is written by Frank-Olaf Schreyer and published by Springer Nature. It's available with International Standard Book Number or ISBN identification 3031848349 (ISBN 10) and 9783031848346 (ISBN 13).

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.