Related papers: Log Canonical Thresholds for Plane Curves in Arbit…
We formulate a refined theory of linear systems, using the methods of a previous paper, "A Theory of Branches for Algebraic Curves", and use it to give a geometric interpretation of the genus of an algebraic curve. Using principles of…
The traditional study of plane and space algebraic curves by looking at their tangent vectors, curvatures and torsions provides geometric, but unfortunately not sufficient information about individual curves in order to be able to…
This is a survey on recent results on counting of curves over finite fields. It reviews various results on the maximum number of points on a curve of genus g over a finite field of cardinality q, but the main emphasis is on results on the…
Let $Y$ be the complement of a plane quartic curve $D$ defined over a number field. Our main theorem confirms the Lang-Vojta conjecture for $Y$ when $D$ is a generic smooth quartic curve, by showing that its integral points are confined in…
In this note we revisit the problem of determining combinatorially the multiplicity at the origin of a toric curve. In addition, we give the exact value of the regularity index of that point for plane toric curves and effective bounds for…
We present the elementary properties of log canonical centers of log varieties.
We show that the resolution graph of a plane curve singularity admits a canonical decomposition into elementary graphs.
In this paper we study the problem of classifying pencils of curves of degree $d$ in $\mathbb{P}^2$ using geometric invariant theory. We consider the action of $SL(3)$ and we relate the stability of a pencil to the stability of its…
We establish upper bounds for tails of order statistics of isotropic log-concave vectors and apply them to derive a concentration of l_r norms of such vectors.
We construct isotrivial and non-isotrivial elliptic curves over $\mathbb{F}_q(t)$ with an arbitrarily large set of separable integral points. As an application of this construction, we prove that there are isotrivial log-general type…
The subject of limit curve theorems in Lorentzian geometry is reviewed. A general limit curve theorem is formulated which includes the case of converging curves with endpoints and the case in which the limit points assigned since the…
In this article we apply the classical method of focal loci of families to give a lower bound for the genus of curves lying on general surfaces. First we translate and reprove Xu's result that any curve C on a general surface in P^3 of…
We present some results concerning the Saito module and the torsion submodule of an analytic plane curve, and we provide a method for computing them. Using this algorithm, we compute analytic invariants for plane curves with multiplicity…
The goal of this paper is a classification theorem of the singularities according to a new invariant, Mather discrepancy. On the other hand, we show some evidences convincing us that Mather discrepancy is a considerable invariant: By…
We give a combinatorial upper bound for the gonality of a curve that is defined by a bivariate Laurent polynomial with given Newton polygon. We conjecture that this bound is generically attained, and provide proofs in a considerable number…
For any locally analytic curve we show that arc length can be complexified and seen as a conformal parameter. As an application, we show that any such curve defines a unique maximal one and that the notions of analytic Jordan curve…
We study the log canonical models of the moduli space MBar_{0,n} of pointed stable genus zero curves with respect to the standard log canonical divisors K+aD, where D denotes the boundary. In particular we show that, as a formal consequence…
We give examples of smooth plane quartics over $\mathbb{Q}$ with complex multiplication over $\overline{\mathbb{Q}}$ by a maximal order with primitive CM type. We describe the required algorithms as we go, these involve the reduction of…
We prove an algebraic canonicity theorem for normal LE-logics of arbitrary signature, in a generalized setting in which the non-lattice connectives are interpreted as operations mapping tuples of elements of the given lattice to closed or…
In this note we study the local negativity for certain configurations of smooth rational curves in smooth surfaces with numerically trivial canonical class. We show that for such rational curves there is a bound for the so-called local…