Let $(X, O)$ be a germ of a normal surface singularity, $\pi : \tilde X \to X$ be the minimal resolution of singularities and let $A = (a_{i, j})$ be the $n \times n$ symmetrical intersection matrix of the exceptional set of $\tilde X$. In an old preprint Nash proves that the set of arcs on a surface singularity is a scheme $\mathcal{H}$, and defines a map $\mathcal{N}$ from the set of irreducible components of $\mathcal{H}$ to the set of exceptional components of the minimal resolution of singularities of $(X, O)$. He proved that this map is injective and ask if it is surjective. In this paper we consider the canonical decomposition $\mathcal{H} = \bigcup_{i=1}^{n} \bar{\mathcal{N}_{i}}$:

In this paper we begin the study of two Rankin-Selberg integrals defined on the exceptional group of type $GE_{6}$. We show that each factorizes and that the contribution from the unramified places is, in one case, the degree 54 Euler product $L^{S}(\pi \times \tau, E_{6} \times GL_{2}, s)$ and in the other case the degree 30 Euler product $L^{S}(\pi \times \tau, \wedge^{2} \times GL_{2}, s)$.

We show that the Poincaré lemma we proved elsewhere in the context of crystalline cohomology of higher level behaves well with regard to the Hodge filtration. This allows us to prove the Poincaré lemma for transversal crystals of level $m$. We interpret the de Rham complex in terms of what we call the Berthelot-Lieberman construction and show how the same construction can be used to study the conormal complex and invariant differential forms of higher level for a group scheme. Bringing together both instances of the construction, we show that crystalline extensions of transversal crystals by algebraic groups can be computed by reduction to the filtered de Rham complexes. Our theory does not ignore torsion and, unlike in the classical case ($m = 0$), not all invariant forms are closed. Therefore, close invariant differential forms of level $m$ provide new invariants and we exhibit some examples as applications.

We establish Noether's inequality for surfaces of general type in positive characteristic. Then we extend Enriques' and Horikawa's classification of surfaces on the Noether line, the so-called Horikawa surfaces. We construct examples for all possible numerical invariants and in arbitrary characteristic, where we need foliations and deformation techniques to handle characteristic 2. Finally, we show that Horikawa surfaces lift to characteristic zero.

Let $f_{\lambda}(z) = \lambda e^{z}$. In this short note, we consider those maps $f_{\lambda}$ with $\lambda$ close to $1$. We show that the probability that $f_{\lambda}$ is hyperbolic approaches $1$ as $\lambda \to 1$.

We introduce a log Picard variety over the complex number field by the method of log geometry in the sense of Fontaine-Illusie, and study its basic properties, especially, its relationship with the group of log version of $\mathbb{G}_{m}$-torsors.