Intersections of the Hermitian Surface with irreducible Quadrics in even Characteristic

We determine the possible intersection sizes of a Hermitian surface $\mathcal H$ with an irreducible quadric of ${\mathrm PG}(3,q^2)$ sharing at least a tangent plane at a common non-singular point when $q$ is even.Comment: 20 pages; extensively revised and corrected version. This paper extends the results of arXiv:1307.8386 to the case q eve

Intersections of the Hermitian surface with irreducible quadrics in PG(3,q2)PG(3,q^2), qq odd

In $PG(3,q^2)$, with $q$ odd, we determine the possible intersection sizes of a Hermitian surface $\mathcal{H}$ and an irreducible quadric $\mathcal{Q}$ having the same tangent plane $\pi$ at a common point $P\in{\mathcal Q}\cap{\mathcal H}$.Comment: 14 pages; clarified the case q=

Kodierungstheorie

[no abstract available

Plane curves giving rise to blocking sets over finite fields

In recent years, many useful applications of the polynomial method have emerged in finite geometry. Indeed, algebraic curves, especially those defined by R\'edei-type polynomials, are powerful in studying blocking sets. In this paper, we reverse the engine and study when blocking sets can arise from rational points on plane curves over finite fields. We show that irreducible curves of low degree cannot provide blocking sets and prove more refined results for cubic and quartic curves. On the other hand, using tools from number theory, we construct smooth plane curves defined over $\mathbb{F}_p$ of degree at most $4p^{3/4}+1$ whose points form blocking sets.Comment: 25 page

Saturating linear sets of minimal rank

Saturating sets are combinatorial objects in projective spaces over finite fields that have been intensively investigated in the last three decades. They are related to the so-called covering problem of codes in the Hamming metric. In this paper, we consider the recently introduced linear version of such sets, which is, in turn, related to the covering problem in the rank metric. The main questions in this context are how small the rank of a saturating linear set can be and how to construct saturating linear sets of small rank. Recently, Bonini, Borello, and Byrne provided a lower bound on the rank of saturating linear sets in a given projective space, which is shown to be tight in some cases. In this paper, we provide construction of saturating linear sets meeting the lower bound and we develop a link between the saturating property and the scatteredness of linear sets. The last part of the paper is devoted to show some parameters for which the bound is not tight.Comment: 26 page

Parahoric Restriction for GSp(4) and the Inner Cohomology of Siegel Modular Threefolds

For irreducible admissible representations of the group of symplectic similitudes GSp(4,F) of genus two over a p-adic number field F, we obtain the parahoric restriction with respect to an arbitrary parahoric subgroup. That means we determine the action of the Levi quotient on the invariants under the pro-unipotent radical in terms of explicit character values. Especially, we get the parahoric restriction of local endoscopic L-packets in terms of lifting data. The inner cohomology of the Siegel modular variety of genus two with an arbitrary l-adic local system admits an endoscopic and a Saito-Kurokawa part under spectral decomposition. For principal congruence subgroups of squarefree level N they define simultaneous representations of the absolute Galois group and the Hecke action of GSp(4;Z/NZ). We decompose them into irreducible constituents and give explicit character values. As an application, we prove the conjectures of Bergström, Faber and van der Geer on level two

Outer Billiards on Kites

Outer billiards is a simple dynamical system based on a convex planar shape. The Moser-Neumann question, first posed by B.H. Neumann around 1960, asks if there exists a planar shape for which outer billiards has an unbounded orbit. The first half of this monograph proves that outer billiards has an unbounded orbit defined relative to any irrational kite. The second half of the monograph gives a very sharp description of the set of unbounded orbits, both in terms of the dynamics and the Hausdorff dimension. The analysis in both halves reveals a close connection between outer billiards on kites and the modular group, as well as connections to self-similar tilings, polytope exchange maps, Diophantine approximation, and odometers.Comment: 296 pages. Essentially, I have added a "second half" to the previous monograph. Parts I-IV are essentially the same as last posted version. Parts V-VI have the new materia