# Number Theory and Algebraic Geometry Seminar

The Number Theory and Algebraic Geometry (NTAG) seminar is a semi-regularly meeting research seminar at SFU dedicated to topics related to number theory and algebraic geometry. It is co-organized by Nils Bruin and Nathan Ilten. We usually meet Thursdays, 3:30pm-4:30pm in K9509.

## Talks for Fall 2019

We will meet most Thursdays (i.e. at least twice a month).

• Shamil Asgarli (UBC)
Thursday, September 12
3:30-4:30pm in K9509

Transversality for algebraic varieties over finite fields

Give a smooth plane curve $C$ defined over the complex numbers, almost all lines on the plane intersect $C$ transversely. What if $C$ is defined over a finite field $F_q$? Can we find at least one transverse line defined over $F_q$? The answer is more subtle because there are only finitely many $F_q$-lines available. In this talk, we will investigate such transversality questions with a focus on the case of plane curves. We will also mention an analogous result for surfaces. This is a joint work with Lian Duan and Kuan-Wen Lai.

• Elina Robeva (UBC)
Thursday, October 3
3:30-4:30pm in K9509

• Adam Topaz (UAlberta)
Thursday, October 10
3:30-4:30pm in K9509

• Greg Smith (Queens)
Thursday, October 24
3:30-4:30pm in K9509

• Nils Bruin (SFU)
Thursday, November 7
3:30-4:30pm in K9509

• Alex Barrios (Carleton College)
Thursday, November 21
3:30-4:30pm in K9509

## Talks for Summer 2019

No regularly scheduled talks.

## Talks for Spring 2019

• Nathan Ilten (SFU)
Friday, January 11
11:30am-12:30pm in K9509

Khovanskii-Finiteness for Rational Plane Curves

Given a projective variety $X$, one may consider full rank (homogeneous) valuations on its coordinate ring $R(X)$. By considering the image of such a valuation, the so-called value semigroup, one is able to recover important information concerning $X$, including its Hilbert function. When does a valuation exist whose value group is finitely generated? This is a fundamental question, which appears very difficult to answer in general. I will discuss joint work with Milena Wrobel in which we partially answer this question for rational plane curves.

• Jim Bryan (UBC)
Friday, January 25
11:30am-12:30pm in K9509

The geometry and arithmetic of the world's tiniest Calabi-Yau threefold

A Calabi-Yau n-fold is a smooth complex projective variety of dimension n admitting a non-vanishing holomorphic n-form. Equivalently, they are complex manifolds admitting a Ricci-flat Kahler metric. Although there is only one topological type of Calabi-Yau manifolds of dimension one or of dimension two, there are hundreds of millions of topological types of Calabi-Yau threefolds. In this talk, we construct a Calabi-Yau threefold which is topologically the smallest known (i.e. has the smallest sum of betti numbers). Our threefold has a nice modular interpretation in terms of elliptic curves with a specified 6-torsion point.

• Imin Chen (SFU)
Friday, February 1
11:30am-12:30pm in K9509

A multi-Frey approach to Fermat equations of signature (7,7,p)

I will report on joint work with Billerey, Dieulefait, and Freitas, where we use multiple Frey hyperelliptic curves to resolve the equation $x^7 + y^7 = 3 z^n$ for all integers $n \ge 2$. A new ingredient which is needed for this equation is the use of a genus 3 Frey hyperelliptic curve over a totally real cubic field. In this talk, I will survey the current approaches to problems of this type, and then explain the new advances.

• Matej Filip (Mainz)
Friday, February 8
11:30am-12:30pm in ASB 10908

Deformations of Gorenstein Toric Varieties

We will study the versal deformation for affine Gorenstein toric varieties in special lattice degrees and give some applications to deformations of Gorenstein Fano toric varieties.

• Milena Hering (Edinburgh)
Friday, March 8
11:30am-12:30pm in ASB 10908

Stability of Toric Tangent Bundles

In this talk I will give a brief introduction to slope stability and present a combinatorial criterion for the tangent bundle on a polarised toric variety to be stable in terms of the lattice polytope corresponding to the polarisation. I will then present a theorem which says that a toric surface admits a polarisation with respect to which the tangent bundle is stable if and only if it is a blow up of the projective plane. This is joint work with Hendrik Süß.

