The Moduli of Singular Curves on K3 Surfaces

In this article we consider moduli properties of singular curves on K3 surfaces. Let $\mathcal{B}_g$ denote the stack of primitively polarized K3 surfaces $(X,L)$ of genus $g$ and let $\mathcal{T}^n_{g,k} \to \mathcal{B}_g$ be the stack parametrizing tuples $[(f: C \to X, L)]$ with $f$ an unramified morphism which is birational onto its image, $C$ a smooth curve of genus $p(g,k)-n$ and $f_*C \in |kL|$. We show that the forgetful morphism $$\eta \; : \; \mathcal{T}^n_{g,k} \to \mathcal{M}_{p(g,k)-n}$$ is generically finite on one component, for all but finitely many values of $p(g,k)-n$. We further study the Brill--Noether theory of those curves parametrized by the image of $\eta$, and find a Wahl-type obstruction for a smooth curve with an unordered marking to have a nodal model on a K3 surface in such a way that the marking is the divisor over the nodes.Comment: Final version. To appear in J. Math. Pures. App

The generic Green-Lazarsfeld secant conjecture

Generalizing the well-known Green Conjecture on syzygies of canonical curves, Green and Lazarsfeld formulated in 1986 the Secant Conjecture predicting that a line bundle L of sufficiently high degree on a curve has a non-linear p-syzygy if and only if L fails to be (p+1)-very ample. Via lattice theory for special K3 surfaces, Voisin's solution of the classical Green Conjecture and calculations on moduli stacks of pointed curves, we prove: (1) The Green-Lazarsfeld Secant Conjecture in various degree of generality, including its strongest possible form in the divisorial case in the universal Jacobian. (2) The Prym-Green Conjecture on the naturality of the resolution of a general Prym-canonical curve of odd genus.Comment: 24 pages. Final version, to appear in Inventiones Mat

A Weighted Correlation Index for Rankings with Ties

Understanding the correlation between two different scores for the same set of items is a common problem in information retrieval, and the most commonly used statistics that quantifies this correlation is Kendall's $\tau$. However, the standard definition fails to capture that discordances between items with high rank are more important than those between items with low rank. Recently, a new measure of correlation based on average precision has been proposed to solve this problem, but like many alternative proposals in the literature it assumes that there are no ties in the scores. This is a major deficiency in a number of contexts, and in particular while comparing centrality scores on large graphs, as the obvious baseline, indegree, has a very large number of ties in web and social graphs. We propose to extend Kendall's definition in a natural way to take into account weights in the presence of ties. We prove a number of interesting mathematical properties of our generalization and describe an $O(n\log n)$ algorithm for its computation. We also validate the usefulness of our weighted measure of correlation using experimental data

The contribution of closed loop tracking control of motion platform on laterally induced postural instability of the drivers at SAAM dynamic simulator

This paper explains the effect of a motion platform closed loop control comparing to the static condition for driving simulators on postural instability. The postural instabilities of the participants (N=18, 15 male and 3 female subjects) were measured as lateral displacements of subject body centre of pressure (YCP ) just before and after each driving session via a balance platform. After having completed the experiments, the two-tailed Mann-Whitney U test was applied to analyze the objective data for merely the post-exposure cases. The objective data analysis revealed that the YCP for the dynamic case indicated a significant lower value than the static situation (U(18), p < 0,0001). It can be concluded that the closed loop tracking control of the hexapod platform of the driving simulator (dynamic platform condition) decreased significantly the lateral postural stability compared to the static operation condition. However the two-tailed Mann-Whitney U test showed that no significant difference was obtained between the two conditions in terms of psychophysical perception