Toric Geometry

Welcome to the course webpage for toric geometry, taught in the 2019 Michaelmas term at Cambridge. This course is intended for Part III students in pure mathematics. 

Instructor: Dhruv Ranganathan (E 1.01). Meeting time: TuThSa 10-11 AM (MR9). 

Overview: Toric varieties occupy an interesting space in algebraic geometry. They are in some sense extremely special (e.g. they are all birational to projective space and have large automorphism groups). On the other hand, they form a sufficiently rich class of examples to be a testing ground for basic theory. Crucially their geometry is controlled by certain combinatorial structures, which makes them places where one can actually calculate things. Our goal will be to develop the basic theory of toric varieties and to get a sense of the geometry of varieties along the way - fundamental concepts of singularity theory, Chow groups, and semistable reduction will all make an appearance. 

Resources:  The canonical texts on the subject are the concise Introduction to Toric Varieties by Fulton and the encyclopedic Toric Varieties by Cox, Little, and Schenck. There are excellent notes by David Cox and Mircea Mustaţă that I also like. 

Lecture schedule (click on the dates for notes, where applicable): Complete handwritten notes are available here.

• October 10: Monomials, binomials, and trinomials, the algebraic torus and toric varieties. 

• October 12: Affine toric varieties, limits of cocharacters, polyhedral cones and semigroups. 

• October 15: More on affine toric varieties, cones, dual cones, and examples. (Notes contained in previous day's)

• October 17: Gluing affine toric varieties, toric varieties are separated. See these notes on gluing.

• October 19: Toric varieties are separated. Points as homomorphisms. Smooth affine toric varieties. 

• October 22: Smoothness and normality of toric varieties. Properness. See this MathOverflow post on normality. 

• At this point, you may want to make sure you're able to work with the basic theory we've developed. See this.

• October 24: Distinguished points, limits of subtori, properness. Discussion of "analytic" valuative criteria.

• October 26: Properness, polytopes and projective toric varieties, morphisms of toric varieties.

• October 29: Morphisms of toric varieties, blowups. 

• October 31: Properness for toric morphisms, properties of blowups, toric surface singularities.

• November 2: Weighted projective spaces and toric surface singularities and resolution.  

• November 5: The orbit-cone correspondence and examples.

• November 7: More on the orbit-cone correspondence, an extended example.

• November 9: Orders of vanishing, Weil and Cartier divisors and class groups. The class group of a toric variety.

• November 12: Proof of the class group formula for toric varieties using the class group exact sequence.

• November 14: The Picard group and class group of a toric variety.

• November 16: Line bundles from piecewise linear functions. 

• November 19: A look ahead to Chow groups. The intersection product of an invariant curve with a Cartier divisor. 

• November 21: Convex piecewise linear functions, polytopes, and toric varieties. 

• November 23: Ampleness and global generation for toric varieties, non-projective toric variety. 

• November 26: Weak factorization and weak semistable reduction. The criterion for equidimensionality. 

• November 28:  Semistable reduction for toric varieties. 

• November 30: The Chow groups and Chow rings of toric varieties.

• December 3:  Toric geometry everywhere: logarithmic structures and tropical geometry. 

What I hope you've gained from this course. My goal was to get you to the point where you've internalized enough of the toric dictionary that a few things have happened. First, you should have access to roughly a bazillion more examples of algebraic varieties than you had before. Second, you will have the ability to compute things of basic interest in algebraic geometry. Finally, you will have seen the range of behaviour and techniques that one sees in algebraic geometry -- how bad singularities can be, the effect they have on things like the class group and Picard group, the resolution of singularities algorithms, the ways in which dimensions of fibers of morphisms change, etc.

What I would have done with two extra weeks. If I had two more weeks, I would have wanted to do a few more topics. First, the is a famous construction of toric varieties due to Cox, which presents every toric variety as a quotient of a complement of a small subset of affine space by a torus action, analogous to the description of projective space as a quotient of affine space minus its origin by the diagonal torus. This also endows any toric variety with generalized homogeneous coordinates. Second, I would have explained the calculation of the cohomology of line bundles on toric varieties, which you're well-equipped to read from the end of Section 3 of Fulton. Finally, I would have liked to cover the Hirzebruch-Riemann-Roch theorem on toric varieties, at the end of Fulton's book.

Where to go now. Hopefully, you are now in a position to read through Fulton's toric varieties book from start to finish, and you will have mastered the material -- you should now go on to master the other elements so that you can defeat Fire Lord Ozai when Sozin's comet arrives! In the development of the subject and in one's algebraic geometry education, perhaps the most natural next direction to get a sense for  algebraic surfaces (see the book by Beauville), where you'll see an entirely different set of phenomena. A careful study of abelian varieties is another natural choice.  Toric geometry grows up into various different beasts. They make a triumphant return as part of the theory degenerations of abelian varieties. The formalism of logarithmic structures and toroidal embeddings is probably the most direct generalization of what we've developed -- the dense torus disappears in this generalization but many features remain. You're also not far from geometric invariant theory or the minimal model program. 

Examples classes: There will be four examples classes. These will be on the following Tuesdays from 330-430 PM in MR15: November 5, November 19, November 26 and (during Lent term) January 21. The associated example sheets appear below.

