Higher ramification and varieties of secant divisors on the generic curve

For a smooth projective curve, the cycles of e-secant k-planes are among the most studied objects in classical enumerative geometry and there are well-known formulas due to Castelnuovo, Cayley and MacDonald concerning them. Despite various attempts, surprisingly little is known about the enumerative validity of such formulas. The aim of this paper is to completely clarify this problem in the case of the generic curve C of given genus. Using degeneration techniques and a few facts about the birational geometry of moduli spaces of stable pointed curves we determine precisely under which conditions the cycle of e-secant k-planes in non-empty and we compute its dimension. We also precisely determine the dimension of the variety of linear series on C carrying e-secant k-planes. In a different direction, in the last part of the paper we study the distribution of ramification points of the powers of a line bundle on C having prescribed ramification at a given point.Comment: 25 pages. Numerous changes suggested by the referee, several proofs explained in more detail. To appear in the Journal of the London Mathematical Societ

An extremal effective survey about extremal effective cycles in moduli spaces of curves

We survey recent developments and open problems about extremal effective divisors and higher codimension cycles in moduli spaces of curves.Comment: Submitted to the Proceedings of the Abel Symposium 2017. Comments are welcom

Tree decompositions with small cost

The f-cost of a tree decomposition ({Xi | i e I}, T = (I;F)) for a function f : N -> R+ is defined as EieI f(|Xi|). This measure associates with the running time or memory use of some algorithms that use the tree decomposition. In this paper we investigate the problem to find tree decompositions of minimum f-cost. A function f : N -> R+ is fast, if for every i e N: f(i+1) => 2*f(i). We show that for fast functions f, every graph G has a tree decomposition of minimum f-cost that corresponds to a minimal triangulation of G; if f is not fast, this does not hold. We give polynomial time algorithms for the problem, assuming f is a fast function, for graphs that has a polynomial number of minimal separators, for graphs of treewidth at most two, and for cographs, and show that the problem is NP-hard for bipartite graphs and for cobipartite graphs. We also discuss results for a weighted variant of the problem derived of an application from probabilistic networks

Syzygies of torsion bundles and the geometry of the level l modular variety over M_g

We formulate, and in some cases prove, three statements concerning the purity or, more generally the naturality of the resolution of various rings one can attach to a generic curve of genus g and a torsion point of order l in its Jacobian. These statements can be viewed an analogues of Green's Conjecture and we verify them computationally for bounded genus. We then compute the cohomology class of the corresponding non-vanishing locus in the moduli space R_{g,l} of twisted level l curves of genus g and use this to derive results about the birational geometry of R_{g, l}. For instance, we prove that R_{g,3} is a variety of general type when g>11 and the Kodaira dimension of R_{11,3} is greater than or equal to 19. In the last section we explain probabilistically the unexpected failure of the Prym-Green conjecture in genus 8 and level 2.Comment: 35 pages, appeared in Invent Math. We correct an inaccuracy in the statement of Prop 2.

On the complexity of color-avoiding site and bond percolation

The mathematical analysis of robustness and error-tolerance of complex networks has been in the center of research interest. On the other hand, little work has been done when the attack-tolerance of the vertices or edges are not independent but certain classes of vertices or edges share a mutual vulnerability. In this study, we consider a graph and we assign colors to the vertices or edges, where the color-classes correspond to the shared vulnerabilities. An important problem is to find robustly connected vertex sets: nodes that remain connected to each other by paths providing any type of error (i.e. erasing any vertices or edges of the given color). This is also known as color-avoiding percolation. In this paper, we study various possible modeling approaches of shared vulnerabilities, we analyze the computational complexity of finding the robustly (color-avoiding) connected components. We find that the presented approaches differ significantly regarding their complexity.Comment: 14 page

Independent Set Reconfiguration in Cographs

We study the following independent set reconfiguration problem, called TAR-Reachability: given two independent sets $I$ and $J$ of a graph $G$, both of size at least $k$, is it possible to transform $I$ into $J$ by adding and removing vertices one-by-one, while maintaining an independent set of size at least $k$ throughout? This problem is known to be PSPACE-hard in general. For the case that $G$ is a cograph (i.e. $P_4$-free graph) on $n$ vertices, we show that it can be solved in time $O(n^2)$, and that the length of a shortest reconfiguration sequence from $I$ to $J$ is bounded by $4n-2k$, if such a sequence exists. More generally, we show that if $X$ is a graph class for which (i) TAR-Reachability can be solved efficiently, (ii) maximum independent sets can be computed efficiently, and which satisfies a certain additional property, then the problem can be solved efficiently for any graph that can be obtained from a collection of graphs in $X$ using disjoint union and complete join operations. Chordal graphs are given as an example of such a class $X$

Preparation and Characterisation of Alumina Template Obtained by OneStep Anodization Method

The goal of this study was to obtain an alumina template (AAO) by one-step anodization method and to evaluate its optical properties correlated with the annealing temperature. AAO was obtained from two different media: sulphuric acid (1.5 M H2SO4) and oxalic acid (0.4 M H2C2O4) at a potential of 15 V and 40 V, respectively. AAO morphology and chemical composition had been investigated by scanning electron microscopy (SEM) and energy dispersive X-ray spectroscopy (EDX). The average pore diameters such as 20 nm for AAO obtained from H2SO4 and 40 nm from H2C2O4 were measured. The crystalline structures of AAO samples annealed at three different temperatures of 150 ºC, 300 ºC and 350 ºC were studied by Xray diffractometry (XRD). The effect of annealing temperature on the optical properties of AAO was studied by UV-VIS spectrophotometry

Copper Plating Corrosion Study in Certain Environments

Copper plating was performed on nickel substrate by means of the potentiostatic electrodeposition method from a sulphate electrolyte solution. The copper coatings morphology was studied by means of the optical and electronic scanning microscopy techniques. The uniform electrodeposited films have a thickness of about 15 µm measured in cross-section. The corrosion behaviours of nickel substrate and copper films in different corrosive environments were studied. The corrosion study was performed by means of the linear polarisation method in four acid environments: 0.5 M H2SO4, HCl, HNO3 and glacial CH3COOH. From the recorded Tafel curves it was possible to obtain some information about the corrosion rate and the polarization resistance. In order to confirm these results, the gravimetric parameter was calculated by means of the “mass loss” method. By means of the X-ray diffraction analysis, the crystallographic structure of the specimens before and after corrosion was revealed. By means of the spectrophotometer device, the optical properties of the specimens were analysed