# Canadian Western Algebraic Geometry Symposium

## October 21-22, 2017 @ UAlberta

# Location and Directions

All talks will be held in CAB 657. Lodging is at the Lister Center. Both are marked on this map.

# Speakers and Abstracts

- Christine Berkesch Zamaere (Minnesota)

*Parametric variation of hypergeometric systems*

We describe the parametric behavior of the series solutions of an A-hypergeometric system. More precisely, we construct explicit stratifications of the parameter space such that, on each stratum, the series solutions of the system are holomorphic. In certain special cases, this leads to an explicit connection between hypergeometric functions at rank jumping parameters and elements in local cohomology modules of a semigroup ring.

- Xi Chen (Alberta)

*Voisin's conjecture on rationally equivalent points of sextic surfaces*

C. Voisin proved that no two distinct points on a very general hypersurface $X$ of degree at least $2n+1$ in ${\mathbb P}^n$ are rationally equivalent. She conjectured that the same holds for $\deg X = 2n$. We give a proof of this conjecture. This is a joint work with James D. Lewis and Mao Sheng.

- María Angélica Cueto (Ohio State)

*Anticanonical tropical del Pezzo surfaces contain exactly 27 lines*

Since the beginning of tropical geometry, a persistent challenge has been to emulate tropical versions of classical results in algebraic geometry. The well-know statement ``any smooth surface of degree 3 in P^3 contains exactly 27 lines'' is known to be false tropically. Work of Vigeland from 2007 provides examples of cubic surfaces with infinitely many lines and gives a classification of tropical lines on general smooth tropical surfaces in TP^3.

In this talk I will explain how to correct this pathology. The novel idea is to consider the embedding of a smooth cubic surface in P^44 via its anticanonical bundle. The tropicalization induced by this embedding contains exactly 27 lines under a mild genericity assumption. More precisely, smooth cubic surfaces in P^3 are del Pezzos, and can be obtained by blowing up P^2 at six points in general position. We identify these points with six parameters over a field with nontrivial valuation. Our genericity assumption involves the valuations of 36 linear expressions in these parameters which give the positive roots of type E_6. Tropical convexity plays a central role in ruling out the existence of extra tropical lines on the tropical cubic surface.

- Charles Doran (Alberta/ICERM)

*On the classification of K3 fibered Calabi-Yau threefolds*

Kodaira’s celebrated classification of elliptic surfaces with section is the “toy model” for a theory of K3 surface fibered threefolds. We will see how each key feature in Kodaira’s story admits an analogue, and use a mix of Hodge theory and geometry to completely classify Calabi-Yau threefolds fibered by certain high-Picard rank K3 surfaces. Time permitting, we will use both classifications to illustrate some key features of a new mirror symmetry conjecture.

- Steven Rayan (Saskatchewan)

*Asymptotic geometry of hyperpolygons*

Nakajima quiver varieties lie at the interface of geometry, representation theory, and combinatorics. I will discuss a particular example, hyperpolygon space, which arises from star-shaped quivers. The simplest of these varieties is a noncompact complex surface admitting the structure of an "instanton", and therefore fits nicely into the Kronheimer-Nakajima classification of ALE hyperkaehler 4-manifolds, which is a geometric realization of the McKay correspondence for finite subgroups of SU(2). For more general hyperpolygon spaces, we speculate on how this classification might be extended by studying the asymptotic geometry of the variety. In moduli-theoretic terms, this involves driving the stability parameter for the quotient to an irregular value. This is joint work in progress with Harmut Weiss, building on previous work with Jonathan Fisher.

- Yongbin Ruan (Michigan)

*A genus two mirror theorem for Gromov-Witten theory of quintic 3-fold*

The computation of Gromov-Witten theory of compact Caleb-Yau 3-fold (symbolized by the quintic 3-fold) is a central problem in geometry and physics where the mirror symmetry plays a key role. Using the mirror symmetry, Candelas and his collaborators proposed a surprising conjectural formula in genus zero in early 90's. The effort to prove the formula in mathematics has leaded the birth of mirror symmetry as a mathematical subject. The genus zero mirror symmetry was established by Givental and Liu-Lian-Yau in 96. As early as 93, Bershadsky, Cecotti,Ooguri and Vafa (BCOV) have already proposed a conjectural formula for genus one and two Gromov Witten invariants of the quintic 3-fold. It took ten years for Zinger to prove the genus one conjecture. Now another ten years have passed. In the talk, I will report a solution to genus two conjecture as well as an approach to go to much higher genus. This is a joint work with Shuai Guo and Felix Janda.

- Greg Smith (Queen's)

*Combinatorics of Toric Vector Bundles*

To each torus-equivariant vector bundle over a smooth complete toric variety, we associate a finite collection of line bundles indexed by vectors in a finite dimensional vector space. In this talk, we will describe how the combinatorics of this data encodes a resolution of the toric vector bundle. We will also indicate some applications of these resolutions.

- Mattia Talpo (Simon Fraser University)

*A logarithmic version of the derived McKay correspondence*

The derived McKay correspondence is an equivalence of derived categories between two different resolutions of certain singular varieties. The simplest example is given by the quotient of the complex 2-dimensional affine space by the action of a finite subgroup of SL(2,C), and the two resolutions in this case are given by the corresponding quotient stack, and by the minimal resolution obtained via blowups.

After recalling some background and history of the topic, I will talk about joint work with Sarah Scherotzke and Nicolò Sibilla, where we prove a "limit" version of this derived equivalence in the context of logarithmic geometry. The two sides of the equivalence in our case are given respectively by the "infinite root stack", corresponding to stacky resolutions, and by the "valuativization", corresponding to resolutions obtained via blowups.

# Tentative Schedule

**SATURDAY**

9:00 - 9:50 Charles Doran

10:15 - 11:05 Angelica Cueto

11:30 - 12:20 Mattia Talpo

LUNCH

2:30 - 3:20 Greg Smith

3:45 - 4:35 Christine Berkesch Zamaere

4:45 - 6:00 POSTERS

6:30 DINNER

**SUNDAY**

9:15 - 10:05 Yongbin Ruan

10:30- 11:20 Steven Rayan

11:30 - 12:20 Xi Chen

# Registration

If you plan to attend, please **register **as soon as possible. Registration is through our PIMS webpage.

# Lodging

Accommodation is available at the **Lister Center **for the nights of October 20th and 21st. If you would like us to book accommodation for you there, please email us at cawagsymposium@gmail.com before September 1st. We will try to cover the costs for students and postdocs; more senior participants will be charged $100/night for staying at Lister.

You are also welcome arranging accommodation on your own (in which case we will not reimburse you). Nonetheless, please don't forget to register for the conference at our PIMS webpage.

Organized by Charles Doran, David Favero, Nathan Ilten, and Jenna Rajchgot. To contact the organizers, please email cawagsymposium@gmail.com.

Support for CWAGS is generously provided by the Pacific Institute for the Mathematical Sciences and the UAlberta Faculty of Science.