Rational nodal curves with no smooth Weierstrass points

Let denote the rational curve with nodes obtained from the Riemann sphere by identifying 0 with and with for , where is a primitive th root of unity. We show that if is even, then has no smooth Weierstrass points, while if is odd, then has smooth Weierstrass points.

     

Galois points for a plane curve in arbitrary characteristic

In 1996, Hisao Yoshihara introduced a new notion in algebraic geometry: a Galois point for a plane curve is a point from which the projection induces a Galois extension of function fields. Yoshihara has established various new approaches to algebraic geometry by using Galois point or generalized notions of it. It is an interesting problem to determine the distribution of Galois points for a given plane curve. In this paper, we survey recent results related to this problem.      

Enriched spin curves on stable curves with two components

In [7], Mainò constructed a moduli space for enriched stable curves, by blowing-up the moduli space of Deligne–Mumford stable curves. We introduce enriched spin curves, showing that a parameter space for these objects is obtained by blowing-up the moduli space of spin curves. The author was partially supported by CNPq (Proc.151610/2005-3) and by Faperj (Proc.E-26/152-629/2005).     

Rational points and Galois points for a plane curve over a finite field

We study the relationship between rational points and Galois points for a plane curve over a finite field. It is known that the set of Galois points coincides with that of rational points of the projective plane if the curve is the Hermitian, Klein quartic or Ballico–Hefez curve. The author proposes a problem: Does the converse hold true? If the curve of genus zero or one has a rational point, we have an affirmative answer.     

Bogomolov conjecture over function fields for stable curves with only irreducible fibers

Let K be a function field and C a non-isotrivial curve of genus g2 overK. In this paper, we will show that if C has a global stable modelwith only geometrically irreducible fibers, then Bogomolov conjecture over function fields holds.     

Completion of G-spectra and stable maps between classifying spaces

We prove structural theorems for computing the completion of a G-spectrum at the augmentation ideal of the Burnside ring of a finite group G. First we show that a G-spectrum can be replaced by a spectrum obtained by allowing only isotropy groups of prime power order without changing the homotopy type of the completion. We then show that this completion can be computed as a homotopy colimit of completions of spectra obtained by further restricting isotropy to one prime at a time, and that these completions can be computed in terms of completion at a prime.As an application, we show that the spectrum of stable maps from BG to the classifying space of a compact Lie group K splits non-equivariantly as a wedge sum of p-completed suspension spectra of classifying spaces of certain subquotients of G×K. In particular this describes the dual of BG.     

A birational embedding with two Galois points for quotient curves

A criterion for the existence of a birational embedding with two Galois points for quotient curves is presented. We apply our criterion to several curves, for example, some cyclic subcovers of the Giulietti–Korchmáros curve or of the curves constructed by Skabelund. New examples of plane curves with two Galois points are described, as plane models of such quotient curves.     

On the number of Galois points for a plane curve in positive characteristic,II

For a smooth plane curve , we call a point a Galois point if the point projection at P is a Galois covering. We study Galois points in positive characteristic. We give a complete classification of the Galois group given by a Galois point and estimate the number of Galois points for C in most cases.      

Codes from curves with total inflection points

The concept of pure gaps of a Weierstrass semigroup at several points of an algebraic curve has been used lately to obtain codes that have a lower bound for the minimum distance which is greater than the Goppa bound. In this work, we show that the existence of total inflection points on a smooth plane curve determines the existence of pure gaps in certain Weierstrass semigroups. We then apply our results to the Hermitian curve and construct codes supported on several points that compare better to one-point codes from that same curve.      

Mond's conjecture for maps between curves

A theorem by D. Mond shows that if is finite and has has degree one onto its image (Y , 0), then the ‐codimension is less than or equal to the image Milnor number , with equality if and only if (Y , 0) is weighted homogeneous. Here we generalize this result to the case of a map germ , where (X , 0) is a plane curve singularity.     

The multiplication map for global sections of line bundles and rank 1 torsion free sheaves on curves

LetX be an integral projective curve andL ∃ Pic^a(X),M ∃ Pic^b (X) with h¹(X, L)= h¹(X, M) = 0 andL, M general. Here we study the rank of the multiplication map μ _L,M :H ⁰(X,L)⊗H ⁰(X,M)→H ⁰(X,L⊗M). We also study the same problem whenL andM are rank 1 torsion free sheaves onX. Most of our results are forX with only nodes as singularities.     

