# Canadian Western Algebraic Geometry Symposium

## March 18-19, 2023 @ Simon Fraser University

# Location and Directions

All talks will take place on the Burnaby Mountain campus of Simon Fraser University.

We respectfully acknowledge that the Burnaby campus of SFU is located in the unceded traditional territories of the Coast Salish peoples, including the səl̓ilw̓ətaʔɬ (Tsleil-Waututh), kʷikʷəƛ̓əm (Kwikwetlem), Sḵwx̱wú7mesh Úxwumixw (Squamish) and xʷməθkʷəy̓əm (Musqueam) Nations.

SFU Burnaby Campus is accessible via public transit (R5 rapid bus from downtown, or skytrain and 145 bus). The nearest major airport is YVR. Public transit from YVR to SFU Burnaby takes roughly 1:20 (Canada line to Waterfront station followed by R5 to SFU). Directions are available via google.

Lectures will take place in AQ 3003. This is one level below "ground level" in the academic quadrangle. Coffee/Poster Session/Conference Dinner will take place one floor up at ground level in AQ 4110 (Poster Session) / AQ 4135 (Conference Dinner)/ AQ 4145 (Coffee).

# Speakers and Abstracts

What is the minimal degree \alpha_X(m) of a homogeneous polynomial in the polynomial ring C[x_0,..., x_N] that vanishes at a finite set of points X={P_1,..., P_s} in P^N_C at least m=(m_1, m_2, ..., m_s) times respectively? This problem is known as the interpolation problem. Though it is daunting to find the exact value, it is feasible to study lower bounds for \alpha_X(m). Chudnovsky gave a conjectural lower bound for the least degree when all the multiplicities are equal, that is, m_1=m_2=...=m_s. Chudnovsky's conjecture has an equivalent statement involving a lower bound of the Waldschmidt constant of the ideal defining points. Harbourne and Huneke proposed a containment conjecture involving the symbolic and the ordinary powers of the ideals, which implies Chudnovsky's conjecture.

We studied the stable version of the containment conjecture, the Cremona transformation, and consequently, we proved the Chudnovsky conjecture for general points over two different projects. In this talk, I will introduce Chudnovsky's conjecture, some other pioneer conjectural bounds, the containment conjectures, and the Cremona transformation.

I will describe the relationship between Chudnovsky's conjecture and the containment conjecture. I will also present some of the tools we used and the results from my joint work with Eloísa Grifo, Tài Huy Hà, and Thài Thnh Nguyen.

Given an abelian surface, one can ask how the Galois action on the n-torsion compares to the action on the n-torsion on the dual abelian variety. It's well-known that they are not the same in general. In work in progress with S. Frei and J. Voight, we produce some examples where they are different. This work has applications to understanding the cohomology of certain varieties of Kummer type.

I will report on joint work with Shiyu Shen. Moduli spaces of SL(n) and PGL(n)-Higgs bundles are conjecturally related by a derived equivalence and. This talk will be devoted to constructing a shadow of this equivalence in complex K-theory.

Abstract: There is a natural map from the symmetric polynomial ring in r_1 variables to the quantum cohomology ring of a type A flag variety Fl(n,r_1,..r_k), given by evaluating Schur polynomials in the Chern roots of the first tautological bundle. I’ll explain how for a large class of Schur polynomials, the result is a Schubert class that can be obtained by dividing the partition into a quantum-hook and smaller partitions. Surprisingly, this is the key result proving a mirror theorem for type A flag varieties. A function W is a mirror of a Fano variety X if enumerative information of X can be determined from W: for example, the Jacobi ring of W should be the quantum cohomology ring of X. Mirrors for Fano toric varieties are well-understood; and more recently Plücker coordinate mirrors have been proposed for a variety of homogeneous spaces. We use quantum hooks to prove that the Plücker coordinate mirror of the flag variety computes quantum cohomology relations. This is joint work with Linda Chen.