• Example Sheet I

• Example Sheet II

• Example Sheet III

• Example Sheet IV

Little Useful Trinkets: 

We have, from time to time, come across and used little facts that are useful in "life", and little examples that will help guide you as you venture deeper into the surrounding mathematics. 

• Affine toric varieties have a good notion of "monomial function". A point on an affine toric variety is equivalent to the data of how to evaluate all of its polynomial functions, or equivalently, all of its monomial functions. This tells us that points on toric varieties valued in a field K are semigroup homomorphisms from the set of monomials to K.

• When we proved separatedness of toric varieties, we did so by finding equations (i.e. establishing closedness) for the diagonal on every toric affine patch of the self product X x X. This used a general fact, that being a closed immersion is local on the target (but not local on the source!). A little more thinking shows that a morphism from X to Y being separated is also local on the target. It also used the general fact that a product of two toric varieties has fan equal to the product of the two fans. 

• A classical construction - consider the polytope in one dimension given by the interval from 0 to d for some integer d. Our construction from October 26 of projective toric varieties from polytopes gives P1 embedded inside Pd by the map sending a complex number to the tuple of all powers of that complex numbers from 0 to d. This is called the rational normal curve. When d is 2 it a conic. When d is 3 it is a cubic curve in space called the twisted cubic.

• Two more classical algebraic constructions - embeddings of projective spaces into larger projective spaces (Veronese embeddings) and embedding products of projective spaces (Segre embeddings). Both of these are toric varieties embedded in an ambient projective space by monomials, and consequently are realized by our polytope construction. In the first case, take the polytope to be appropriate dilations of the lattice simplex. In standard the second case, take products of lattices simplices in different lattices. We did this for the trivial (first) Veronese of P2 and for the Segre embedding of P1xP1. Hit up Wikipedia for a few minutes.

• Another reason that some people might like the polytope construction -- polytopes don't just give you toric varieties, not even projective toric varieties, but projective toric varieties with a chosen embedding in projective space. Projective space has a symplectic form on it (the symplectic part of the Fubini-Study Kähler metric), and if the toric variety is smooth, this gives a symplectic manifold structure to the toric variety. So you now have lots of examples of symplectic manifolds. This is a big subject.

• When we considered properness of toric varieties, we used the valuative criterion and saw a general phenomenon. If the scheme in question is irreducible, then it suffices to check the valuative criterion in the case where the the spectrum of the valued field maps to any pre-chosen non-empty open. 

• In many ways, simplicial toric varieties work almost as well as smooth toric varieties. One example is that their Chow groups inherit a ring structure with rational coefficients just as smooth toric varieties have ring structrues on their Chow groups with integer coefficients. Broadly, varieties with orbifold singularities, which including simplicial toric varieties, form the world of smooth Deligne-Mumford stacks, which have become a part of the algebraic geometry canon. Weighted projective spaces are the most fundamental examples. 

• Toric varieties are "stratified", in the sense that they are decomposed into finitely many locally closed subsets, where the closure of each such subset is a union of the subsets. In our case, this stratification is formed by the torus orbits of the dense torus action. Stratified spaces more generally are gadgets that come up constantly -- keep the toric examples in mind to picture these.

• The class group of a toric variety is equal to the Picard group in the smooth case. They are both finitely generated abelian groups, and the difference between them is one measure of the singularities of a variety. We've seen examples of varieties with torsion in the class group as well as in the Picard group (though the latter under some strange circumstances -- it doesn't often happen). More generally, I hope you internalize how if one tries to specify a slope on every ray of a fan, non-smooth cones impose global consistencies that detect the difference between these groups. That's an instance of what a singularity "does" to a variety.

• More on the above -- we've seen that there is a resolution of singularities for toric varieties. This means that we can think of singular varieties as smooth varieties that have been "compressed down", thereby squishing some of the information into the singularities. The discrepancy between the class and Picard groups is one instance of the information hiding inside the singularities. 

• On a variety that is stratified, it is natural to ask to what extent the Picard group, Class group, or Chow groups are generated by the strata. This is true for toric varieties, and some other natural varieties, but crucially, not always. The ability of boundary strata to detect properties on the variety, such as positivity of line bundles, has generated a remarkable amount of attention: the F-conjecture is a famous example.

• We've also seen that the positivity properties of line bundles (global generation, ampleness) are reflected in the convexity properties of the function. A very natural generalization to consider is to replace line bundles with vector bundles. Unlike line bundles, not all vector bundles are given by combinatorial data. The ones that are form a very interesting object of study though -- google "Toric vector bundles" to get started.

• We constructed a variety that is proper but does not admit any ample line bundles, so it is not projective. Things can get crazier -- there are toric varieties that admit not non-constant maps to projective space. See this. However, blowups are a way to fix this -- the Chow lemma states that after a sequence of blowups, a proper variety can be made projective. Reducing statements to the projective case by blowing up is a crucial technique in algebraic geometry.

• Just as singularities can be "fixed" by a sequence of blowups, bad properties of a morphism (non-flatness, non-reducedness) can be "fixed" by two procedures: a projective birational morphism and a ramified base change. This process is called semistable reduction and it is the "relative" or "families" version of resolution of singularities. A good place to start is Dan Abramovich's ICM talk notes and video.

• Intersection theory and Chow groups are a beautiful and technical subject. The canonical reference is Fulton's book Intersection Theory. In the toric situation, things are much less mysterious. We talked about this very briefly, but the treatments in both the references for the course contain lots of details.