Compactified Picard stacks over the moduli stack of stable curves with marked points

In this paper we give a construction of algebraic (Artin) stacks endowed with a modular map onto the moduli stack of stable curves of genus g with n marked points. The stacks we construct are smooth, irreducible and have dimension 4g−3+n, yielding a geometrically meaningful compactification of the universal Picard stack parametrizing n-pointed smooth curves together with a line bundle.     

Central limit theorem and stable laws for intermittent maps

In the setting of abstract Markov maps, we prove results concerning the convergence of renormalized Birkhoff sums to normal laws or stable laws. They apply to one-dimensional maps with a neutral fixed point at 0 of the form x+x¹⁺, for (0, 1). In particular, for >1/2, we show that the Birkhoff sums of a Hölder observable f converge to a normal law or a stable law, depending on whether f(0)=0 or f(0)0. The proof uses spectral techniques introduced by Sarig, and Wieners Lemma in non-commutative Banach algebras.Mathematics Subject Classification (2000):37A30, 37A50, 37C30, 37E05, 47A56, 60F05     

Border collision bifurcation curves and their classification in a family of 1D discontinuous maps

In this paper we consider a one-dimensional piecewise linear discontinuous map in canonical form, which may be used in several physical and engineering applications as well as to model some simple financial markets. We classify three different kinds of possible dynamic behaviors associated with the stable cycles. One regime (i) is the same existing in the continuous case and it is characterized by periodicity regions following the period increment by 1 rule. The second one (ii) is the regime characterized by periodicity regions of period increment higher than 1 (we shall see examples with 2 and 3), and by bistability. The third one (iii) is characterized by infinitely many periodicity regions of stable cycles, which follow the period adding structure, and multistability cannot exist. The analytical equations of the border collision bifurcation curves bounding the regions of existence of stable cycles are determined by using a new approach.     

Limit theorems for a class of critical superprocesses with stable branching

We consider a critical superprocess $\{X;{\mathbf{P}}_{\mu }\}$ with general spatial motion and spatially dependent stable branching mechanism with lowest stable index ${\gamma }_0>1$. We first show that, under some conditions, ${\mathbf{P}}_{\mu }(\left|X_t\right|\ne 0)$ converges to 0 as $t\to \infty$ and is regularly varying with index ${({\gamma }_0-1)}^{-1}$. Then we show that, for a large class of non-negative testing functions $f$, the distribution of $\{X_t\left(f\right);{\mathbf{P}}_{\mu }\left(\cdot \right|6X_{t}6\ne 0)\}$, after appropriate rescaling, converges weakly to a positive random variable ${\mathbf{z}}^{({\gamma }_0-1)}$ with Laplace transform $E\left[e^{-u{\mathbf{z}}^{({\gamma }_0-1)}}\right]=1-{(1+u^{-({\gamma }_0-1)})}^{-1∕({\gamma }_0-1)}.$     

On the Number of Galois Points for a Plane Curve in Positive Characteristic

We study Galois points for a plane smooth curve C ? P ² of degree d ≥ 4 in characteristic p > 2. We generalize Yoshihara's result on the number of inner (resp., outer) Galois points to positive characteristic under the assumption that d ? 1 (resp., d ? 0) modulo p. As an application, we also find the number of Galois points in the case that d = p.     

Invariant curves for planar competitive and cooperative maps

In this paper we present results on the existence of invariant curves for planar maps that are monotone with respect to either the south-east or north-east ordering. Some of these curves are the stable or unstable manifolds of hyperbolic fixed points (saddle points) or non-hyperbolic fixed points, and are also the boundary of basins of attraction of such points.     

Bootstrap confidence interval for a correlation curve

A correlation curve measures the strength of the association between two variables locally at different values of x. The purpose of this study is to obtain point-wise confidence intervals for a correlation curve using wild bootstrap techniques. Empirical coverage probabilities are found to be close to the specified nominal level. Bootstrapping is an attractive alternative to confidence intervals based on asymptotic expressions that have slow rate of convergence.