arXiv subjects
Publications and source records attributed to Beata Kocel-Cynk.
We give a proof of the o-minimal version of the Whitney Extension Theorem simplified as compared to the original ones. A new simplifying ingredient is a definable variant of Urysohn's lemma for class $\mathcal{C}^q$ (see Section 3).
We present a complete classification of all arrangements of eight planes in projective threespace that give rise to double octic Calabi-Yau threefolds. Building on earlier work, we determine all 455 combinatorial types and describe the projective automorphism groups associated with each arrangement. These automorphisms allow us to identify geometrically distinguished subfamilies and elements, and to construct a rich collection of elliptic and K3 fibrations arising naturally from the singularities of the arrangements. As applications, we exhibit two one-parameter families of Calabi-Yau threefolds that each possess three pairwise non-equivalent maximally unipotent monodromy points-phenomena that are rare in known examples. We further construct two Calabi-Yau threefolds whose third cohomology decomposes, via complex multiplication, into complementary two-dimensional Hodge structures of types (1,0,1,0) and (0,1,0,1). This decomposition yields new instances of modularity not previously observed in the Calabi-Yau setting.
We study fibrations by elliptic curves and K3 surfaces of double octic Calabi-Yau threefolds determined by singular lines and points of multiplicity at least 4 of the defining octic arrangement. As a consequence we conclude that every double octic admits many elliptic and K3 fibrations. We apply descriptions of some K3 fibrations to construct a birational map between elements of two families of double octics.
We prove that, for any closed semialgebraic subset $W$ of $\mathbb{R}^n$ and for any positive integer $p$, there exists a Nash function $f:\mathbb{R}^n\setminus W\longrightarrow (0, \infty)$ which is equivalent to the distance function from $W$ and at the same time it is $Λ_p$-regular in the sense that $|D^αf(x)|\leq C d(x, W)^{1- |α|}$, for each $x\in \mathbb{R}^n\setminus W$ and each $α\in \mathbb{N}^n$ such that $1\leq |α|\leq p$, where $C$ is a positive constant. In particular, $f$ is Lipschitz. Some applications of this result are given.
In the present paper we propose a combinatorial approach to study the so called double octic Clabi--Yau threefolds. We use this description to give a complete classification of double octics with $h^{1,2}\le1$ and to derive their geometric properties (Kummer surface fibrations, automorphisms, special elements in families).