Computational Algebraic Geometry

M2 course at LJAD, Thursday morning, 9:00am-12:00am, academic year 2020-2021.

Laurent Busé

Summary:

The classical results and open problems surrounding free resolutions, regularity and syzygies, topics that lie at the interface between commutative algebra and algebraic geometry, have undergone a striking evolution over the last quarter of a century, aided in large part by computer algebra calculations. Several new techniques have emerged and led to important theoretical developments with new results and new conjectures attracting of a lot of interest. In the same time, the applications of these techniques have been successfully applied in many fields such as combinatorics, geometric modeling, optimization, statistics, and it is now a very active area of research. The aim of this course is precisely to introduce students to some fundamental techniques and recent developments on effective methods in commutative algebra, with a view toward applications in computational algebraic geometry. Many examples will be treated and the students will be trained to the free computer algebra system Macaulay2 which is widely used and developed by a large community of mathematicians.

The first part of this course will be devoted to some of the main tools and concepts in commutative algebra that are used to derive effective methods: associated primes and primary decomposition, graded rings and modules, finite free resolutions, regular sequences, Hilbert functions and Grobner basis. The second part of the lectures will focus on syzygies and their geometric content. We will develop material in relation with the Castelnuovo-Mumford regularity, a central and very active research topic, and modern elimination theory, with applications to the study of fibers of rational maps between projective spaces and the solving of polynomial systems. For that purpose, some classical material from homological algebra will be presented from a computational perspective, including Koszul complexes, Cech complexes, local cohomology, Tor and Ext functors, spectral sequences. Blowup algebras (symmetric and Rees algebras) will also be introduced and studied, in particular the shape and structure of their defining equations.

Macaulay 2 webpage is here.

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