• Emanuele Ventura (Texas A&M)
Friday, March 15
11:30am-12:30pm in K9509

Symmetry groups of tensors

To analyze the complexity of the matrix multiplication tensor, Strassen introduced a class of tensors that vastly generalize it, the tight tensors. Tight tensors are essentially tensors with positive dimensional symmetry group. Besides the motivation from algebraic complexity, the study of symmetry groups of vectors in a representation of an algebraic group is a classical topic in algebraic geometry and invariant theory. It is then natural to investigate tensors with large symmetry groups, under a genericity assumption (1-generic). In this talk, we discuss some combinatorial consequences of tightness, and sketch the geometry behind the classification of 1-generic tensors with maximal symmetry groups. This is based on joint work with A. Conner, F. Gesmundo, JM Landsberg, and Y. Wang.

• Dan Lewis (SFU)
Friday, March 29
11:30am-12:30pm in K9509

Two-cover descent on genus 3 curves

I will report upon my MSc thesis work with Nils Bruin, wherein we have provided a MAGMA implementation of the above, based upon a 2016 paper of Bruin, Poonen and Stoll. In this talk, I will start by providing the necessary background to interpret the title, remark upon the p-adic analysis that is fundamental to our method, and close by presenting a few small results that have emerged in our investigations.

• Sasha Zotine (SFU)
Friday, April 5
11:30am-12:30pm in ASB 10908

Global Generation of Vector Bundles over Elliptic Curves

Algebraic vector bundles are a construction useful for studying the geometry of varieties; they are objects which associate a vector space to each point of the variety in a "polynomial" fashion. When a vector bundle is "globally generated" it can be viewed as the quotient of the trivial vector bundle, making it easy to work with. In 1957, Sir Michael Atiyah showed that every indecomposable bundle over a smooth elliptic curve was determined by a point on the curve, and two invariants; the rank and degree. We provide a sharp bound on the degree of the indecomposable bundles associated to the base point to exactly determine when they are globally generated. This work was done using explicit representations of the bundles via transition matrices, which we obtained using Atiyah's results.

## Talks for Summer 2018

No regularly scheduled talks.

## Talks for Spring 2018

• Charles Turo (SFU)
Friday, April 6
11am-12pm in K9509

Vanishing cup product for smooth complete toric varieties

The geometry of a toric variety $X$ is determined completely by the combinatorics of the associated fan. In fact, when $X$ is smooth, there is a generalization of the Euler sequence for projective space which gives a particularly nice description of the tangent sheaf $T_X$ in terms of sheaves of boundary divisors. The first cohomology group of the tangent sheaf $H^1(X,T_X)$ describes first-order deformations of $X$. There is a "cup product" map $H^1(X,T_X)\times H^1(X,T_X) \to H^2(X,T_X)$ which encodes the possibility of combining two first-order deformations. In this talk, I use the Euler sequence to (partially) show the vanishing of the cup product map for smooth complete toric varieties.

• Claudiu Raicu (Notre Dame)
Friday, March 9
11am-12pm in K9509

Koszul modules

The Cayley-Chow form of a projective variety X is an equation that detects when a given linear space intersects X non-trivially. I will explain how it can be computed in the case when X is the Grassmannian of lines in its Plücker embedding, by relating it to a fascinating class of modules called Koszul modules. Despite the elementary definition of Koszul modules, their study has close ties to that of syzygies of generic canonical curves, but also important implications to the structure of Alexander invariants of finitely presented groups. Joint work with M. Aprodu, G. Farkas, S. Papadima, and J. Weyman.

• Nathan Ilten (SFU)
Friday, February 23
11am-12pm in K9509

Divisors, Positivity, and Fujita's Freeness Conjecture

Given an abstract projective variety, how do you find a embedding of this variety in projective space? The standard approach is use a "divisor" to produce a collection of rational functions which induce a rational map to projective space. If the divisor has sufficiently nice properties, this map will be a morphism, or even an embedding. Fujita's freeness conjecture provides a conjectural criterion guaranteeing that this map is indeed a morphism.

In this talk, I will provide an overview of this approach to embedding varieties in projective space, and discuss the various positivity notions surrounding divisors. I will then report on joint work with Klaus Altmann, in which we prove the freeness conjecture for complexity-one T-varieties.

