On the hypersurface of Luroth quartics

The hypersurface of Luroth quartic curves inside the projective space of plane quartics has degree 54. We give a proof of this fact along the lines outlined in a paper by Morley, published in 1919. Another proof has been given by Le Potier and Tikhomirov in 2001, in the setting of moduli spaces of vector bundles on the projective plane. Morley's proof uses the description of plane quartics as branch curves of Geiser involutions and gives new geometrical interpretations of the 36 planes associated to the Cremona hexahedral representations of a nonsingular cubic surface

Arithmétique des espaces de modules des courbes hyperelliptiques de genre 3 en caractéristique positive

The aim of this thesis is to provide an explicit description of the moduli spaces of genus 3 hyperelliptic curves in positive characteristic. Over a field of odd characteristic, a parameterization of these moduli spaces is given via the algebra of invariants of binary forms of degree 8 under the action of the special linear group. Following the work of Lercier and Ritzenthaler, the case of fields of characteristic 3, 5 and 7 are still open. However, in these remaining cases, the classical methods in characteristic zero do not work in providing generators for these algebra of invariants. Hence we provide only separating invariants in characteristic 3 and 7. Furthermore our results in characteristic 5 show that this approach is not suitable.From these results, we describe the stratification of the moduli spaces of genus 3 hyperelliptic curves in characteristic 3 and 7 according to the automorphism groups of the curves and implement algorithms to reconstruct a curve from its invariants. For this reconstruction step, we paid attention to arithmetic issues, like the obstruction to be a field of definition for the field of moduli.Finally, in the case of characteristic 2, we use a different approach where the curves are defined by their Artin-Schreier models. The arithmetic structure of the ramification points of these curves stratifies the moduli space in 5 cases and we define in each case invariants that characterize the isomorphism class of hyperelliptic curves.L’objet de cette thèse est une description effective des espaces de modules des courbes hyper- elliptiques de genre 3 en caractéristique positive. En caractéristique nulle ou impaire, on obtient une paramétrisation de ces espaces de modules par l’intermédiaire des algèbres d’invariants pour l’action du groupe spécial linéaire sur les espaces de formes binaires de degré 8, qui sont de type fini. Suite aux travaux de Lercier et Ritzenthaler, les cas des caractéristiques 3, 5 et 7 restaient ouverts. Pour ces derniers, les méthodes classiques de la caractéristique nulle sont inopérantes pour l’obtention de générateurs pour les algèbres d’invariants en jeu. Nous nous sommes donc contenté d’exhiber des invariants séparants en caractéristiques 3 et 7. En outre, nos résultats concernant la caractéristique 5 suggèrent l’inadéquation de cette approche pour ce cas.À partir de ces résultats, nous avons pu expliciter la stratification des espaces de modules des courbes hyperelliptiques de genre 3 en caractéristiques 3 et 7 selon les groupes d’automorphismes et implémenter divers algorithmes, dont celui de Mestre, pour la reconstruction d’une courbe à partir de son module, i.e. la valeur de ses invariants. Pour cette phase de reconstruction, nous nous sommes notamment attachés aux questions arithmétiques, comme l’existence d’une obstruction à être un corps de définition pour le corps de modules et, dans le cas contraire, à l’obtention d’un modèle de la courbe sur ce corps de définition minimal.Enfin pour la caractéristique 2, notre approche est différente, dans la mesure où les courbes sont étudiées via leurs modèles d’Artin-Schreier. Nous exhibons pour ceux-ci des invariants bigradués qui dépendent de la structure arithmétique des points de ramifications des courbes

Newton polytopes and numerical algebraic geometry

We develop a collection of numerical algorithms which connect ideas from polyhedral geometry and algebraic geometry. The first algorithm we develop functions as a numerical oracle for the Newton polytope of a hypersurface and is based on ideas of Hauenstein and Sottile. Additionally, we construct a numerical tropical membership algorithm which uses the former algorithm as a subroutine. Based on recent results of Esterov, we give an algorithm which recursively solves a sparse polynomial system when the support of that system is either lacunary or triangular. Prior to explaining these results, we give necessary background on polytopes, algebraic geometry, monodromy groups of branched covers, and numerical algebraic geometry.Comment: 150 pages, 65 figures, contains content from arXiv:1811.12279 and arXiv:2001.0422

Newton Polytopes and Numerical Algebraic Geometry

We develop a collection of numerical algorithms which connect ideas from polyhedral geometry and algebraic geometry. The first algorithm we develop functions as a numerical oracle for the Newton polytope of a hypersurface and is based on ideas of Hauenstein and Sottile. Additionally, we construct a numerical tropical membership algorithm which uses the former algorithm as a subroutine. Based on recent results of Esterov, we give an algorithm which recursively solves a sparse polynomial system when the support of that system is either lacunary or triangular. Prior to explaining these results, we give necessary background on polytopes, algebraic geometry, monodromy groups of branched covers, and numerical algebraic geometry

Rational approximations, multidimensional continued fractions and lattice reduction

We first survey the current state of the art concerning the dynamical properties of multidimensional continued fraction algorithms defined dynamically as piecewise fractional maps and compare them with algorithms based on lattice reduction. We discuss their convergence properties and the quality of the rational approximation, and stress the interest for these algorithms to be obtained by iterating dynamical systems. We then focus on an algorithm based on the classical Jacobi--Perron algorithm involving the nearest integer part. We describe its Markov properties and we suggest a possible procedure for proving the existence of a finite ergodic invariant measure absolutely continuous with respect to Lebesgue measure.Comment: 30 pages, 4 figure

Rationally connected rational double covers of primitive Fano varieties

We show that for a Zariski general hypersurface $V$ of degree $M+1$ in ${\mathbb P}^{M+1}$ for $M\geqslant 5$ there are no rational maps $X\dashrightarrow V$ of degree 2, where $X$ is a rationally connected variety. This fact is true for many other families of primitive Fano varieties, either. It generalizes easily for rationally connected Galois rational covers with an abelian Galois group and motivates a conjecture on absolute rigidity of primitive Fano varieties