Date | Speaker | Title | Abstract |
January 10, 2019 Start: 11:10 AM Location: Surge 277 |
| Planning meeting. | Planning meeting. |
Thursday, January 17, 2019 Start: 11:10 AM Location: Surge 277 |
Ethan Kowalenko UC Riverside | Introduction to toric varieties, lattices, and cones. Part 1. | Toric varieties give a wonderful entry point to algebraic geometry through their associated combinatorics, allowing one to compute many examples. Following a lecture series of David Cox, we will learn the basic theory of toric varieties. This first lecture will define toric varieties with some examples. Two important lattices will be introduced, the lattice of characters and the lattice of one-parameter subgroups. We'll show through examples how these lattices come naturally into play for toric varieties. |
Thursday, January 24, 2019 Start: 11:10 AM Location: Surge 277 |
Ethan Kowalenko UC Riverside | Introduction to toric varieties, lattices, and cones. Part 2. | Toric varieties give a wonderful entry point to algebraic geometry through their associated combinatorics, allowing one to compute many examples. Following a lecture series of David Cox, we will learn the basic theory of toric varieties. This first lecture will define toric varieties with some examples. Two important lattices will be introduced, the lattice of characters and the lattice of one-parameter subgroups. We'll show through examples how these lattices come naturally into play for toric varieties. |
Thursday, January 31, 2019 Start: 11:10 AM Location: Surge 277 |
Humberto Diaz UC Riverside | The toric variety of a fan. Part 1. | Continuing our study of toric varieties, we define a fan and show how this gives the information to glue together affine toric varieties to form a new toric variety. We will also give some examples of well-known projective varieties obtained in this way. If time permits, we will state the orbit-cone correspondence for toric varieties. |
Thursday, February 7, 2019 Start: 11:10 AM Location: Surge 277 |
Humberto Diaz UC Riverside | The toric variety of a fan. Part 2. | Continuing our study of toric varieties, we give a characterization of fundamental algebro-geometric notions for toric varieties in terms of cone/fan data. We will also describe the orbit-cone correspondence in detail. This gives a bijection between the cones in a given fan and the torus orbits of the corresponding toric variety. Finally, we discuss quotient singularities on toric varieties, how they arise and how to resolve them. |
Thursday, February 14, 2019 Start: 11:10 AM Location: Surge 277 |
Zhixian Zhu UC Riverside | The homogeneous coordinate ring. | We will study the divisor class group for toric varieties in terms of fan data. Then we construct the homogeneous coordinate ring graded by the divisor class group and use the quotient construction to recover the toric variety. |
Thursday, February 21, 2019 Start: 11:10 AM Location: Surge 277 |
Christopher Lyons California State University, Fullerton | A simple formula for the Picard number of K3 surfaces of BHK type. | The Berglund-Hubsch-Krawitz (BHK) mirror symmetry rule takes as its input a special type of Calabi-Yau variety X with accompanying group of automorphisms G, and it returns another such pair of "mirror" objects X^T and G^T. It's an intriguing construction that has gained a lot of attention in the past decade, and its description only requires some basic algebra. When X has dimension 2, the quotient X/G yields an algebraic surface Y called a K3 surface of BHK type. Building upon work of T. Shioda, a result of T. Kelly shows that the Picard number of Y can be computed via a careful investigation of the elements in the mirror group G^T. We discuss how to refine this to yield a simple formula for this Picard number in terms of only the degree of X^T. This is joint work with Bora Olcken. |