Computational Algebraic Geometry
M2 course at LJAD, Thursday morning, 9:00am-12:00am, academic year 2021-2022.
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.
Macaulay 2 is a free software for computations in algebraic geometry. Macaulay 2 webpage is here.
Lectures:
- Sept. 16th: Ideals, algebraic varieties, primary decomposition
(Chapter 1, Sections 1-4, M2 file).
- Sept. 23rd: Nullstellensatz, projective spaces and varieties, graded rings and modules, Hilbert functions, polynomials and series.
(Chapter 1, Section 5, Chapter 2 M2 file).
- Sept. 30th: Hilbert and series polynomial, finite free resolution, regular sequences.
(Chapter 3, M2 file).
- Oct. 7th: Grobner bases.
(Chapter 4, Sections 1-3).
- Oct. 8th: Computation of syzygy modules, elimination ideal.
(Chapter 4, Sections 4-5 and Chapter 5, Section 1, M2 file).
- Oct. 21th: Applications of elimination ideals, Sylvester resultant.
(Chapter 5, Sections 2-3; midterm-exam).
- Oct. 28th (2 lectures): Cokernel of the Sylvester matrix, Bézout theorem, implicitization of plane curves.
(Chapter 5, Sections 3-5; M2 file).
- Nov. 17th: The Elimination Theorem, Inertia forms and saturation, definition of resutants.
(Chapter 6, Sections 1-3).
- Nov. 19th: Properties of resutants, applications
(Chapter 6, Sections 4-5).
- Nov. 30th: final exam.
Notes:
References:
- David Cox, John Little, Donald O'Shea, Ideals Varieties and Algorithms. Undergraduate texts in Maths, Third Edition, 2007.
- David Cox, John Little, Donald O'Shea, Using Algebraic Geometry. Graduate Texts in Maths, 2005.
- David Eisenbud. Commutative Algebra with a view toward algebraic geometry. Graduate Texts in Mathematics, volume 150. Springer-Verlag, New York, 1995.
- David Eisenbud. The Geometry of Syzygies: A Second Course in Algebraic Geometry and Commutative Algebra. Graduate Texts in Mathematics, Vol.~229, Springer, 2005.
- Robin Hartshorne. Algebraic Geometry. Graduate Texts in Mathematics, volume 52, Springer, 1977.
- Serge Lang, Algebra. Graduate Texts in Mathematics, volume 211, Springer, 2002.
- Mateusz Michalek and Bernd Sturmfels. Invitation to Nonlinear Algebra, volume 211 of Graduate Studies in Mathematics. AMS, 2020.
- Hal Schenck. Computational Algebraic Geometry. Cambridge University Press, 2003.