K 6-Minors in Triangulations on the Nonorientable Surface of Genus 3

In this paper, we shall give a constructive characterization of triangulations on the nonorientable surface of genus 3 without K ₆-minors. Our characterization implies that every 5-connected triangulation and every 4-representative triangulation on the surface has a K ₆-minor.     

Irreducible Triangulations of Surfaces with Boundary

A triangulation of a surface is irreducible if no edge can be contracted to produce a triangulation of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of vertices of an irreducible triangulation of a (possibly non-orientable) surface of genus g ≥ 0 with b ≥ 0 boundary components is O(g + b). So far, the result was known only for surfaces without boundary (b = 0). While our technique yields a worse constant in the O(.) notation, the present proof is elementary, and simpler than the previous ones in the case of surfaces without boundary.     

Optimal 1-planar graphs which triangulate other surfaces

We show that, for any given non-spherical orientable closed surface F², there exists an optimal 1-planar graph which can be embedded on F² as a triangulation. On the other hand, we prove that there does not exist any such graph for the nonorientable closed surfaces of genus at most 3.     

The locally 2-arc transitive graphs admitting an almost simple group of Suzuki type

In this paper, seven families of vertex-intransitive locally (G,2)-arc transitive graphs are constructed, where Sz(q)?G?Aut(Sz(q)), q=²2k+1 for some k∈N. It is then shown that for any graph Γ in one of these families, Sz(q)?Aut(Γ)?Aut(Sz(q)) and that the only locally 2-arc transitive graphs admitting an almost simple group of Suzuki type whose vertices all have valency at least three are (i) graphs in these seven families, (ii) (vertex transitive) 2-arc transitive graphs admitting an almost simple group of Suzuki type, or (iii) double covers of the graphs in (ii). Since the graphs in (ii) have been classified by Fang and Praeger (1999) [6], this completes the classification of locally 2-arc transitive graphs admitting a Suzuki simple group     

Maximal green sequences for cluster algebras associated with the <Emphasis Type="Italic">n</Emphasis>-torus with arbitrary punctures

It is well known that any triangulation of a marked surface produces a quiver. In this paper we will provide a triangulation for orientable surfaces of genus n with an arbitrary number interior marked points (called punctures) whose corresponding quiver has a maximal green sequence.     

Symplectic Lefschetz fibrations of curves as gonal coverings

We show that after a finite base change every symplectic Lefschetz fibration ${f \colon X \rightarrow B}$ of genus g >  3 curves over a closed oriented surface becomes a finite covering of degree ${\frac{g}{2} + 1}$ or ${\frac{g}{2} + \frac{3}{2}}$ of a family of spheres over a Riemann surface, with a branch locus admitting complex algebraic curves as local models. In the case of fibers of genus 4, it is shown that after a 2:1 base change the family admits a trigonal covering to a symplectic ruled surface, with symplectic branch locus.     

Computational Complexity of the Vertex Cover Problem in the Class of Planar Triangulations

We study the computational complexity of the vertex cover problem in the class of planar graphs (planar triangulations) admitting a plane representation whose faces are triangles. It is shown that the problem is strongly NP-hard in the class of 4-connected planar triangulations in which the degrees of vertices are of order O(log n), where n is the number of vertices, and in the class of plane 4-connected Delaunay triangulations based on the Minkowski triangular distance. A pair of vertices in such a triangulation is adjacent if and only if there is an equilateral triangle ?(p, λ) with p ∈ R² and λ > 0 whose interior does not contain triangulation vertices and whose boundary contains this pair of vertices and only it, where ?(p, λ) = p + λ? = {x ∈ R²: x = p + λa, a ∈ ?}; here ? is the equilateral triangle with unit sides such that its barycenter is the origin and one of the vertices belongs to the negative y-axis. Keywords: computational complexity, Delaunay triangulation, Delaunay TD-triangulation.     

Some new 3-designs from PSL(2, q) with q ≡ 1 (mod4)

We determine the sizes of orbits from the action of subgroups of PSL(2,q) on projective line X = GF(q) ∪ {∞} with q a prime power and congruent to 1 modulo 4.As an example of its application,we construct some new families of simple 3-designs admitting PSL(2,q) as automorphism group.     

Le famiglie misurabili di iperboli equilatere del piano

We determine the subgroups of the group of general similarity transformations of the planeR ². This result then allows us to classify the measurable families (following the definition of M. I. Stoka [3]) of non degenerate equilater hyperbolas ofR ². At the same time we give an example of a family of varieties not admitting any measure.     

