The arithmetic of certain quartic curves

We give a sufficient condition using the Brauer?CManin obstruction under which certain quartic curves have no rational points. Using this sufficient condition, we construct two families of genus one quartic curves violating the Hasse principle explained by the Brauer?CManin obstruction.     

On the normal bundle of curves on nodal quartic surfaces

Let k be an algebraically closed field, char k = 0. Let C be an irreducible nonsingular curve such that 2C = S ? F, where S and F are two surfaces and all the singularities of F are rational double points (if any). We prove that C can never pass through rational singularities of types A _2n n∈N, E₆ and E₈. We give conditions for C to pass through rational singularities of types. A _2k+1 k∈Z⁺ D_n n≥4 and E₇, (0.8).     

Rough Marcinkiewicz integrals along certain smooth curves

This paper is devoted to the study of the multiple-parameter rough Marcinkiewicz integral operators associated with certain smooth curves. It is shown that the Grafakos-Stefanov type size condition F_α (S^m-1 × S^n-1) of the kernel implies the L^p-boundedness of these Marcinkiewicz integral operators for some α>1/2 and 1+12α<p<1+2α, which is an essential improvement of certain previous results.     

A remark on the Hilbert scheme of smooth complex space curves

Partially supported by Korea Science & Engineering Foundation. The author is grateful to the Max-Planck-Institut für Mathematik for the support, where a major part of this paper was written during his stay there.     

On the computation of the Picard group for certain singular quartic surfaces

We test the methods for computing the Picard group of a K3 surface in a situation of high rank. The examples chosen are resolutions of quartics in P ³ having 14 singularities of type A ₁. Our computations show that the method of R. van Luijk works well when sufficiently large primes are used.     

Billiards and Teichmüller curves on Hilbert modular surfaces

This paper exhibits an infinite collection of algebraic curves isometrically embedded in the moduli space of Riemann surfaces of genus two. These Teichmüller curves lie on Hilbert modular surfaces parameterizing Abelian varieties with real multiplication. Explicit examples, constructed from L-shaped polygons, give billiard tables with optimal dynamical properties.

     

The quartic characters of certain quadratic units

Criteria are obtained for the quartic residue character of the fundamental unit of the real quadratic field $\text{Q((2q)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$, where q is prime and either q ≡ 7(mod 8), or q ≡ 1(mod 8) and X² ? 2qY² = ?2 is solvable in integers X and Y.     

Enriques diagrams and log-canonical thresholds of curves on smooth surfaces

It is shown that, on a smooth surface, the log-canonical threshold of a curve with an isolated singularity is computed by the term ideal of the curve in a suitable system of local parameters at the singularity. The proof uses the Enriques diagram of the singularity and shows that the log-canonical threshold depends only on a non-degenerate path of that diagram.     

The Hilbert function of a curve lying on a smooth cubic surface

Summary Let C be any reduced and irreducible curve, lying on a smooth cubic surface S P ³. In this paper we determine the Hilbert function of C. Moreover we characterize some kinds of curves on S: the arithmetically Cohen-Macaulay curves, the maximal rank curves and the extremal ones.Work done with financial support of M.P.I., while the author was a member of C.N.R.     

Base point free pencils of small degree on certain smooth curves

We give sufficient conditions on numbers d and m such that a linear system of degree m on the normalization C of a plane curve [`(C)]\overline {C} of degree d which is in a certain sense not too singular is in the natural way induced by either a pencil of lines or a pencil of conics in the plane. Those results generalize results on nodal and cuspidal plane curves and seem to complement the recent results of [2]. We present a new approach via the geometry of curves in \Bbb P₁×\Bbb P₂{\Bbb P}_1\times {\Bbb P}_2.     

Bitangents of tropical plane quartic curves

Mathematische Zeitschrift - We study smooth tropical plane quartic curves and show that they satisfy certain properties analogous to (but also different from) smooth plane quartics in algebraic...     

2-Descent on elliptic curves and rational points on certain Kummer surfaces

The first part of this paper further refines the methodology for 2-descents on elliptic curves with rational 2-division points which was introduced in [J.-L. Colliot-Thélène, A.N. Skorobogatov, Peter Swinnerton-Dyer, Hasse principle for pencils of curves of genus one whose Jacobians have rational 2-division points, Invent. Math. 134 (1998) 579-650]. To describe the rest, let E⁽¹⁾ and E⁽²⁾ be elliptic curves, D⁽¹⁾ and D⁽²⁾ their respective 2-coverings, and X be the Kummer surface attached to D⁽¹⁾×D⁽²⁾. In the appendix we study the Brauer-Manin obstruction to the existence of rational points on X. In the second part of the paper, in which we further assume that the two elliptic curves have all their 2-division points rational, we obtain sufficient conditions for X to contain rational points; and we consider how these conditions are related to Brauer-Manin obstructions. This second part depends on the hypothesis that the relevent Tate-Shafarevich group is finite, but it does not require Schinzel's Hypothesis.