Algorithms for Del Pezzo Surfaces of Degree 5 (Construction, Parametrization)

It is well known that every Del Pezzo surface of degree 5 defined over k is parametrizable over k. In this paper we give an efficient construction for parametrizing, as well as algorithms for constructing examples in every isomorphism class and for deciding equivalence.Comment: 15 page

Shear coordinate description of the quantised versal unfolding of D_4 singularity

In this paper by using Teichmuller theory of a sphere with four holes/orbifold points, we obtain a system of flat coordinates on the general affine cubic surface having a D_4 singularity at the origin. We show that the Goldman bracket on the geodesic functions on the four-holed/orbifold sphere coincides with the Etingof-Ginzburg Poisson bracket on the affine D_4 cubic. We prove that this bracket is the image under the Riemann-Hilbert map of the Poisson Lie bracket on the direct sum of three copies of sl_2. We realise the action of the mapping class group by the action of the braid group on the geodesic functions . This action coincides with the procedure of analytic continuation of solutions of the sixth Painlev\'e equation. Finally, we produce the explicit quantisation of the Goldman bracket on the geodesic functions on the four-holed/orbifold sphere and of the braid group action.Comment: 14 pages, 2 picture

Noncommutative del Pezzo surfaces and Calabi-Yau algebras

The hypersurface in a 3-dimensional vector space with an isolated quasi-homogeneous elliptic singularity of type E_r,r=6,7,8, has a natural Poisson structure. We show that the family of del Pezzo surfaces of the corresponding type E_r provides a semiuniversal Poisson deformation of that Poisson structure. We also construct a deformation-quantization of the coordinate ring of such a del Pezzo surface. To this end, we first deform the polynomial algebra C[x,y,z] to a noncommutative algebra with generators x,y,z and the following 3 relations (where [u,v]_t = uv- t.vu): [x,y]_t=F_1(z), [y,z]_t=F_2(x), [z,x]_t=F_3(y). This gives a family of Calabi-Yau algebras A(F) parametrized by a complex number t and a triple F=(F_1,F_2,F_3), of polynomials in one variable of specifically chosen degrees. Our quantization of the coordinate ring of a del Pezzo surface is provided by noncommutative algebras of the form A(F)/(g) where (g) stands for the ideal of A(F) generated by a central element g, which generates the center of the algebra A(F) if F is generic enough.Comment: The statement and proof of Theorem 2.4.1 corrected, Introduction expanded, several misprints fixe

Rational points of bounded height on general conic bundle surfaces

A conjecture of Manin predicts the asymptotic distribution of rational points of bounded height on Fano varieties. In this paper we use conic bundles to obtain correct lower bounds or a wide class of surfaces over number fields for which the conjecture is still far from being proved. For example, we obtain the conjectured lower bound of Manin's conjecture for any del Pezzo surface whose Picard rank is sufficiently large, or for arbitrary del Pezzo surfaces after possibly an extension of the ground field of small degree.Comment: 35 pages; final versio

Brauer-Manin obstruction for Erd\H{o}s-Straus surfaces

We study the failure of the integral Hasse principle and strong approximation for the Erd\H{o}s-Straus conjecture using the Brauer-Manin obstruction.Comment: 17 page

Rational points on del Pezzo surfaces of degree four

We study the distribution of the Brauer group and the frequency of the Brauer--Manin obstruction to the Hasse principle and weak approximation in a family of smooth del Pezzo surfaces of degree four over the rationals.Comment: 26 page

Cubic and quartic transformations of the sixth Painleve equation in terms of Riemann-Hilbert correspondence

A starting point of this paper is a classification of quadratic polynomial transformations of the monodromy manifold for the 2x2 isomonodromic Fuchsian systems associated to the Painleve VI equation. Up to birational automorphisms of the monodromy manifold, we find three transformations. Two of them are identified as the action of known quadratic or quartic transformations of the Painleve VI equation. The third transformation of the monodromy manifold gives a new transformation of degree 3 of Picard's solutions of Painleve VI.Comment: Added: classification of quadratic transformations of the Monodromy manifold; new cubic (and quartic) transformations for Picard's case. 26 Pages, 3 figure