Curve Of Positive Genus Research Articles

On the section conjecture and Brauer\u2013Severi varieties

J. Stix proved that a curve of positive genus over mathbb {Q} which maps to a non-trivial Brauer–Severi variety satisfies the section conjecture. We prove that, if X is a curve of positive genus over a number field k and the Weil restriction R_{k/mathbb {Q}}X admits a rational map to a non-trivial Brauer–Severi variety, then X satisfies the section conjecture. As a consequence, if X maps to a Brauer–Severi variety P such that the corestriction {text {cor}}_{k/mathbb {Q}}([P])in {text {Br}}(mathbb {Q}) is non-trivial, then X satisfies the section conjecture.

Elliptic fibrations on K3 surfaces with a non-symplectic involution fixing rational curves and a curve of positive genus

In this paper we complete the classification of the elliptic fibrations on K3 surfaces which admit a non-symplectic involution acting trivially on the Néron–Severi group. We use the geometric method introduced by Oguiso and moreover we provide a geometric construction of the fibrations classified. If the non-symplectic involution fixes at least one curve of genus 1, we relate all the elliptic fibrations on the K3 surface with either elliptic fibrations or generalized conic bundles on rational elliptic surfaces. This description allows us to write the Weierstrass equations of the elliptic fibrations on the K3 surfaces explicitly and to study their specializations.

Higher distances for constant dimensions codes: the case of osculating spaces to a Veronese variety

In linear network coding the information is encoded in terms of a basis of a vector space and it is received as a basis of a possible altered vector space. In the constant dimension case Koetter and Kschischang introduced a metric on the Grassmannian and proved efficient and correct decoding in terms of this metric. Here we introduce a second order invariant of the code: the minimum dimension of the linear span of 3 different linear subspaces belonging to the code. This is the case \(s=3\) of a family \(d_s\) and \(d'_s\), \(s\ge 3\), of invariants of network codes. We study these invariants in a case recently proposed by Hansen (the set of all osculating spaces of a Veronese embedding of a finite projective space) and for a related case (the set of osculating spaces to curves of positive genus) with a complete description of the case of elliptic curves and the ones related to the Hermitian curve.

Symmetries of order four on K3 surfaces

In this paper we study automorphisms of order four on K3 surfaces. We give a classification of the non-symplectic ones when either the square of the automorphism is symplectic, or its fixed locus contains a curve of positive genus, or its fixed locus contains at least a curve and all the curves fixed by its square are rational. We provide partial results in the other cases.

The Fano surface of the Klein cubic threefold

We prove that the Klein cubic threefold $F$ is the only smooth cubic threefold which has an automorphism of order $11$. We compute the period lattice of the intermediate Jacobian of $F$ and study its Fano surface $S$. We compute also the set of fibrations of $S$ onto a curve of positive genus and the intersection between the fibres of these fibrations. These fibres generate an index $2$ sub-group of the Néron-Severi group and we obtain a set of generators of this group. The Néron-Severi group of $S$ has rank $25=h^{1,1}$ and discriminant $11^{10}$.

Galois structure and de Rham invariants of elliptic curves

Let K be a number field with ring of integers O K . Suppose a finite group G acts numerically tamely on a regular scheme X over O K . One can then define a de Rham invariant class in the class group Cl ( O K [ G ] ) , which is a refined Euler characteristic of the de Rham complex of X . Our results concern the classification of numerically tame actions and the de Rham invariant classes. We first describe how all Galois étale G-covers of a K-variety may be built up from finite Galois extensions of K and from geometric covers. When X is a curve of positive genus, we show that a given étale action of G on X extends to a numerically tame action on a regular model if and only if this is possible on the minimal model. Finally, we characterize the classes in Cl ( O K [ G ] ) which are realizable as the de Rham invariants for minimal models of elliptic curves when G has prime order.

SPECIAL RAMIFICATION LOCI ON THE DOUBLE PRODUCT OF A GENERAL CURVE

Let C be a general connected, smooth, projective curve of positive genus g. For each integer i≥0, we give formulae for the number of pairs (P, Q) ∈ C × C off the diagonal such that (g + i − 1)Q - (i + 1)P is linearly equivalent to an effective divisor, and the number of pairs (P, Q) ∈ C × C off the diagonal such that (g + i + 1)Q − (i + 1)P is linearly equivalent to a moving effective divisor.

On the square-free sieve

We improve on the best available bounds for the square-free sieve and provide a general framework for its applicability. The failure of the local-to-global principle allows us to obtain results better than those reached by a classical sieve-based approach. Techniques involving sphere-packing yield upper bounds on the number of integer and rational points on curves of positive genus.

On model complete differential fields

We develop a geometric approach to definable sets in differentially closed fields, with emphasis on the question of orthogonality to a given strongly minimal set. Equivalently, within a family of ordinary differential equations, we consider those equations that can be transformed, by differential-algebraic transformations, so as to yield solutions of a given fixed first-order ODEXX. We show that this sub-family is usually definable (in particular ifXXlives on a curve of positive genus). As a corollary, we show the existence of many model-complete, superstable theories of differential fields.

RESULTS CONCERNING THINNESS OF D0L LANGUAGES

A language L is called thin if there exists an integer n0such that for all n≥n0L contains at most one word of length n. We show that thinness is decidable for exponential D0L languages. We show also that Siegel's result concerning integral points on algebraic curves of positive genus can often be used to prove that a polynomially bounded HD0L language is thin.