Related papers: Nodal surfaces in $\mathbb{P}^3$ and coding theory
It is well known since Noether that the gonality of a smooth plane curve of degree d>3 is d-1. Given a k-dimensional complex projective variety X, the most natural extension of gonality is probably the degree of irrationality, that is the…
We study the maximal number of triangulations that a planar set of $n$ points can have, and show that it is at most $30^n$. This new bound is achieved by a careful optimization of the charging scheme of Sharir and Welzl (2006), which has…
In the present paper, we study arrangements of smooth plane conics having only nodes and tacnodes as the singularities. We provide an interesting estimation on the number of nodes and tacnodes that depends only on a linear function of the…
Using elementary methods of algebraic geometry, we present constructions of hyperelliptically fibred surfaces containing nodal fibres.
We consider K3 surfaces which are double cover of rational elliptic surfaces. The former are endowed with a natural elliptic fibration, which is induced by the latter. There are also other elliptic fibrations on such K3 surfaces, which are…
Topologically protected states can be found in physical systems, that show singularities in some energy contour diagram. These singularities can be characterized by winding numbers, defined on a classification surface, which maps physical…
A quadratic point on a surface in $RP^3$ is a point at which the surface can be approximated by a quadric abnormally well (up to order 3). We conjecture that the least number of quadratic points on a generic compact non-degenerate…
The generating functions of the Severi degrees for sufficiently ample line bundles on algebraic surfaces are multiplicative in the topological invariants of the surface and the line bundle. Recently new proofs of this fact were given for…
We nearly triple the number of logical qubits per physical qubit of surface codes in the teraquop regime by concatenating them into high-density parity check codes. These "yoked surface codes" are arrayed in a rectangular grid, with parity…
We give bounds on the number of non-simple closed curves on a negatively curved surface, given upper bounds on both length and self-intersection number. In particular, it was previously known that the number of all closed curves of length…
Maximally recoverable codes are codes designed for distributed storage which combine quick recovery from single node failure and optimal recovery from catastrophic failure. Gopalan et al [SODA 2017] studied the alphabet size needed for such…
We construct examples of flat surfaces in $\mathbb{H}^3$ which are graphs over a two-punctured horosphere and classify complete embedded flat surfaces in $\mathbb{H}^3$ with only one end and at most two isolated singularities.
We establish the slope equality and give an upper bound of the slope for finite cyclic covering fibrations of an elliptic surface including bielliptic fibrations of genus greater than 5. We also give an upper bound of the slope for triple…
We show the existence of infinitely many knot exteriors where each of which contains meridional essential surfaces of any genus and (even) number of boundary components. That is, the compact surfaces that have a meridional essential…
We show the finiteness of the N\'eron-Severi lattices of complex projective K3 surfaces whose automorphism groups are non-elementary hyperbolic with explicit descriptions, under the assumption that the Picard number $\ge 6$ which is optimal…
Linear codes and $t$-designs are interactive with each other. It is well known that some $t$-designs have been constructed by using certain linear codes in recent years. However, only a small number of infinite families of the extended…
We give a lower bound on the number of non-simple closed curves on a hyperbolic surface, given upper bounds on both length and self-intersection number. In particular, we carefully show how to construct closed geodesics on pairs of pants,…
We investigate configurations of rational double points with the total Milnor number 21 on supersingular $K3$ surfaces. The complete list of possible configurations is given. As an application, we also give the complete list of extremal…
We present a systematic study of threefolds fibred by K3 surfaces that are mirror to sextic double planes. There are many parallels between this theory and the theory of elliptic surfaces. We show that the geometry of such threefolds is…
The so-called {\it kissing number} for hyperbolic surfaces is the maximum number of homotopically distinct systoles a surface of given genus $g$ can have. These numbers, first studied (and named) by Schmutz Schaller by analogy with lattice…