On quartics with three-divisible sets of cusps

 We study the geometry and codes of quartic surfaces with many cusps. We apply Gr?bner bases to find examples of various configurations of cusps on quartics. Received: 10 May 2002 / Revised version: 11 November 2002 Published online: 3 March 2003 Permanent address: Institute of Mathematics, Jagiellonian University, ul. Reymonta 4, 30-059 Kraków, Poland. e-mail: rams@mi.uni-erlangen.de, rams@im.uj.edu.pl Research partially supported by the Schwerpunktprogramm ``Global methods in complex geometry' of the Deutsche Forschungsgemeinschaft, and by EAGER. Mathematics Subject Classification (2000): 14J25, 14J17     

Remez-type inequality on sets with cusps

A Remez-type inequality is proved for a large family of sets with cusps in R^N${\mathbb{R}}^N$, including compact, fat and semialgebraic (or subanalytic) sets.     

On convex projective manifolds and cusps

This study of properly or strictly convex real projective manifolds introduces notions of parabolic, horosphere and cusp. Results include a Margulis lemma and in the strictly convex case a thick–thin decomposition. Finite volume cusps are shown to be projectively equivalent to cusps of hyperbolic manifolds. This is proved using a characterization of ellipsoids in projective space.     

Ricci flow on surfaces with cusps

We consider the normalized Ricci flow ? _t g = (ρ ? R)g with initial condition a complete metric g ₀ on an open surface M where M is conformal to a punctured compact Riemann surface and g ₀ has ends which are asymptotic to hyperbolic cusps. We prove that when χ(M) < 0 and ρ < 0, the flow g(t) converges exponentially to the unique complete metric of constant Gauss curvature ρ/2 in the conformal class.     

Finite projective ordered sets

We show that quasiprojectivity and projectivity are equivalent properties for finite ordered sets of more than two elements.     

Large cardinals and projective sets

We investigate measure and category in the projective hierarchie in the presence of large cardinals. Assuming a measurable larger than Woodin cardinals we construct a model where every -set is measurable, but some -set does not have Baire property. Moreover, from the same assumption plus a precipitous ideal on we show how a model can be forced where every set is measurable and has Baire property. Received October 12, 1994     

Two theorems about projective sets

In this paper we prove two (rather unrelated) theorems about projective sets. The first one asserts that subsets of ℵ₁ which are in the codes are constructible; thus it extends the familiar theorem of Shoenfield that subsets of ω are constructible. The second is concerned with largest countable sets and establishes their existence under the hypothesis of Projective Determinacy and the assumption that there exist only countably many ordinal definable reals. Y. N. Moschovakis is a Sloan Foundation Fellow. During the preparation of this paper, both authors were partially supported by NSF Grant GP-27964.     

Homotopy classification of projective convex sets

A subset of projective space is called convex if its intersection with every line is connected. The complement of a projective convex set is again convex. We prove that for any projective convex set there exists a pair of complementary projective subspaces, one contained in the convex set and the other in its complement. This yields their classification up to homotopy.     

Condensations of projective sets onto compacta

For a coanalytic-complete or -complete subspace of a Polish space we prove that there exists a continuous bijection of onto the Hilbert cube . This extends results of Pytkeev. As an application of our main theorem we give an answer to some questions of Arkhangelskii and Christensen.

Under the assumption of Projective Determinacy we also give some generalizations of these results to higher projective classes.

     

Planar Riemann surfaces with uniformly distributed cusps: parabolicity and hyperbolicity

We consider a planar Riemann surface R made of a non‐compact simply connected plane domain from which an infinite discrete set of points is removed. We give several conditions for the collars of the cusps in R caused by these points to be uniformly distributed in R in terms of Euclidean geometry. Then we associate a graph G with R by taking the Voronoi diagram for the uniformly distributed cusps and show that G represents certain geometric and analytic properties of R .     

Linear sets on the projective line with complementary weights

Linear sets on the projective line have attracted a lot of attention because of their link with blocking sets, KM-arcs and rank-metric codes. In this paper, we study linear sets having two points of complementary weight, that is, with two points for which the sum of their weights equals the rank of the linear set. As a special case, we study those linear sets having exactly two points of weight greater than one, by showing new examples and studying their equivalence issue. Also, we determine some linearized polynomials defining the linear sets recently introduced by Jena and Van de Voorde [30].     

On the asymptotic behaviour of capillary surfaces in cusps

Concus and Finn [2] discovered that capillary surfaces rise to infinity in corners with sufficiently small opening angle. They also found the leading term of an asymptotic expansion. Miersemann [5] improved this result to obtain a complete asymptotic expansion. In the present paper we will apply the methods of the above authors to discuss asymptotic behaviour of capillarities in cusps, which is a corner with opening angle 0. A large variety of asymptotic formulas will be provided. The general comparison theorem from Concus and Finn will play an important role in the proofs.     

Regular sets on the projective line

We show that if G is the group PL(2,q)(for q a prime-power) acting on the points of the projective line in the usual way, then for q>27 there is a set of 5 points such that no non-trivial element of G fixes .     

Linear sets in finite projective spaces

In this paper linear sets of finite projective spaces are studied and the “dual” of a linear set is introduced. Also, some applications of the theory of linear sets are investigated: blocking sets in Desarguesian planes, maximum scattered linear sets, translation ovoids of the Cayley Hexagon, translation ovoids of orthogonal polar spaces and finite semifields. Besides “old” results, new ones are proven and some open questions are discussed.     

On complex projective surfaces with trigonal hyperplane sections

Let S be a complex projective surface endowed with an ample and spanned line bundle L. Assume that (S,L) does not belong to some special classes and that c_l(L)²10. We prove that(K_SL)·K_S–3 and |L| contains a trigonal curve (of genus4) iff either (S,L) is a rational surface ruled by cubics, or the g¹ ₃ of C is cut out by |K_S ^–1|. This result applies to surface having a hyperplane section which is a trigonal curve.Partially supported by the M.P.I. of the Italian Government