Related papers: Algebraic k-systems of curves

In this article, we study subloci of solvable curves in $\mathcal{M}_g$ which are contained in either a K3-surface or a quadric or a cubic surface. We give a bound on the dimension of such subloci. In the case of complete intersection genus…

For an integer $k\geq 1$, a graph is called a $k$-circulant if its automorphism group contains a cyclic semiregular subgroup with $k$ orbits on the vertices. We show that, if $k$ is even, there exist infinitely many cubic arc-transitive…

A meander can be seen as a pair of transversally intersecting simple closed curves on a 2-sphere. We consider pairs of transversally intersecting simple closed curves on a closed oriented surface of arbitrary genus g. The number of such…

Let C be an algebraic curve in a power of an elliptic curve, both defined over the algebraic numbers. We show that the set of algebraic points of C which satisfy certain conditions is a finite set. This result has implications with the…

Given a graph $G=\big{(}V(G),E(G)\big{)}$, a set $S\subseteq V(G)$ is called a $k$-dominating set if every vertex in $V(G)\setminus S$ has at least $k$ neighbors in $S$. Two disjoint sets $A,B\subset V(G)$ form a $k$-coalition in $G$ if…

Let $L$ be a very ample line bundle on a smooth curve $C$ of genus $g$ with $\frac{3g+3}{2}<\deg L\le 2g-5$. Then $L$ is normally generated if $\deg L>\max\{2g+2-4h^1(C,L), 2g-\frac{g-1}{6}-2h^1(C,L)\}$. Let $C$ be a triple covering of…

Three intersection theorems are proved. First, we determine the size of the largest set system, where the system of the pairwise unions is l-intersecting. Then we investigate set systems where the union of any s sets intersect the union of…

In this paper we prove a theorem stated by Castelnuovo which bounds the dimension of linear systems of plane curves in terms of two invariants, one of which is the genus of the curves in the system. Then we classify linear systems whose…

Let $\mathcal{G}$ be a minor-closed graph class. We say that a graph $G$ is a $k$-apex of $\mathcal{G}$ if $G$ contains a set $S$ of at most $k$ vertices such that $G\setminus S$ belongs to $\mathcal{G}.$ We denote by $\mathcal{A}_k…

A \emph{geometric graph} is a graph whose vertex set is a set of points in general position in the plane, and its edges are straight line segments joining these points. We show that for every integer $k \ge 2$, there exists a constat $c>0$…

We give upper bounds on the genus of a curve with general moduli assuming that it can be embedded in a projective nonsingular surface $Y$ so that $\dim(|C|) > 0$. We find such bounds for all types of surfaces of intermediate Kodaira…

Let $\mathcal{X}$ be an algebraic curve of genus $g$ defined over an algebraically closed field $K$ of characteristic $p \geq 0$, and $q$ a prime dividing $|\mbox{Aut}(\mathcal{X})|$. We say that $\mathcal{X}$ is a $q$-curve. Homma proved…

Let $G$ be a simple $n$-vertex graph and $W\subseteq\V(G)$. We say that $W$ is a $\delta_k$-small set if $$ \sqrt[k]{\frac{\sum_{v\in W}d^k(v)}{\abs W}}\leq n-\abs W. $$ Let $\varphi^{(k)}(G)$ denote the smallest natural number $r$ such…

Let $S_{g,p}$ denote the genus $g$ orientable surface with $p \ge 0$ punctures, and let $\omega(g,p)= 3g+p-4$. We prove the existence of infinitely long geodesic rays $\left\{v_{0},v_{1}, v_{2}, ...\right\}$ in the curve graph satisfying…

We consider relations between the size, treewidth, and local crossing number (maximum number of crossings per edge) of graphs embedded on topological surfaces. We show that an $n$-vertex graph embedded on a surface of genus $g$ with at most…

We prove that if $X$ is a complex projective K3 surface and $g>0$, then there exist infinitely many families of curves of geometric genus $g$ on $X$ with maximal, i.e., $g$-dimensional, variation in moduli. In particular every K3 surface…

In this paper, we analyze the structure of maximal sets of $k$-dimensional spaces in $\mathrm{PG}(n,q)$ pairwise intersecting in at least a $(k-2)$-dimensional space, for $3 \leq k\leq n-2$. We give an overview of the largest examples of…

A unit cube in $k$ dimensions ($k$-cube) is defined as the the Cartesian product $R_1\times R_2\times...\times R_k$ where $R_i$(for $1\leq i\leq k$) is a closed interval of the form $[a_i,a_i+1]$ on the real line. A graph $G$ on $n$ nodes…

We prove that any non-isotrivial elliptic K3 surface over an algebraically closed field $k$ of arbitrary characteristic contains infinitely many rational curves. In the case when $\mathrm{char}(k)\neq 2,3$, we prove this result for any…

Let $G=(V,E)$ be a graph. A (proper) $k$-edge-coloring is a coloring of the edges of $G$ such that any pair of edges sharing an endpoint receive distinct colors. A classical result of Vizing ensures that any simple graph $G$ admits a…