Related papers: Algebraic intersection in regular polygons

We define an infinite series of translation coverings of Veech's double-n-gon for odd n greater or equal to 5 which share the same Veech group. Additionally we give an infinite series of translation coverings with constant Veech group of a…

A theta surface in affine 3-space is the zero set of a Riemann theta function in genus 3. This includes surfaces arising from special plane quartics that are singular or reducible. Lie and Poincar\'e showed that theta surfaces are precisely…

Let $d \geq 2$. The Cheeger constant of a graph is the minimum surface-to-volume ratio of all subsets of the vertex set with relative volume at most 1/2. There are several ways to define surface and volume here: the simplest method is to…

We explicitly compute the limiting gap distribution for slopes of saddle connections on the flat surface associated to the regular octagon with opposite sides identified. This is the first such computation where the Veech group of the…

In this article, we derive estimates of Teichm\"uller modular forms, and associated invariants. Let $\mathcal{M}_{g}$ denote the moduli space of compact hyperbolic Riemann surfaces of genus $g\geq 2$, and let $\overline{M}_{g}$ be the…

A convex lattice polygon Delta determines a pair (S,L) of a toric surface together with an ample toric line bundle on S. The Severi degree N^{Delta,delta} is the number of delta-nodal curves in the complete linear system |L| passing through…

For two convex discs $K$ and $L$, we say that $K$ is $L$-convex if it is equal to the intersection of all translates of $L$ that contain $K$. In $L$-convexity the set $L$ plays a similar role as closed half-spaces do in the classical notion…

Let $P$ be a polytope defined by the system $A x \leq b$, where $A \in R^{m \times n}$, $b \in R^m$, and $\text{rank}(A) = n$. We give a short geometric proof of the following tight upper bound on the number of vertices of $P$: $$ n! \cdot…

We give the values of the Siegel-Veech constants associated with saddle connections having distinct endpoints on translation surfaces in Prym eigenform loci in $\Omega \mathcal{M}_3(2,2)^{\rm odd}$. In particular, we show that these…

We state a general formula to compute the volume of the intersection of the regular $n$-simplex with some $k$-dimensional subspace. It is known that for central hyperplanes the one through the centroid containing $n-1$ vertices gives the…

We study the topology of the set X of the solutions of a system of two quadratic inequalities in the real projective space RP^n (e.g. X is the intersection of two real quadrics). We give explicit formulae for its Betti numbers and for those…

We study constructing an algebraic curve from a Riemann surface given via a translation surface, which is a collection of finitely many polygons in the plane with sides identified by translation. We use the theory of discrete Riemann…

We study the symmetries and geodesics of an infinite translation surface which arises as a limit of translation surfaces built from regular polygons, studied by Veech. We find the affine symmetry group of this infinite translation surface,…

Given an affine variety X and a finite dimensional vector space of regular functions L on X, we associate a convex body to (X, L) such that its volume is responsible for the number of solutions of a generic system of functions from L. This…

We consider hyperbolic links that admit alternating projections on surfaces in compact, irreducible 3-manifolds. We show that, under some mild hypotheses, the volume of the complement of such a link is bounded below in terms of a Kauffman…

We form a generating series of regularized volumes of intersections of special cycles on a non-compact unitary Shimura variety with a fixed base change cycle. We show that it is a Hilbert modular form by identifying it with a theta…

Let $(X,\Delta)$ be an $n$-dimensional $\epsilon$-klt log $\QQ$-Fano pair. We give an upper bound for the volume ${\rm Vol}(-(K_X+\Delta))=(-(K_X+\Delta))^n$ when $n=2$ or $n=3$ and $X$ is {$\QQ$-factorial} of $\rho(X)=1$. This bound is…

We obtain exact lower bounds for Kolmogorov $n$-widths in spaces $C$ and $L$ of classes of convolutions with Neumann kernel $N_{q,\beta}(t)=\sum\limits_{k=1}^{\infty}\dfrac{q^k}{k}\cos\left(kt-\dfrac{\beta\pi}{2}\right)$, ${q\in(0,1)}$,…

Given integer $k$ and a $k$-graph $F$, let $t_{k-1}(n,F)$ be the minimum integer $t$ such that every $k$-graph $H$ on $n$ vertices with codegree at least $t$ contains an $F$-factor. For integers $k\geq3$ and $0\leq\ell\leq k-1$, let…

A collection $ \Delta $ of simple closed curves on an orientable surface is an algebraic $ k $-system if the algebraic intersection number $\langle \alpha,\beta \rangle$ is equal to $k $ in absolute value for every $ \alpha , \beta \in…