Toric varieties: an introduction to algebraic geometry

Summary: the course is aimed at graduate students but with low prerequisites, the task is to make an easy introduction of basic concepts of Algebraic Geometry, to varieties over complex numbers, from the standpoint of toric varieties.

Main text:

• notes to 1 week intensive course in Grenoble Geometry of Toric Varieties by Cox, Brion, Barthel and Bonavero, available at Bonavero's web page.

Additional books: (R denotes the book is on the reserve in the library)

• Harsthorne: Algebraic Geometry, (R);
• Harris: Algebraic Geometry: a first course;
• Eisenbud, Harris: The geometry of schemes;
• Mumford: Projective algebraic geometry;
• Shafarevich: Osnowy algebraiczeskoj geometrii;
• Fulton Introduction to toric varieties, (R);
• Reid: Undergraduate algebraic geometry.

Tentative contents of the course.

1. Complex affine varieties: Nullstellensatz, Zariski topology, regular functions. Relation: affine varieties <-> finitely generated C-algebras without nilpotents. Irreducible varieties, field of rational functions. Maps of affine varieties. Products of affine varieties.
2. Affine toric varieties. Lattices and cones. Faces of cones and related open affine subsets. Maps of affine toric varieties. Big torus and its action on affine toric varieties, one parameter subgroups, fixed points.
3. Zariski tangent space, smoothness and Jacobian criterion for affine varieties. Regular cones and smooth toric varieties.
4. Sheaves. Abstract varieties; sheaf of regular functions. Gluing varieties. Separability. Projective varieties. Fans and construction of toric varieties. Separability for toric varieties.
5. Morphisms of varieties. Veronese and Segre maps. Birational morphisms. Dividing fans and birational morphisms of toric varieties. Blowups. Sub-latices and finite morphisms. Singularities of toric surfaces, quotients via abelian groups. Weighted projective space and fake weighted projective space.
6. Torus action on a toric variety. Orbit decomposition and relation to cones in the fan. Action of 1-parameter group, sources and sinks, Bialynicki-Birula decomposition for smooth toric varieties.
7. Vector bundles on varieties: transition functions, \v{C}ech cocycles and sections of vector bundles. Line bundles, Cartier divisors, principal divisors, linear equivalence, Picard group. Sections of line bundles, linear systems and maps into projective space. Picard group of toric varieties, sections of line bundles on toric varieties

Assignments.
By BBBC we mean Geometry of Toric Varieties by BBBC , other texts are referred to by the name of the author.

1. [BBBC], lect. 1: excer. 1.6, 1.8, 1.10 and 1.11; [Hartshorne], Ch. I: excer. 1.2 and 1.3; due Th, Jan 22th
2. [BBBC], lect. 2: excer. 1.10, 1.11, 1.12; [Fulton], sect. 1.3: excercises at pp. 18, 19 and 20; due Tue, Feb 10th.
3. Problems about Hirzebruch surfaces (due Th Feb 24th): a Hirzebruch surface F_m is defined by a complete fan on a plane with an intergal lattice N divided by the rays spanned by the vectors: (1,0), (0,1), (0,-1), (-1,m). Find the projection of F_m onto P^1, show that it is a P^1 bundle and find its sections: [Fulton], p23. Next describe the action of 1-parameter subgroups of T_N on F_m: find fixed points of the action and orbits. What is the real homology/cohomology of F_m? Prove that any toric surface whose fan is complete and contains four 2-dimensional regular cones (i.e. spanned by a basis of the lattice) is a Hirzebruch surface.
4. Weil divisors (due March 23): please read sections 1 - 3 of Lect. 9 from [BBBC], prepare all excercises which are in the text.

Projects assigned and provisional dates:

• Toric surface singularities and resolutions: Parsa Bakhtary, Apr 13
• The magic number 12 for toric surfaces: Bogume Jang, Apr 15
• 1-cycles on toric varieties and toric cone theorem: Sandeep Varma, Apr 20
• Toric contraction theorem: Shang Ning, Apr 22
• Toric varieties with orbifold contact structure: Valeria Grant Perez and Manish Kumar, Apr 27