The cubic surfaces with twenty-seven lines over finite fields

In this thesis, we classify the cubic surfaces with twenty-seven lines in three dimensional projective space over small finite fields. We use the Clebsch map to construct cubic surfaces with twenty-seven lines in PG(3; q) from 6-arcs not on a conic in PG(2; q). We introduce computational and geometrical procedures for the classification of cubic surfaces over the finite field Fq. The performance of the algorithms is illustrated by the example of cubic surfaces over F13, F17 and F19

Markoff Triples and Strong Approximation

We investigate the transitivity properties of the group of morphisms generated by Vieta involutions on the solutions in congruences to the Markoff equation as well as to other Markoff type affine cubic surfaces. These are dictated by the finite $\bar{\mathbb Q}$ orbits of these actions and these can be determined effectively. The results are applied to give forms of strong approximation for integer points, and to sieving, on these surface

Hilbert's fourteenth problem over finite fields, and a conjecture on the cone of curves

We give examples over arbitrary fields of rings of invariants that are not finitely generated. The group involved can be as small as three copies of the additive group, as in Mukai's examples over the complex numbers. The failure of finite generation comes from certain elliptic fibrations or abelian surface fibrations having positive Mordell-Weil rank. Our work suggests a generalization of the Morrison-Kawamata cone conjecture from Calabi-Yau varieties to klt Calabi-Yau pairs. We prove the conjecture in dimension 2 in the case of minimal rational elliptic surfaces.Comment: 26 pages. To appear in Compositio Mathematic

Massless Higher Spins and Holography

We treat free large N superconformal field theories as holographic duals of higher spin (HS) gauge theories expanded around AdS spacetime with radius R. The HS gauge theories contain massless and light massive AdS fields. The HS current correlators are written in a crossing symmetric form including only exchange of other HS currents. This and other arguments point to the existence of a consistent truncation to massless HS fields. A survey of massless HS theories with 32 supersymmetries in D=4,5,7 (where the 7D results are new) is given and the corresponding composite operators are discussed. In the case of AdS_4, the cubic couplings of a minimal bosonic massless HS gauge theory are described. We examine high energy/small tension limits giving rise to massless HS fields in the Type IIB string on AdS_5 x S^5 and M theory on AdS_{4/7} x S^{7/4}. We discuss breaking of HS symmetries to the symmetries of ordinary supergravity, and a particularly natural Higgs mechanism in AdS_5 x S^5 and AdS_4 x S^7 where the HS symmetry is broken by finite g_{YM}. In AdS_5 x S^5 it is shown that the supermultiplets of the leading Regge trajectory cross over into the massless HS spectrum. We propose that g_{YM}=0 corresponds to a critical string tension of order 1/R^2 and a finite string coupling of order 1/N. In AdS_7 x S^4 we give a rotating membrane solution coupling to the massless HS currents, and describe these as limits of Wilson surfaces in the A_{N-1}(2,0) SCFT, expandable in terms of operators with anomalous dimensions that are asymptotically small for large spin. The minimal energy configurations have semi-classical energy E=s for all s and the geometry of infinitely stretched strings with energy and spin density concentrated at the endpoints.Comment: 77 pages, latex, minor corrections to eqs 4.26-30, a refined discussion of long strings in Sec

Construction of Rational Surfaces Yielding Good Codes

In the present article, we consider Algebraic Geometry codes on some rational surfaces. The estimate of the minimum distance is translated into a point counting problem on plane curves. This problem is solved by applying the upper bound "\`a la Weil" of Aubry and Perret together with the bound of Homma and Kim for plane curves. The parameters of several codes from rational surfaces are computed. Among them, the codes defined by the evaluation of forms of degree 3 on an elliptic quadric are studied. As far as we know, such codes have never been treated before. Two other rational surfaces are studied and very good codes are found on them. In particular, a [57,12,34] code over $\mathbf{F}_7$ and a [91,18,53] code over $\mathbf{F}_9$ are discovered, these codes beat the best known codes up to now.Comment: 20 pages, 7 figure