Related papers: Kempe's Universality Theorem for Rational Space Cu…

Motivated by problems arising in the relative trace formula and arithmetic invariant theory we prove the existence of rational points on orbits arising from certain infinitesimal symmetric spaces. As an application, we prove analogous…

The traditional Riemann Mapping Theorem can be proved with circle packing techniques. We prove the Combinatorial Riemann Mapping Theorem for tilings of bounded size using circle packings.

We define a graph algebra version of the stationary phase integration over the coadjoint orbits in the Reshetikhin formula for the colored Jones-HOMFLY polynomial. As a result, we obtain a `universal' U(1)-RCC invariant of links in rational…

We study properties of rational curves on complete intersections in positive characteristic. It has long been known that in characteristic 0, smooth Calabi-Yau and general type varieties are not uniruled. In positive characteristic,…

We study embedded rational curves in projective toric varieties. Generalizing results of the first author and Zotine for the case of lines, we show that any degree $d$ rational curve in a toric variety $X$ can be constructed from a special…

We propose an approach for showing rationality of an algebraic variety $X$. We try to cover $X$ by rational curves of certain type and count how many curves pass through a generic point. If the answer is $1$, then we can sometimes reduce…

We determine all modular curves $X_0(N)$ with infinitely many quartic points. To do this, we define a pairing that induces a quadratic form representing all possible degrees of a rational morphism from $X_0(N)$ to a positive rank elliptic…

A variety is rationally connected if two general points can be joined by a rational curve. A higher version of this notion is rational simple connectedness, which requires suitable spaces of rational curves through two points to be…

In the open problem of classification of rational cuspidal plane curves it is essential to find good necessary conditions on the type of singularities of a curve C in order C to exit. Motivated by the study of the Seiberg-Witten invariant…

In this article formulas for the quantum product of a rational surface are given, and used to give an algebro-geometric proof of the associativity of the quantum product for strict Del Pezzo surfaces, those for which $-K$ is very ample. An…

We present a conjecture for the power-law exponent in the asymptotic number of types of plane curves as the number of self-intersections goes to infinity. In view of the description of prime alternating links as flype equivalence classes of…

The Newton polygon of the implicit equation of a rational plane curve is explicitly determined by the multiplicities of any of its parametrizations. We give an intersection-theoretical proof of this fact based on a refinement of the…

For each link type $K$ in the 3-sphere, we show that there is a polynomial $p_K$ such that any two diagrams of $K$ with $c_1$ and $c_2$ crossings differ by at most $p_K(c_1) + p_K(c_2)$ Reidemeister moves. As a consequence, the problem of…

We consider all genus 2 curves over Q given by an equation y^2 = f(x) with f a squarefree polynomial of degree 5 or 6, with integral coefficients of absolute value at most 3. For each of these roughly 200000 isomorphism classes of curves,…

We completely classify all plane curves of degree at most 30 with a unique cuspidal (locally unibranch) singular point and rational normalization in terms of the Newton pairs parameterizing the cusp. We distinguish between prime and…

We examine pairs of closed plane curves that have the same closing property as two conic sections in Poncelet's porism. We show how the vertex curve can be computed for a given envelope and vice versa. Our formulas are universal in the…

In this paper, we establish the rationality conjecture raised in \cite{FKS} for any $(r-1)$-connected ($r\geq 2$) $kr$-dimensional CW-complex $X$ ($k\geq 2$) having a unique spherical cohomology class $u\in \tilde{H}^r(X, \mathbb{Z})$ such…

We construct an irreducible rational curve of degree 10 in $CP^2$ which has 12 triple points and a union of three rational quartics with 19 triple points. This gives counter-examples to a conjecture by Dimca, Harbourne, and Sticlaru. We…

We prove that any $C^{1+BV}$ degree $d \geq 2$ circle covering $h$ having all periodic orbits weakly expanding, is conjugate in the same smoothness class to a metrically expanding map. We use this to connect the space of parabolic external…

A cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. The paper is concerned with the linkedness of the graphs of cubical polytopes. A graph with at least $2k$ vertices is $k$-linked if, for every…