Frank-Olaf Schreyer – författare

In the last decade, there has been a burgeoning of activity in the design and implementation of algorithms for algebraic geometric compuation. Some of these algorithms were originally designed for abstract algebraic geometry, but now are of interest for use in applications and some of these algorithms were originally designed for applications, but now are of interest for use in abstract algebraic geometry.

The workshop on Algorithms in Algebraic Geometry that was held in the framework of the IMA Annual Program Year in Applications of Algebraic Geometry by the Institute for Mathematics and Its Applications on September 18-22, 2006 at the University of Minnesota is one tangible indication of the interest. One hundred ten participants from eleven countries and twenty states came to listen to the many talks; discuss mathematics; and pursue collaborative work on the many faceted problems and the algorithms, both symbolic and numberic, that illuminate them.

This volume of articles captures some of the spirit of the IMA workshop.

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.

The series is aimed specifically at publishing peer reviewed reviews and contributions presented at workshops and conferences. Each volume is associated with a particular conference, symposium or workshop. These events cover various topics within pure and applied mathematics and provide up-to-date coverage of new developments, methods and applications.