• Yoav Len (Waterloo)
Friday, February 9
11am-12pm in K9509

Tangent Lines and the Equation 28=7x4

I will discuss algebraic and combinatorial aspects of tangent lines to curves. As shown by Plücker in the 19th century, every smooth plane quartic curve has 28 lines that are tangent at two points. This quantity hides rich combinatorics, and is related to other phenomena such as lines on a cubic surface and double covers of curves. I will begin with a brief introduction to tropical geometry, and show that similar phenomena occur for tropical curves. I will then explain how tropical techniques allow us to recover Plücker's count by degenerating algebraic curves.

• Imin Chen (SFU)
Friday, January 26
11am-12pm in K9509

A multi-Frey approach to a Fermat equation of signature (13,13,7)

I will report on joint work with Billerey, Dembele, Dieulefait, and Freitas, where we use a refined application of the multi-Frey method to resolve the generalized Fermat equation $x^{13} + y^{13} = 3 z^n$ for all integers $n \ge 2$ using modular Frey elliptic curves over totally real fields. The case $n = 7$ caused some difficulty because of an obstructing Hilbert newform. In this talk, I will give some background to the problem and explain how this particular case was resolved.

• Kalle Karu (UBC)
Friday, January 12
11am-12pm in K9509

Cox Rings and Mori Dream Spaces

The Cox ring of a variety generalizes the homogeneous coordinate ring of a projective space. The main problem in the theory of Cox rings is to determine if they are finitely generated, in which case the variety is called a Mori dream space. In this talk I will concentrate on blowups of toric varieties at a point, and more precisely, blowups of weighted projective planes. The Cox rings of such blowups have been studied extensively in commutative algebra. However, the toric varieties point of view allows us to simplify many of the algebra techniques and prove new results.

## Talks for Fall 2017

• Avi Kulkarni (SFU)
Monday December 4
1:30pm-2pm in MathWest 2830

An arithmetic invariant theory of curves from E8

We identify a family of algebraic curves over Q such that the arithmetic of these curves is related to the orbits of an algebraic group of type E8 acting on a specific algebraic variety. Our results are analogous to Jack Thorne's results relating the arithmetic of plane curves of degree 4 to the orbits of an algebraic group of type E7. Our results are also analogous to those of Bhargava and Gross, who use it determine average ranks of hyperelliptic Jacobians.

• Mattia Talpo (SFU)
Friday December 1
11am-12pm in K9509

Logarithmic algebraic geometry, algebraic stacks, and their interactions

Algebraic stacks and logarithmic geometry are nowadays important tools in algebraic geometry, the branch of mathematics that studies the geometry of solution sets of polynomial equations. Both theories are enhancements of ordinary algebraic geometry, where the varieties are equipped with extra structure: stacks have stabilizer groups that keep track of "intrinsic symmetry" and log schemes have a specified subset of "monomial" functions. I will start the talk with an overview of these concepts, and then focus on my work concerning interactions between the two areas.

• Frank Sottile (Texas A&M)
Friday November 24
11am-12pm in K9509

Some algebraic geometry in applications

Algebraic geometry is the study of sets which arise as the common zeroes to a collection of polynomials. It is a deep and powerful subject, combining geometric intuition with algebraic precision. It is also increasingly a useful tool in applications of mathematics, for whenever polynomials arise, the methods of algebraic geometry may be brought to bear on the problem at hand. I will illustrate this growing trend through a series of interrelated examples of algebraic geometry arising in applications.

• Klaus Altmann (FU Berlin)
Friday October 27
11am-12pm in AQ4100

Immaculate line bundles on smooth toric varieties

We call a line bundle L on an algebraic variety immaculate if it has no cohomology at all. In contrast to the notion of acyclic bundles, we do not even allow global sections. If L is written as a difference of two globally generated bundles, i.e. if it is expressed as the formal difference of two polytopes D+ and D-, then the immaculacy of L corresponds to the contractibility of the difference D+ \ (D- +r) for all integral shift vectors r. Immaculate bundles come into focus as differences between members of exceptional sequences. However, the structure of the immaculate locus becomes interesting in itself. It is a finite union of affine planes of different dimensions within the Picard group. We know the structure in detail for Picard rank 2 (or more general for splitting fans) and for Picard rank 3. This is work in progress together with Jarek Buczynsky (Warszaw) and Lars Kastner and Anna-Lena Winz (Berlin).