Related papers: A note on elliptic curves on toric surfaces

From a finite set in a lattice, we can define a toric variety embedded in a projective space. In this paper, we give a combinatorial description of the dual defect of the toric variety using the structure of the finite set as a Cayley sum…

Langlands has described the irreducible admissible representations of $T$, when $T$ is the group of points of an algebraic torus over a local field. Also, Langlands described the automorphic representations of $T_{\mathbb A}$ when…

We study topological properties of automorphisms of 4-dimensional torus generated by integer symplectic matrices. The main classifying element is the structure of the topology of a foliation generated by unstable leaves of the automorphism.…

We prove that any compact surface with constant positive curvature and conical singularities can be decomposed into irreducible components of standard shape, glued along geodesic arcs connecting conical singularities. This is a spherical…

Let $D$ be an irreducible plane curve. In this article, we first introduce a notion of a quadratic residue curve mod $D$, and study quadratic residue concis $C$ mod an irreducible quartic curve $Q$. As an application, we study a dihedral…

A Laurent polynomial $f$ in two variables naturally describes a projective curve $C(f)$ on a toric surface. We show that if $C(f)$ is a smooth curve of genus at least 7, then $C(f)$ is not Brill-Noether general. To accomplish this, we…

We study the Hodge structure of elliptic surfaces which are canonically defined from bielliptic curves of genus three. We prove that the period map for the second cohomology has one dimensional fibers, and the period map for the total…

Let ${\mathcal {B}}$ be a reducible reduced plane curve. We introduce a new point of view to study the topology of $(\PP^2, {\mathcal {B}})$ via Galois covers and Alexander polynomials. We show its effectiveness through examples of Zariski…

This note presents two observations which have in common that they lie at the boundary of toric geometry. The first one because it concerns the deformation of affine toric varieties into non toric germs in order to understand how to avoid…

We show that the isolated invariant branches globalize to algebraic curves, when we consider weak toric type complex hyperbolic foliations on projective toric ambient surfaces. To do it, we pass through a characterization of weak toric type…

We study equisingular deformation problems for curves and surfaces in algebraic families, with particular emphasis on situations where nodal behavior is no longer generic. Extending classical Severi theory, we develop deformation--theoretic…

We study parameter spaces of linear series on projective curves in the presence of unibranch singularities, i.e. {\it cusps}; and to do so, we stratify cusps according to value semigroup. We show that {\it generalized Severi varieties} of…

An abelian variety admits only a finite number of isomorphism classes of principal polarizations. The paper gives an interpretation of this number in terms of class numbers of definite Hermitian forms in the case of a product of elliptic…

There is a close relationship between the embedded topology of complex plane curves and the (group-theoretic) arithmetic of elliptic curves. In a recent paper, we studied the topology of some arrangements of curves which include a special…

Let $X$ be a smooth irreducible projective curve of genus $g$ and gonality 4. We show that the canonical model of $X$ is contained in a uniquely defined surface, ruled by conics, whose geometry is deeply related to that of $X$. This surface…

We study the space of rational curves on del Pezzo surfaces in positive characteristic. For most primes p we prove the irreducibility of the moduli space of rational curves of a given nef class, extending results of Testa in characteristic…

This paper focuses on the classification of all toric log Del Pezzo surfaces with exactly one singularity up to isomorphism, and on the description of how they are embedded as intersections of finitely many quadrics into suitable projective…

We recast elliptic surfaces over the projective line in terms of the non-commutative tori and one-parameter families of the periodic continued fractions. The correspondence is used to study the Picard numbers, the ranks and the minimal…

Our main result is an effective version of the Torelli theorem in genus $3$ and any characteristic not $2$: the configuration of the odd theta characteristics of a curve $C$ of genus $3$ determines a del Pezzo surface $S$ of degree two and…

We determine explicit birational models over Q for the modular surfaces parametrising pairs of N-congruent elliptic curves in all cases where this surface is an elliptic surface. In each case we also determine the rank of the Mordell-Weil…