Arcs in Minkowski planes

The aim of this paper is to study some properties of k-arcs in Minkowski planes focalizing the attention on problems of existence and completness.Work done under the auspicies of G.N.S.A.G.A. supported by 40% grants of M.U.R.S.T.In memoriam Giuseppe Tallini     

Generalized Hyperfocused Arcs in

A generalized hyperfocused arc in is an arc of size k with the property that the secants can be blocked by a set of points not belonging to the arc. We show that if q is a prime and is a generalized hyperfocused arc of size k, then or 4. Interestingly, this problem is also related to the (strong) cylinder conjecture ( 2 , 5 Problem 919), as we point out in the last section.     

Arcs in the plane

Assuming PFA, every uncountable subset E of the plane meets some C¹ arc in an uncountable set. This is not provable from MA(ℵ₁), although in the case that E is analytic, this is a ZFC result. The result is false in ZFC for C² arcs, and the counter-example is a perfect set.     

Rationally 4-periodic biquotients

We give an explicit construction of a maximal torsion-free finite-index subgroup of a certain type of Coxeter group. The subgroup is constructed as the fundamental group of a finite and non-positively curved polygonal complex. First we consider the special case where the universal cover of this polygonal complex is a hyperbolic building, and we construct finite-index embeddings of the fundamental group into certain cocompact lattices of the building. We show that in this special case the fundamental group is an amalgam of surface groups over free groups. We then consider the general case, and construct a finite-index embedding of the fundamental group into the Coxeter group whose Davis complex is the universal cover of the polygonal complex. All of the groups which we embed have minimal index among torsion-free subgroups, and therefore are maximal among torsion-free subgroups.     

A New Approach to Arcs

Some properties of arcs in PG(2,q) are discussed via its cyclic model. A covering number of a set of points in PG(2,q) is the smallest number of lines needed to cover the set -, where the set of points of PG(2,q) is identified with Z . It is proved that the covering number of a conic is either 1 or greater than q/4. Hyperovals with covering number 2 are characterized for q even. Also, a possible method for constructing nonclassical hyperovals having small covering numbers is given.     

Arcs and resolution of singularities

For a certain class of varieties X, including the toric surfaces, we derive a formula for the valuation d_X on the arc space of a smooth ambient space Y, in terms of an embedded resolution of singularities. A simple transformation rule yields a formula for the geometric Poincaré series. Furthermore, we prove that for this class of varieties, the arithmetic and the geometric Poincaré series coincide. We also study the geometric valuation for plane curves.Research Assistant of the Fund for Scientific Research – Flanders (Belgium) (F.W.O.)Mathematics Subject Classification (2000): 14J17, 14E15, 14B20, 14M25, 03C98Acknowledgement The author would like to thank the referee, whose remarks have greatly improved the structure of this paper.     

Rigidity Theorem for Self-Affine Arcs

Doklady Mathematics - It has been known for more than a decade that, if a self-similar arc $$gamma $$ can be shifted along itself by similarity maps that are arbitrarily close to identity, then...     

Arcs in the Hall Planes

Using a transformation tecnique for designs introduced in [1], I construct a class of arcs embeddeable in the Hall plane and in the dual of the Hall plane of order q proving also their completeness in the unital of Grüning. Math. Subj. Class.: 51A35 Non Desarguesian affine and projective planes. 51E22 Blocking sets, ovals, k-arcs.     

Rationally smooth algebraic monoids

A reductive monoid M is called rationally smooth if it has sufficiently mild singularities as a topological space. We characterize this class of monoids in combinatorial terms. We then use our results to calculate the Betti numbers of certain projective, rationally smooth group embeddings using the “monoid BB-decomposition”.     

On Small Complete Arcs and Transitive ‐Invariant Arcs in the Projective Plane

Let q be an odd prime power such that q is a power of 5 or (mod 10). In this case, the projective plane admits a collineation group G isomorphic to the alternating group A₅. Transitive G‐invariant 30‐arcs are shown to exist for every . The completeness is also investigated, and complete 30‐arcs are found for . Surprisingly, they are the smallest known complete arcs in the planes , and . Moreover, computational results are presented for the cases and . New upper bounds on the size of the smallest complete arc are obtained for .     

Arcs fixed byA 5 andA 6

The complete list of thek-arcsK inPG(n, q) fixed by a projective group isomorphic toA ₅ orA ₆, which acts primitively on the points ofK, is presented. This leads to new classes of 10-arcs inPG(n, q), 3 n 5. Our results also show that the non-classical 10-arc inPG(4, 9), discovered by D.G. Glynn [3], belongs to an infinite class of 10-arcs inPG(4, 3^h),h 2, fixed by a projective group isomorphic toA ₆.The first author wishes to thank the Belgian National Fund for Scientific Research for financial supportSenior research assistant of the Belgian National Fund for Scientific ResearchDedicated to Professor M. Scafati Tallini on the occasion of her sixtyfifth birthday     

Complete Arcs in Steiner Triple Systems

A complete arc in a design is a set of elements which contains no block and is maximal with respect to this property. The spectrum of sizes of complete arcs in Steiner triple systems is determined without exception here.     

New Maximal Arcs in Desarguesian Planes

Constructions are described of maximal arcs in Desarguesian projective planes utilizing sets of conics on a common nucleus in PG(2, q). Several new infinite families of maximal arcs in PG(2, q) are presented and a complete enumeration is carried out for Desarguesian planes of order 16, 32, and 64. For each arc we list the order of its stabilizer and the numbers of subarcs it contains. Maximal arcs may be used to construct interesting new partial geometries, 2-weight codes, and resolvable Steiner 2-designs.