### Mohsen Kharkeiran (Alberta): Heterotic/IIA String Duality From Mirror Symmetry

It is well-known that Heterotic string theory compactified on a K3 \times T^2 is dual to Type IIA string theory compactified on a Calabi-Yau threefold which admits a K3-fibration. The fibration data is determined by the K3 and the vector bundles on it in Heterotic side. Even though this is already known for more than 25 years, the actual prescription for constructing the Heterotic dual of a given K3-fibered Calabi-Yau threefold is not known, except when the Calabi-Yau admits elliptic fibration in addition to the K3-fibration. We propose a new way to construct the Heterotic dual systematically based on the fact that the mirror Calabi-Yau threefold admits a Tyurin degeneration. In other words, we try to extract the Heterotic data from the Tyurin degeneration data.

Based on ongoing work with Aaron Bertram, we will explore properties of toric varieties constructed from root systems, their moduli space interpretations due to Losev-Manin and Batyrev-Blume, and decompositions of their bounded derived categories of coherent sheaves.

Geometric vertex decomposition (a degeneration technique) and Gorenstein liaison are two frameworks that have been used to produce similar results about similar families of algebraic varieties. In this talk, I will describe an explicit connection between these two approaches. In particular, I will show that each geometrically vertex decomposable ideal is linked by a sequence of ascending elementary G-biliaisons of height 1 to an ideal of indeterminates and, conversely, that each elementary G-biliaison of a certain type gives rise to a geometric vertex decomposition. As an application, I will show that several well-known families of ideals are glicci. I will provide some applications to the study of various classes of toric ideals.

This talk will include results from joint works with Mike Cummings, Sergio Da Silva, Patricia Klein, Thai Nguyen, and Adam Van Tuyl.

Essential dimension is an interesting numerical invariant of a finite group. In this talk I will survey the properties of this invariant and explain how advances in the Minimal Model Program lead to new insights into its asymptotic behavior.

# Schedule

Saturday, March 18

9:00- 9:05 Welcome, land acknowledgement, and announcements

9:05 - 9:55 Rajchgot

Coffee break

10:20 - 11:10 Bisui

11:20 - 12:10 Kalashnikov

Lunch Break

1:45 - 2:35 Groechenig

2:50 - 3:40 Reichstein

Coffee break

4:05 - 4:55 EDI Cafe

5:10 - 6:10 Poster Session

6:30 - 8:00 Conference Dinner

Sunday, March 19

9:15 - 10:05 Lamarche

Coffee break

10:30- 11:20 Kharkeiran

11:30 - 12:20 Honigs

Schedule is subject to change.

# Poster Session

There will be a poster session Saturday afternoon. All students and postdocs are encourage to bring posters.

# Accommodations

If you are a student or postdoc and hoping for us to pay for your lodging, please register and wait for us to contact you (sometime in early February). We will be coordinating lodging for you.

If you do not fall in the above category, please book your lodging yourself. Here are some options:

The Simon, located on campus. This is our top recommendation. There are limited rooms, so book now!

The Atrium Inn, a 22 minute bus ride (on the R5) from campus.

Hotel at the Waldorf, a 30 minute bus ride (on the R5) from campus.

Numerous hotels in Downtown Vancouver. Depending on location and time of day, these are 45 minutes to an hour from campus.

# Registration and Support

Although there is no registration fee, we request that all participants register here.

The deadline for applying for funding has passed.

This event is being organized in accordance with the PIMS code of conduct. We encourage all participants to familiarize themselves with this code.

# Reimbursements

If you are getting reimbursed for your travel or hotel costs, please fill out the form linked to below. Send a signed copy of it with scans of all relevant receipts and boarding passes to Nathan (nilten@sfu.ca) by no later than April 1st.

On the form, for Research Grant Affiliation please write "Conference Speaker/Participant". On the second page of the form in the Research Funding box, please write Canadian Western Algebraic Geometry Symposium Speaker/Particiant.

Organized by Susan Cooper, Charles Doran, Nathan Ilten, and Steve Rayan. To contact the organizers, please email cawagsymposium@gmail.com.

Support for CWAGS is generously provided by the Pacific Institute for the Mathematical Sciences.