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.

Current Term

The schedule for the current semester is here.

Talks for Spring 2020

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

• Filippo Nuccio (Lyon/UBC)
  Thursday, January 23
  3:30-4:30pm in K9509
  
  Rational elliptic curves with supersingular reduction, and Iwasawa theory
  

  Iwasawa theory is a combination of (p-adic) analytic and algebraic methods affording good control on the arithmetic of Galois representations, especially if they "come from geometry". I will start with a brief account of classical Iwasawa theory for number fields, as pioneered by Iwasawa himself, and then move on to the Iwasawa theory of elliptic curves defined over Q. A crucial dichotomy in this approach comes from the reduction type of the elliptic curve modulo a certain prime number p. I will recall what "ordinary" and "supersingular" reductions are and how the Iwasawa theory of the curve is affected by the reduction type. Finally, I will report on a recent joint work with Sujatha (UBC) in the supersingular case and will present some numerical examples.

• Ahmad Mokhtar (SFU)
  Thursday, January 30
  3:30-4:30pm in K9509
  
  Toric Degenerations of Rational Quintic Space Curves
  

  In algebraic geometry, a common technique for studying projective varieties relies on finding a flat degeneration from the variety in question to a toric variety which can be described by combinatorial data. However, it is not clear in general how to find such a degeneration or even if one exists. In this talk, I investigate the existence of toric degeneration for the special case of quintic rational space curves. Using work of Buczyński, Ilten, and Ventura, I obtain explicit parameterizations for all quintic rational curves in $\mathbb{P}^3$ and characterize which of these admit such a degeneration.

• Valentina Di Proietto (Exeter/UBC)
  Thursday, February 27
  3:30-4:30pm in K9509
  
  A non-abelian algebraic criterion for good reduction of curves
  

  For a family of proper hyperbolic complex curves $f:X\to\Delta^*$ over a punctured disc $\Delta^*$ with semistable reduction at the center, Oda proved, with transcendental methods, that the outer monodromy action of $\pi_1(\Delta)^*\cong\mathbb{Z}$ on the classical unipotent fundamental group of the generic fiber of $f$ is trivial if and only if $f$ has good reduction at the center. In this talk I explain a joint work with B. Chiarellotto and A. Shiho in which we give a purely algebraic proof of Oda's result.

• Sharon Robins (SFU)
  Thursday, March 12
  3:30-4:30pm in K9509
  
  Algebraically Hyperbolic Surfaces
  

  Algebraic hyperbolicity serves as a bridge between differential geometry and algebraic geometry. Generally, it is difficult to show that a given projective variety is algebraically hyperbolic. However, it was established recently that a very general surface of degree at least five in projective space is algebraically hyperbolic. In this talk, we are interested in generalizing the study of surfaces in the projective space to the surfaces in toric threefolds. In exploring the combinatorial description of toric threefolds with Picard rank 2 by Kleinschmidt and then following the method finding algebraic hyperbolic surfaces in toric threefolds by Haase and Ilten, I will discuss my results on algebraically hyperbolic and non-hyperbolic surfaces in each of these varieties.

• Klaus Altmann (FU Berlin)
  Thursday, March 19
  3:30-4:30pm in K9509
  
  Cancelled
  

• Milena Wrobel (Oldenburg)
  Thursday, April 2
  3:30-4:30pm in K9509
  
  Cancelled
  

• Alia Hamieh (UBC)
  Thursday, April 9
  3:30-4:30pm on ZOOM (contact organizer for link)
  
  Mean Values of Long Dirichlet Polynomials with Higher Divisor Coefficients
  

  In this talk, I report on recent work with Nathan Ng. Assuming a conjectural formula for a certain family of additive divisor sums, we prove an asymptotic formula for mean values of long Dirichlet polynomials with higher order shifted divisor functions as coefficients. This establishes a conjecture of Coney-Keating under the assumption of an additive divisor conjecture. As a consequence, we prove a special case of a conjecture of Conrey-Gonek when the additive divisor conjecture is known.

