Related papers: Units, polyhedra, and a conjecture of Satake

We introduce the notion of a categorical cone, which provides a categorification of the classical cone over a projective variety, and use our work on categorical joins to describe its behavior under homological projective duality. In…

For an algebraic function field $F/K$ and a discrete valuation $v$ of $K$ with perfect residue field $k$, we bound the number of discrete valuations on $F$ extending $v$ whose residue fields are algebraic function fields of genus zero over…

Let $\mathcal{F}$ be a plane singular curve defined over a finite field $\mathbb{F}_q$. The linear system of plane curves of a given degree passing through the singularities of $\cF$ provides potentially good bounds for the number of points…

We prove, assuming resolution of singularities in positive characteristic, an analogue of Siegel's theorem on sum of squares in positive characteristic. The method of proof combines techniques from central simple algebras with model theory…

Rational quartic spectrahedra in 3-space are semialgebraic convex subsets in $\mathbb{R}^3$ of semidefinite, real symmetric $(4 \times 4)$-matrices, whose boundary admits a rational parameterization. The Zariski closure in…

For $X$ a smooth cubic threefold we study the Pl\"ucker embedding of the Fano surface of lines $S$ of $X$. We prove that if $X$ is general then the minimal gonality of a covering family of curves of $S$ is four and that this happens for a…

Let $X$ be a curve of genus $g\geq 2$ over a number field $F$ of degree $d = [F:Q]$. The conjectural existence of a uniform bound $N(g,d)$ on the number $\#X(F)$ of $F$-rational points of $X$ is an outstanding open problem in arithmetic…

In the large rank limit, for any nonexceptional affine algebra, the graded branching multiplicities known as one-dimensional sums, are conjectured to have a simple relationship with those of type A, which are known as generalized Kostka…

We study the counting function of topological Poincar\'e series associated with rational homology sphere plumbed 3-manifold with connected negative definite tree, interpreting as an alternating sum of coefficient functions associated with…

The aim of this paper is to show how a conjectural lower bound on the canonical height function in the spirit of Lang and Silverman leads to an explicit uniform bound on the number of rational points on curves of genus $g\geq 2$ over a…

We give a systematic method of providing numerical evidence for higher order Stark-type conjectures such as (in chronological order) Stark's conjecture over $\mathbb{Q}$, Rubin's conjecture, Popescu's conjecture, and a conjecture due to…

For each positive integer k, we investigate the L-function attached to the k-th symmetric power of the F-crystal associated to the family of cubic exponential sums of x^3 + \lambda x. We explore its rationality, field of definition, degree,…

This is meant to be a survey article for the Cubo Journal. We discuss the existence and number of rational points over a finite field, the Hodge type over the complex numbers, and the motivic conjectures which are controlling those…

We study spherical tetrahedra with rational dihedral angles and rational volumes. Such tetrahedra occur in the Rational Simplex Conjecture by Cheeger and Simons, and we supply vast families, discovered by computational efforts, of positive…

We study the distribution of partial sums of Rademacher random multiplicative functions $(f(n))_n$ evaluated at polynomial arguments. We show that for a polynomial $P\in \mathbb Z[x]$ that is a product of at least two distinct linear…

The general purpose of this paper is to investigate the notion of "pluriharmonics" for the general potential theory associated to a convex cone $F\subset {\rm Sym}^2({\bf R}^n)$. For such $F$ there exists a maximal linear subspace $E\subset…

We prove that every polytope described by algebraic coordinates is the face of a projectively unique polytope. This provides a universality property for projectively unique polytopes. Using a closely related result of Below, we construct a…

We associate to a real projective variety $X$ two convex cones which are fundamental in real algebraic geometry: the cone $P_X$ of quadratic forms nonnegative on $X$, and the cone $\Sigma_X$ of sums of squares of linear forms. The dual cone…

Let F be a polarized irreducible holomorphic symplectic fourfold, deformation equivalent to the Hilbert scheme parametrizing length-two zero-dimensional subschemes of a K3 surface. The homology group H^2(F,Z) is equipped with an integral…

We revisit results of Fujino--Sato on complete non-projective $\mathbb Q$-factorial toric varieties and their conjectural factorization by flips. We show that their main results admit short conceptual proofs, avoiding any restriction on the…