An Introduction to Algebraic Geometry

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

Autorentext

Frank-Olaf Schreyer is a German mathematician, specializing in algebraic geometry and algorithmic algebraic geometry. Schreyer received in 1983 his PhD from Brandeis University with thesis Syzygies of Curves with Special Pencils under the supervision of David Eisenbud. Schreyer was a professor at University of Bayreuth and is since 2002 a professor for mathematics and computer sciences at Saarland University. He has been a visiting professor at the Simons Laufer Mathematical Sciences Institute at Berkeley and at KAIST in South Korea. He is involved in the development of (algorithmic) algebraic geometry, and is well-known for his contributions to the theory of syzygies.

Inhalt

1. Hilbert's Nullstellensatz.- 2. The algebra-geometry dictionary.- 3. Noetherian rings and primary decomposition.- 4. Localization.- 5. Rational functions and dimension.- 6. Integral ring extensions and Krull dimension.- 7. Constructive ideal and module theory.- 8. Projective algebraic geometry.- 9. Bézout's theorem.- 10. Local rings and power series.- 11. Products and morphisms of projective varieties.- 12. Resolution of curve singularities.- 13. Families of varieties.- 14. Bertini's theorem and applications.- 15. The geometric genus of a plane curve.- 16. Riemann-Roch.- A. A glimpse of sheaves and cohomology.- B. Code for Macaulay2 computation.- References.- Glossary.- Index.