Room: INF 205/ SR 3, Time: Wednesday, 16-18 (suspended)
Office hour: please drop me an email; since the course is delivered through a set of notes and recordings, it is particularly important that you feel free to get in touch with me if you have a question or want to discuss the material.
Toric varieties are nice compactifications of algebraic tori. Their geometry can be described entirely in combinatorial terms, either via rational cone complexes embedded in a fixed vector space (fans) or via polytopes. They provide the geometer with infinitely many computable examples (don't worry, there's some good software that comes to our help!). Their study has inspired many theories and constructions throughout geometry and mathematical physics.
Content summary:
22.04: a crash course on complex algebraic schemes;
29.04: affine toric varieties from characters and binomial ideals;
06.05: strongly convex rational polyhedral cones; local properties (normality and smoothness);
13.05: toric morphisms with examples: normalisation, faces (localisation), simplicial cones and quotient singularities;
20.05: projective toric varieties from characters; normal separated toric varieties from fans;
27.05: limits of 1 PS and the orbit-cone correspondence; every normal separated toric variety has a fan;
03.06: a reminder on the classical topology, separatedness and properness; toric morphisms in general;
10.06: a reminder on Weil and Cartier divisors; exact sequences for Cl(X_\Sigma); every Cartier divisor on an affine toric variety is principal;
17.06: Cartier divisors as piecewise linear functions, their global sections as lattice points of a polyhedron; projective toric varieties with a(n ample) line bundle from polytopes;
24.06: basepoint-free (bpf) and ample line bundles in terms of convexity of their support functions;
01.07: the quotient construction of a toric variety with no torus-factors, the functor of a smooth toric variety (\Sigma-collections), homogeneous vs local coordinates;
08.07: the quotient construction from the point of view of GIT; an overview of the secondary fan;
15.07: the intersection product, the cone of curves and the cone of nef divisors, the structure of bpf divisors;
22.07: Kähler differentials and the tangent bundle, the Euler sequence, the dualising line bundle;
29.07: toric varieties of Picard rank one (i.e. weighted projective spaces and fakes).
I strongly recommend taking a look at Dhruv's little useful trinkets.
Plan of the course:
toric varieties from fans, local and global properties;
line bundles and their cohomology, polytopes;
the GIT construction and the functor of points of a smooth toric variety;
cohomology/Chow ring;
possible extra material: the secondary fan (aka toric VGIT), resolutions, deformations of toric singularities.
Essential bibliography:
Books: Cox-Little-Schenck "Toric varieties", Fulton "Introduction to toric varieties".
Notes: T. Reichelt (in German), D. Ranganathan, M. Mustaţă.
Prerequisites: useful would be an acquaintance with the concept that (algebraic) varieties can be constructed by gluing (affine) charts (but we can start by reviewing Spec and Proj if required), and some other fundamental concepts in geometry, such as line bundles and (singular) cohomology. Toric varieties are great because they provide you with a ton of examples on which to apply your algebraic geometry, but they also constitute a playground where you can learn some algebro-geometric terminology if you are not familiar with it yet.
Oral exam