On Parameterizations of plane rational curves and their syzygies

Let $C$ be a plane rational curve of degree $d$ and $p:\tilde C \rightarrow C$ its normalization. We are interested in the splitting type $(a,b)$ of $C$, where $\mathcal{O}_{\mathbb{P}^1}(-a-d)\oplus \mathcal{O}_{\mathbb{P}^1}(-b-d)$ gives the syzigies of the ideal $(f_0,f_1,f_2)\subset K[s,t]$, and $(f_0,f_1,f_2)$ is a parameterization of $C$. We want to describe in which cases $(a,b)=(k,d-k)$ ($2k\leq d)$, via a geometric description; namely we show that $(a,b)=(k,d-k)$ if and only if $C$ is the projection of a rational curve on a rational normal surface in $\mathbb{P}^{k+1}$.Comment: 12 Page

Exploiting c\mathbf{c}-Closure in Kernelization Algorithms for Graph Problems

A graph is c-closed if every pair of vertices with at least c common neighbors is adjacent. The c-closure of a graph G is the smallest number such that G is c-closed. Fox et al. [ICALP '18] defined c-closure and investigated it in the context of clique enumeration. We show that c-closure can be applied in kernelization algorithms for several classic graph problems. We show that Dominating Set admits a kernel of size k^O(c), that Induced Matching admits a kernel with O(c^7*k^8) vertices, and that Irredundant Set admits a kernel with O(c^(5/2)*k^3) vertices. Our kernelization exploits the fact that c-closed graphs have polynomially-bounded Ramsey numbers, as we show

The support theorem for the complex Radon transform of distributions

The complex Radon transform $\hat F$ of a rapidly decreasing distribution $F\in\mathscr{O}_C^{\prime}(\mathbb{C}^n)$ is considered. A compact set $K\subset\mathbb{C}^n$ is called linearly convex if the set $\mathbb{C}^n \setminus K$ is a union of complex hyperplanes. Let $\hat K$ denote the set of complex hyperplanes which meet $K$. The main result of the paper establishes the conditions on a linearly convex compact $K$ under which the support theorem for the complex Radon transform is true: from the relation $\hbox{supp}(\hat F)\subset\hat K$ it follows that $F\in\mathscr{O}^{\prime}_C(\mathbb{C}^n)$ is compactly supported and $\hbox{supp}(F)\subset K$.Comment: 8 page