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By the work of J.Huh, one can interpret binomial coefficients as a solution to an intersection problem on a permutohedral variety $X_E$. Applying Hirzebruch-Riemann-Roch, this intersection problem is equivalent to computing Euler…
The classical Prym construction associates to a smooth, genus $g$ complex curve $X$ equipped with a nonzero cohomology class $\theta \in H^1(X,\mathbb{Z}/2\mathbb{Z})$, a principally polarized abelian variety (PPAV) $\mbox{Prym}(X,\theta)$.…
We find expressions of the polynomials defining the dual varieties of Grassmannians $Gr(3,9)$ and $Gr(4,8)$ both in terms of the fundamental invariants and in terms of a generic semi-simple element. We project the polynomial defining the…
In 2013, Abo and Wan studied the analogue of Waring's problem for systems of skew-symmetric forms and identified several defective systems. Of particular interest is when a certain secant variety of a Segre-Grassmann variety is expected to…
We briefly review previous work on the invariant theory of 3 x 3 x 3 arrays. We then recall how to generate arrays of arbitrary size m_1 x ... x m_k with hyperdeterminant 0. Our main result is an explicit formula for the 3 x 3 x 3…
This paper is based on the first author's lectures at the 2012 University of Regina Workshop "Connections Between Algebra and Geometry". Its aim is to provide an introduction to the theory of higher secant varieties and their applications.…
Techniques from representation theory, symbolic computational algebra, and numerical algebraic geometry are used to find the minimal generators of the ideal of the trifocal variety. An effective test for determining whether a given tensor…
We determine the minimal generators of the ideal of the tangential variety of a Segre-Veronese variety, as well as the decomposition into irreducible GL-representations of its homogeneous coordinate ring. In the special case of a Segre…
The hyperdeterminant of a polynomial (interpreted as a symmetric tensor) factors into several irreducible factors with multiplicities. Using geometric techniques these factors are identified along with their degrees and their…
A Waring decomposition of a (homogeneous) polynomial f is a minimal sum of powers of linear forms expressing f. Under certain conditions, such a decomposition is unique. We discuss some algorithms to compute the Waring decomposition, which…
By using a result from the numerical algebraic geometry package Bertini we show that (up to high numerical accuracy) a specific set of degree 6 and degree 9 polynomials cut out the secant variety $\sigma_{4}(\mathbb{P}^{2}\times \mathbb{P}…
We prove a set-theoretic version of the Landsberg--Weyman Conjecture on the defining equations of the tangential variety of a Segre product of projective spaces. We introduce and study the concept of exclusive rank. For the proof of this…
We prove a motivic enhancement of the classical Picard--Lefschetz formula. Our proof is completely motivic, and yields a description of the motivic nearby cycles at a quasi-homogeneous singularity, as well as its monodromy, in terms of an…
The aim of these notes is to present an accessible overview of some topics in classical algebraic geometry which have applications to aspects of discrete integrable systems. Precisely, we focus on surface theory on the algebraic geometry…
We define q-analogues of Mirzakhani's recursion for Weil-Petersson volumes and the Stanford-Witten recursion for super Weil-Petersson volumes. Okuyama recently introduced a q-deformation of the Gaussian Hermitian matrix model which produces…
Given varieties $X, Y, W$ and dominant morphisms $\phi:X\to Y$ and $f:X\to W$ such that $f$ is constant on fibres of $\phi$ , we give sufficient conditions to guarantee that $f$ descends to a rational map or a morphism $Y\to W.$ We pay…
Gale duality is an involution of point configurations in projective spaces. Goppa duality extends this concept to a duality between linear series on a Gorenstein curve passing through prescribed points. We generalize this classical result…
We give a more detailed description of the new system of Pl\"ucker-like equations from [4], discuss how it relates to the usual Pl\"ucker equations, and correct a mistake in that article.
The purpose of this note is to show that the subvarieties of small degree inside a general hypersurface of large degree come from intersecting with linear spaces or other varieties.
We study integral models of some Shimura varieties with bad reduction at a prime $p$, namely the Siegel modular variety and Shimura varieties associated with some unitary groups. We focus on the case where the level structure at $p$ is…