We prove a global Torelli theorem for pairs (Y,D) where Y is a smooth projective rational surface and D ∈ |−KY | is a cycle of rational curves, as conjectured by Friedman in 1984. In addition, we construct natural universal families for such pairs.

This paper proposes the computation of the Tate pairing, Ate pairing and its variations on the special Jacobi quartic elliptic curve Y 2 = dX +Z. We improve the doubling and addition steps in Miller’s algorithm to compute the Tate pairing. We use the birational equivalence between Jacobi quartic curves and Weierstrass curves, together with a specific point representation to obtain the best resu...

The analyticity domains of the Lindstedt series for the standard map are studied numerically using Padé approximants to model their natural boundaries. We show that if the rotation number is a Diophantine number close to a rational value p/q, then the radius of convergence of the Lindstedt series becomes smaller than the critical threshold for the corresponding KAM curve, and the natural bounda...

Introduction. In this paper a point correspondence is introduced which is proving to be very helpful in the study of a general non-ruled analytic surface in ordinary space. If on a surface S tangents to the curves of an asymptotic family are constructed at the points of two curves of 5 which are not members of the family and which intersect in a point y of S, two ruled surfaces are thereby form...

Let X be a smooth complex projective variety, and let Y ⊂ X be a smooth very ample hypersurface such that −KY is nef. Using the technique of relative Gromov-Witten invariants, we give a new short and geometric proof of (a version of) the “mirror formula”, i.e. we show that the generating function of the genus zero 1-point Gromov-Witten invariants of Y can be obtained from that of X by a certain...

Algebraic geometry is a powerful combination of algebra and geometry, with a rich history and many applications, both theoretical and practical. It also has an intimidating reputation. Hence it is important to have a variety of introductory texts at different levels of sophistication. We will see that Schenck's book offers an interesting path into this wonderful subject. 1. What is algebraic ge...

The seismic performance of four pile-supported models is studied for two conditions: (i) transient to full liquefaction condition, i.e. the phase when excess pore pressure gradually increases during the shaking; (ii) full liquefaction condition, i.e. defined as the state where the seismically induced excess pore pressure equalises to the overburden stress. The paper describes two complementary ...

We study locally Cohen-Macaulay space curves lying on normal surfaces. We prove some theorems on the behaviour of the cohomology functions and initial ideals of such space curves, which give a basic distinction between those curves and curves lying on non-normal surfaces. A natural step in the study and classification of space curves is to classify the Hilbert functions of space curves, or, mor...

These are notes from a talk in the Arithmetic Geometry Learning Seminar on Drinfeld modules during the fall of 2017. In this talk, we discuss the uniformization of the Drinfeld modular curves (of fixed rank, with level structure), following [Dri74, §6]. In addition, we introduce the relevant rigid-analytic background. The exposition draws heavily on [DH87, Gek86]. 1. Uniformization of Complex M...

The goal of this paper is to investigate the crystalline properties of families of elliptic curves or, more precisely, to study the families of crystals (elliptic crystals) attached to families of elliptic curves. Elliptic crystals over a point can be defined and classified very simply in terms of semilinear algebra, and this is how we begin (1.1). Our definition is cast in the logarithmic sett...