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
  
  Orthogonal Tensor Decomposition
  

  Tensor decomposition has many applications. However, it is often a hard problem. In this talk we will discuss a family of tensors, called orthogonally decomposable, which retain some of the properties of matrices that general tensors don't. A symmetric tensor is orthogonally decomposable if it can be written as a linear combination of tensor powers of $n$ orthonormal vectors. As opposed to general tensors, such tensors can be decomposed efficiently. We study the spectral properties of symmetric orthogonally decomposable tensors and give a formula for all of their eigenvectors. We also give polynomial equations defining the set of all such tensors. Analogously, we study nonsymmetric orthogonally decomposable tensors, describing their singular vector tuples and giving polynomial equations that define them. To extend the definition to a larger set of tensors, we define tight-frame decomposable tensors and study their properties. Finally, we conclude with some open questions and future research directions.

• Adam Topaz (UAlberta)
  Thursday, October 10
  3:30-4:30pm in K9509
  
  Parameterization of divisors using cohomology
  

  In this talk, I will discuss two cohomological methods which parameterize all prime divisors on regular models of given geometric function field. The technical core of these methods is fairly elementary, using mainly ideas from classical projective geometry. Time permitting, I will discuss how these methods can be used to prove new results in the context of birational anabelian geometry.

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

  In algebraic geometry, Hilbert schemes are the prototypical parameter spaces: their points correspond to closed subschemes in a projective space with a fixed Hilbert polynomial. Without assuming any prior knowledge about Hilbert schemes, we will first survey some of their known features. We will then present new numerical conditions on the polynomial that completely characterize when the associated Hilbert scheme is smooth. In this smooth situation, our explicit description of the subschemes being parametrized also provides new insights into the global geometry of Hilbert schemes. This talk is based on joint work with Roy Skjelnes (KTH).

• Nils Bruin (SFU)
  (Postponed)
  
  
  What is Chabauty's method?
  

  With a whole Arizona Winter School being dedicated to Nonabelian Chabauty, some of you may wonder what this Chabauty business entails. I will try to satisfy your curiosity a little bit, and hopefully leave you wanting for more. I will discuss Chabauty's original partial proof of Mordell's Conjecture, where under certain conditions he links rational points to zeros of a (p-adic!) analytic function on a compact domain. I will also draw a parallel with a method of Skolem for solving certain diophantine equations, and discuss how Chabauty's method is still attractive and valuable from a computational point of view, even after Faltings's full proof of Mordell's Conjecture.

• Alex Barrios (Carleton College)
  Thursday, November 21
  3:30-4:30pm in K9509
  
  Lower Bounds on the Modified Szpiro Conjecture
  

  The modified Szpiro conjecture, which is equivalent to the $ABC$ Conjecture, states that for each $\epsilon>0$ there are finitely many rational elliptic curves $E$ satisfying $N_{E}^{6+\epsilon}<\max\left\{ \left\vert c_{4}^{3}\right\vert ,c_{6}^{2}\right\} $ where $N_{E}$ is the conductor of $E$ and $c_{4}$ and $c_{6}$ are the invariants associated to a minimal model of $E$. In this talk we will show that for a rational elliptic curve $E$ with torsion subgroup $E\!\left(\mathbb{Q}\right) _{\text{tors}}\cong T$, there is an explicit lower bound on the modified Szpiro ratio which depends only on $T$, i.e., $l_{T}<\frac{\log \max\left\{ \left\vert c_{4}^{3}\right\vert ,c_{6}^{2}\right\} }{\log N_{E}}$ for all $E/ \mathbb{Q}$ with $T\hookrightarrow E\!\left( \mathbb{Q}\right) $. The techniques of the proof rely on a careful analysis of the minimal models of E and Tate's algorithm.

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).