96768 - Toric Geometry and Applications

Learning outcomes

At the end of the course, students have a firm knowledge of the geometry of toric varieties and they know some of their applications. In particular, they understand the connection between toric varieties and combinatorics. They are able to use this knowledge in their research and for the construction of mathematical models.

Course contents

Toric Geometry is the study of toric varieties, a special class of algebraic varieties constructed combinatorially from discrete geometric objects such as polytopes or cones. There exists a rich dictionary translating algebro-geometric properties of a toric variety into properties of its combinatorial counterpart. This has made the setting of toric varieties a valuable testing ground for conjectures in algebraic geometry. Conversely, the power of algebraic geometry has enabled the proof of results in combinatorics through the use of toric varieties.

The course will provide an introduction to toric varieties: affine toric varieties constructed from cones, abstract toric varieties built from fans, their properties (orbits, compactness, smoothness), projective toric varieties derived from polytopes, and some topological/geometric invariants of toric varieties (fundamental group, singular cohomology). In the final part, applications of toric varieties to combinatorics will be presented; depending on the audience's interests, topics may include the Kähler package associated with a matroid, following the work of Adiprasito-Huh-Katz, and/or Ehrhart theory.

 

Prerequisites:

• Mandatory undergraduate courses in algebra and geometry (in particular: topology and the fundamental group, polynomial rings and their ideals).
• Basic knowledge of algebraic topology (specifically: singular homology and cohomology).
• Basic knowledge of algebraic geometry (specifically: Zariski-closed sets in affine or projective space over an algebraically closed field).

Advanced topics in algebraic geometry (such as schemes, locally free/invertible sheaves, vector bundles, divisors) and in commutative algebra are not required.

Readings/Bibliography

Cox, Little, Schenck, Toric varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society.

Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press.

Additional references for consultation:

Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 15, Springer-Verlag.

Cox, What is a toric variety?, Topics in algebraic geometry and geometric modeling, 2003, pp. 203–223.

Kempf, Knudsen, Mumford, Saint-Donat, Toroidal embeddings 1, Lecture Notes in Mathematics, volume 339, Springer.

C. Eur, Essence Of Independence: Hodge Theory Of Matroids Since June Huh, section 4.

Teaching methods

Lectures will be delivered at the blackboard. Occasionally, there will be computer-based exercises using computational algebra software capable of handling toric varieties.

Assessment methods

Homeworks assigned during the semester. Final oral exam.

 

 

Students with learning disorders and\or temporary or permanent disabilities: please, contact the office responsible (https://site.unibo.it/studenti-con-disabilita-e-dsa/en/for-students ) as soon as possible so that they can propose acceptable adjustments. The request for adaptation must be submitted in advance (15 days before the exam date) to the lecturer, who will assess the appropriateness of the adjustments, taking into account the teaching objectives.

Office hours

See the website of Roberto Pagaria

See the website of Andrea Petracci