Depth of cohomology support loci for quasi-projective varieties via orbifold pencils

The present paper describes a relation between the quotient of the fundamental group of a smooth quasi-projective variety by its second commutator and the existence of maps to orbifold curves. It extends previously studied cases when the target was a smooth curve. In the case when the quasi-projective variety is a complement to a plane algebraic curve this provides new relations between the fundamental group, the equation of the curve, and the existence of polynomial solutions to certain equations generalizing Pell's equation. These relations are formulated in terms of the depth which is an invariant of the characters of the fundamental group discussed in detail here.Comment: 22 page

Free quotients of fundamental groups of smooth quasi-projective varieties

We study the fundamental groups of the complements to curves on simply connected surfaces, admitting non-abelian free groups as their quotients. We show that given a subset of the Néron-Severi group of such a surface, there are only finitely many classes of equisingular isotopy of curves with irreducible components belonging to this subset for which the fundamental groups of the complement admit surjections onto a free group of a given sufficiently large rank. Examples of subsets of the Néron-Severi group are given with infinitely many isotopy classes of curves with irreducible components from such a subset and fundamental groups of the complements admitting surjections on a free group only of a small rank. © The Author(s) 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

Invariants of Combinatorial Line Arrangements and Rybnikov's Example

Following the general strategy proposed by G.Rybnikov, we present a proof of his well-known result, that is, the existence of two arrangements of lines having the same combinatorial type, but non-isomorphic fundamental groups. To do so, the Alexander Invariant and certain invariants of combinatorial line arrangements are presented and developed for combinatorics with only double and triple points. This is part of a more general project to better understand the relationship between topology and combinatorics of line arrangements.Comment: 27 pages, 2 eps figure

Characterization and regulation of wild‐type and mutant TASK‐1 two pore domain potassium channels indicated in pulmonary arterial hypertension

Key points The TASK-1 channel gene (KCNK3) has been identified as a possible disease-causing gene in heritable pulmonary arterial hypertension (PAH). In the present study, we show that novel mutated TASK-1 channels, seen in PAH patients, have a substantially reduced current compared to wild-type TASK-1 channels. These mutated TASK-1 channels are located at the plasma membrane to the same degree as wild-type TASK-1 channels. ONO-RS-082 and alkaline pH 8.4 both activate TASK-1 channels but do not recover current through mutant TASK-1 channels. We show that the guanylate cyclase activator, riociguat, a novel treatment for PAH, enhances current through TASK-1 channels but does not recover current through mutant TASK-1 channels. Pulmonary arterial hypertension (PAH) affects ∼15–50 people per million. KCNK3, the gene that encodes the two pore domain potassium channel TASK-1 (K2P3.1), has been identified as a possible disease-causing gene in heritable PAH. Recently, two new mutations have been identified in KCNK3 in PAH patients: G106R and L214R. The present study aimed to characterize the functional properties and regulation of wild-type (WT) and mutated TASK-1 channels and determine how these might contribute to PAH and its treatment. Currents through WT and mutated human TASK-1 channels transiently expressed in tsA201 cells were measured using whole-cell patch clamp electrophysiology. Localization of fluorescence-tagged channels was visualized using confocal microscopy and quantified with in-cell and on-cell westerns. G106R or L214R mutated channels were located at the plasma membrane to the same degree as WT channels; however, their current was markedly reduced compared to WT TASK-1 channels. Functional current through these mutated channels could not be restored using activators of WT TASK-1 channels (pH 8.4, ONO-RS-082). The guanylate cyclase activator, riociguat, enhanced current through WT TASK-1 channels; however, similar to the other activators investigated, riociguat did not have any effect on current through mutated TASK-1 channels. Thus, novel mutations in TASK-1 seen in PAH substantially alter the functional properties of these channels. Current through these channels could not be restored by activators of TASK-1 channels. Riociguat enhancement of current through TASK-1 channels could contribute to its therapeutic benefit in the treatment of PAH

Albanese varieties of cyclic covers of the projective plane and orbifold pencils

The paper studies a relation between fundamental group of the complement to a plane singular curve and the orbifold pencils containing it. The main tool is the use of Albanese varieties of cyclic covers ramified along such curves. Our results give sufficient conditions for a plane singular curve to belong to an orbifold pencil, i.e. a pencil of plane curves with multiple fibers inducing a map onto an orbifold curve whose orbifold fundamental group is non trivial. We construct an example of a cyclic cover of the projective plane which is an abelian surface isomorphic to the Jacobian of a curve of genus 2 illustrating the extent to which these conditions are necessary

Delta invariant of curves on rational surfaces I. An analytic approach

We prove that if (C, 0) is a reduced curve germ on a rational surface singularity (X, 0) then its delta invariant can be recovered by a concrete expression associated with the embedded topological type of the pair C X. Furthermore, we also identify it with another (a priori) embedded analytic invariant, which is motivated by the theory of adjoint ideals. Finally, we connect our formulae with the local correction term at singular points of the global Riemann-Roch formula, valid for projective normal surfaces, introduced by Blache

Nodal degenerations of plane curves and Galois covers

Globally irreducible nodes (i.e. nodes whose branches belong to the same irreducible component) have mild effects on the most common topological invariants of an algebraic curve. In other words, adding a globally irreducible node (simple nodal degeneration) to a curve should not change them a lot. In this paper we study the effect of nodal degeneration of curves on fundamental groups and show examples where simple nodal degenerations produce non-isomorphic fundamental groups and this can be detected in an algebraic way by means of Galois coverings.Comment: 16 pages, 3 figure