# Part III Toric Geometry (Lent 2022)

Complete handwritten notes

Course outline

Example sheets: 1, 2, 3, 4

**Lecturer**: Dr. Navid Nabijou**Schedule**: MWF 12:00-13:00, MR13**Examples Classes**: Monday 7th February, 21st February, 7th March, 14th March, 13:30-14:30, MR9. Hand in solutions to up to three questions (of your own choosing) to my pigeonhole in Central Core by 14:00 the preceding Friday.

**Overview**: The main goal of this course is for you to develop intuition and fluency in handling algebraic varieties. We will achieve this by focusing on a special roster of examples: toric varieties. These are a class of highly symmetric varieties, with tightly constrained geometry. Despite their special nature, toric varieties exhibit a wide range of phenomena, and are the perfect "starter pack" for building up your own mental stable of examples. The central theme of the course will be that calculations which are difficult-to-impossible for arbitrary varieties reduce, in the case of toric varieties, down to combinatorics and linear algebra.

**References**: Fulton's book (*Introduction to Toric Varieties*) is a wonderful guide which quickly takes you to the heart of the action (please don't be put off by the slightly old-fashioned typography). Cox-Little-Schenck (*Toric Varieties*) is vast in its scope - if at any point you have a gap in your understanding, you will probably find the answer somewhere in there. Both books contain ample exercises, and you should engage with these.

**Exam notes**: The style of these lectures is closer to that of a graduate course. Gaps are left for you to fill, there is no "set text" and you are encouraged to engage with additional references. However, for the purposes of the exam it is helpful and reassuring to have a more narrow and "official" path through the course. Here it is:

The handwritten notes can be considered a complete reference. No exam question will require you to know definitions or theorems not contained in those notes.

If the proof of a result is left as an exercise, it should be considered examinable. On the other hand, if a proof is given solely as a reference (usually to Fulton or Cox-Little-Schenck), or omitted entirely, then it is not examinable.

If you can solve all the example sheets to a high standard, then you should be in a good position to do well on the exam. Starred questions are either (a) more difficult, or (b) more computationally lengthy than a typical exam question.

A general note. Your mathematical arguments should be written in full sentences. Try to emulate the style of proofs in textbooks. Do not leave it to the examiner to try to piece together the logic behind your thought process.

**Lecture schedule**:

21/01: Introduction. Algebraic tori, toric varieties [Fulton 1.1] and 1-parameter subgroups [Fulton 2.3].

24/01: Construction of affine toric varieties from cones [Fulton 1.1, 1.3].

26/01: Binomial equations and semigroups [Fulton 1.3].

28/01: Affine semigroups giving rise to toric varieties [Cox-Little-Schenck 1.1].

31/01: Saturation and normality [Fulton 1.3]. Smoothness [Fulton 2.1].

02/02: Smoothness continued [Fulton 2.1]. Construction of toric varieties from fans via gluing [Fulton 1.1, 1.4].

04/02: Gluing continued. Worked examples [Fulton 1.1].

07/02: Orbit-cone correspondence [Fulton 3.1, Cox-Little-Schenck 3.2].

09/02: Closed toric strata and the star fan [Fulton 3.1].

11/02: Properness [Fulton 2.4].

14/02: Toric morphisms [Cox-Little-Schenck 3.3].

16/02: Examples of toric morphisms: projections, blowups, ramified covers.

18/02: Surface singularities and cyclic quotients [Fulton 2.2].

21/02: Resolution of singularities [Fulton 2.6]. Simplicial toric varieties as quotients [Fulton 2.2].

23/02: Weil divisors [Cox-Little-Schenck 4.1, Fulton 3.3-3.4].

25/02: Cartier divisors [Cox-Little-Schenck 4.2, Fulton 3.3-3.4].

28/02: Cartier divisors continued: piecewise-linear functions.

02/03: More Cartier divisors. Self-intersection numbers [Fulton 5.1]. Pullbacks.

04/03: Global sections of line bundles via lattice points [Cox-Little-Schenck 4.3].

07/03: Basepoint-free and ample line bundles [Cox-Little-Schenck 6.1].

09/03: Toric varieties as global quotients [Cox-Little-Schenck 5.1].

11/03: Consequences of the quotient description: homogeneous co-ordinates and functors of points.

14/03: Topology: fundamental group and singular cohomology [Cox-Little-Schenck 12.1-12.4, Fulton 3.2, 4.1, 5.1, 5.2].

16/03: Our destiny is in the stars.