Rational curves and rational singularities

 We study rational curves on algebraic varieties, especially on normal affine varieties endowed with a ℂ^*-action. For varieties with an isolated singularity, covered by a family of rational curves with a general member not passing through the singular point, we show that this singularity is rational. In particular, this provides an explanation of classical results due to H. A. Schwartz and G. H. Halphen on polynomial solutions of the generalized Fermat equation. Received: 7 May 2002 / Published online: 16 May 2003 Mathematics Subject Classification (2000): 14J17, 14L30, 13H10     

Rational spectral transformations and orthogonal polynomials

We show that generic rational transformations of the Stieltjes function with polynomial coefficients (ST) can be presented as a finite superposition of four fundamental elementary transforms: Christoffel transform (CT), Geronimus transform (GT) and forward and backward associated transformations A⁺T, A⁻T. It is shown that the Laguerre-Hahn polynomials (LHP) on arbitrary nonuniform lattice are covariant with respect to ST (i.e., ST of a LHP yields another LHP), whereas the semi-classical polynomials are covariant with respect to a subclass of linear ST. Some applications of these results to the theory of the semi-classical polynomials are considered.     

Rational points of algebraic curves

It is proved that the number of k-points of certain curves of genus g>1 is finite.     

A distance on curves modulo rigid transformations

We propose a geometric method for quantifying the difference between parametrized curves in Euclidean space by introducing a distance function on the space of parametrized curves up to rigid transformations (rotations and translations). Given two curves, the distance between them is defined as the infimum of an energy functional which, roughly speaking, measures the extent to which the jet field of the first curve needs to be rotated to match up with the jet field of the second curve. We show that this energy functional attains a global minimum on the appropriate function space, and we derive a set of first-order ODEs for the minimizer.     

Rational curves on K3 surfaces

We show that projective K3 surfaces with odd Picard rank contain infinitely many rational curves. Our proof extends the Bogomolov-Hassett-Tschinkel approach, i.e., uses moduli spaces of stable maps and reduction to positive characteristic.     

Rational connectivity and sections of families over curves

A “pseudosection” of the total space X of a family of varieties over a base variety B is a subvariety of X whose general fiber over B is rationally connected. We prove a theorem which is a converse, in some sense, of the main result of [T. Graber, J. Harris, J. Starr, Families of rationally connected varieties, J. Amer. Math. Soc. 16 (2003) 69-90]: a family of varieties over B has a “pseudosection” if its restriction to each one-parameter subfamily has a “pseudosection” (which, due to [T. Graber, J. Harris, J. Starr, Families of rationally connected varieties, J. Amer. Math. Soc. 16 (2003) 69-90], holds if and only if each one-parameter subfamily has a section). This is used to give a negative answer to a question posed by Serre to Grothendieck: There exists a family of O-acyclic varieties (a family of Enriques surfaces) parametrized by P¹ with no section.     

Rational points near planar curves and Diophantine approximation

In this paper, we establish asymptotic formulae with optimal errors for the number of rational points that are close to a planar curve, which unify and extend the results of Beresnevich–Dickinson–Velani [6] and Vaughan–Velani [22]. Furthermore, we complete the Lebesgue theory of Diophantine approximation on weakly non-degenerate planar curves that was initially developed by Beresnevich–Zorin [5] in the divergence case.     

Elementary transformations of pfaffian representations of plane curves

Let C be a smooth curve in P² given by an equation F=0 of degree d. In this paper we consider elementary transformations of linear pfaffian representations of C. Elementary transformations can be interpreted as actions on a rank 2 vector bundle on C with canonical determinant and no sections, which corresponds to the cokernel of a pfaffian representation. Every two pfaffian representations of C can be bridged by a finite sequence of elementary transformations. Pfaffian representations and elementary transformations are constructed explicitly. For a smooth quartic, applications to Aronhold bundles and theta characteristics are given.     

On singular quadratic complexes,quintic curves and Cremona transformations

Let X be a quadratic complex given by the intersection of two nonsingular quadrics in a projective space of dimension five. Let L be a line contained in X, and π the projection from X to a projective three space with center L. When X is nonsingular the map π is birational and the base locus scheme of π ^?1 is a smooth quintic curve of genus 2. Now assume X is a singular irreducible and reduced quadratic complex and consider the same set up. The purpose of this work is to classify quintic curves arising as the base locus scheme of π ^?1 in the case where π is birational and the Cremona transformations obtained by composing π ^?1 with another projection of the same type.     

Splitting of vector bundles on algebraic curves and elementary transformations

Let Ebe a rank nvector bundle on a smooth projective curve X. It is known that Emay be obtained from a splitted bundle +1≤i≤ L_i;, rank(L_i) = 1, by a finite number of elementary transformations. Here we give upper bounds for their minimal number. If n= 2 this is related to the order of stability of E.     

Rational classes and divisors on curves of genus 2

We describe a method of looking for rational divisor classes on a curve of genus 2. We have an algorithm to decide if a given class of divisors of degree 3 contains a rational divisor. It is known that the shape of the kernel of Cassel’s morphism (X − T) is related to the existence of rational classes of degree 1. Our key tool is the dual Kummer surface.V. G. L. Neumann supported by CNPq, Brazil     

Rational torsion on optimal curves and rank-one quadratic twists

When an elliptic curve E^′/Q of square-free conductor N has a rational point of odd prime order l?N, Dummigan (2005) in [Du] explicitly constructed a rational point of order l on the optimal curve E, isogenous over Q to E^′, under some conditions. In this paper, we show that his construction also works unconditionally. And applying it to Heegner points of elliptic curves, we find a family of elliptic curves E^′/Q such that a positive proportion of quadratic twists of E^′ has (analytic) rank 1. This family includes the infinite family of elliptic curves of the same property in Byeon, Jeon, and Kim (2009) [B-J-K].     

Rational curves on moduli spaces of vector bundles

We identify the spaces Hom_i(ℙ¹,M) fori = 1, 2, whereM is the moduli space of vector bundles of rank 2 and determinant isomorphic to ,x ₀ ∈X, on a compact Riemann surface of genusg ≥ 2.     

Rational nodal curves with no smooth Weierstrass points

Let denote the rational curve with nodes obtained from the Riemann sphere by identifying 0 with and with for , where is a primitive th root of unity. We show that if is even, then has no smooth Weierstrass points, while if is odd, then has smooth Weierstrass points.

     

Rational curves of minimal degree and characterizations of projective spaces

In this paper we investigate complex uniruled varieties X whose rational curves of minimal degree satisfy a special property. Namely, we assume that the tangent directions to such curves at a general point x ∈ X form a linear subspace of T_xX. As a first application of our main result, we give a unified geometric proof of Mori's, Wahl's, Campana-Peternell's and Andreatta-Wiśniewski's characterizations of . We also give a characterization of products of projective spaces in terms of the geometry of their families of rational curves of minimal degree.