Families of fixed-point cellular rotations

A cellular rotation is a pseudofree cellular automorphism, with no non-fixed pseudofixed points, of a graph embedded in an orientable surface. A family of cellular rotations is a collection of cellular rotations having one embedding of each genus above some fixed minimum genus, all sharing the same quotient embedding and, in an appropriate sense, the same voltage-assignment data. We provide a complete catalog of all families of cellular rotations having at least one fixed point, and provide preliminary results regarding families of cellular rotations having no fixed points.     

Space models of degree g+1 of curves of genus g

It is the aim of this note to determine all smooth curves of genus g admitting a birational space model of degree g + 1 whose plane sections define a complete linear series. And we add criteria when there is a smooth such model.     

Propagation of boundary CR foliations and Morera type theorems for manifolds with attached analytic discs

We prove that homologically nontrivial generic smooth (2n−1)-parameter families of analytic discs in Cn, n?2, attached by their boundaries to a CR-manifold Ω, test CR-functions in the following sense: if a smooth function on Ω analytically extends into any analytic discs from the family, then the function satisfies tangential CR-equations on Ω. In particular, we give an answer (Theorem 1) to the following long standing open question, so called strip-problem, earlier solved only for special families (mainly for circles): given a smooth one-parameter family of Jordan curves in the plane and a function f admitting holomorphic extension inside each curve, must f be holomorphic on the union of the curves? We prove, for real-analytic functions and arbitrary generic real-analytic families of curves, that the answer is “yes,” if no point is surrounded by all curves from the family. The latter condition is essential. We generalize this result to characterization of complex curves in C² as real 2-manifolds admitting nontrivial families of attached analytic discs (Theorem 4). The main result implies fairly general Morera type characterization of CR-functions on hypersurfaces in C² in terms of holomorphic extensions into three-parameter families of attached analytic discs (Theorem 2). One of the applications is confirming, in real-analytic category, the Globevnik-Stout conjecture (Theorem 3) on boundary values of holomorphic functions. It is proved that a smooth function on the boundary of a smooth strictly convex domain in Cn extends holomorphically inside the domain if it extends holomorphically into complex lines tangent to a given strictly convex subdomain. The proofs are based on a universal approach, namely, on the reduction to a problem of propagation, from the boundary to the interior, of degeneracy of CR-foliations of solid torus type manifolds (Theorem 2.2).     

Fixed point free involutions on Riemann surfaces

In this paper, involutions without fixed points on hyperbolic closed Riemann surface are discussed. For an orientable surface X of even genus with an arbitrary Riemannian metric d admitting an involution τ, it is known that min _p∈X d(p, τ(p)) is bounded by a constant which depends on the area of X. The corresponding claim is proved to be false in odd genus, and the optimal constant for hyperbolic Riemann surfaces is calculated in genus 2. The author was supported in part by the Swiss National Science Foundation grants 20-68181.02 and PBEL2-106180.     

An Arakelov inequality in characteristic p and upper bound of p-rank zero locus

In this paper we show an Arakelov inequality for semi-stable families of algebraic curves of genus g?1 over characteristic p with nontrivial Kodaira-Spencer maps. We apply this inequality to obtain an upper bound of the number of algebraic curves of p-rank zero in a semi-stable family over characteristic p with nontrivial Kodaira-Spencer map in terms of the genus of a general closed fiber, the genus of the base curve and the number of singular fibres. The parallel results for smooth families of Abelian varieties over k with W₂-lifting assumption are also obtained.     

On the moduli of surfaces admitting genus two fibrations over elliptic curves

In this paper, we study the structure, deformations and the moduli spaces of complex projective surfaces admitting genus two fibrations over elliptic curves. We observe that a surface admitting a smooth fibration as above is elliptic, and we employ results on the moduli of polarized elliptic surfaces to construct moduli spaces of these smooth fibrations. In the case of nonsmooth fibrations, we relate the moduli spaces to the Hurwitz schemes of morphisms of degree n from elliptic curves to the modular curve X(d), d ≥ 3. Ultimately, we show that the moduli spaces in the nonsmooth case are fiber spaces over the affine line with fibers determined by the components of . Received: 30 August 2006     

Tautological rings of stable map spaces

We find a set of generators and relations for the system of extended tautological rings associated to the moduli spaces of stable maps in genus zero, admitting a simple geometrical interpretation. In particular, when the target is Pn, these give a complete presentation for the cohomology and Chow rings in the cases with/without marked points.     

Closed transversals and the genus of closed leaves in foliated 3-manifolds

The classical local theory of integrable 2-plane fields in 3-space leads to interesting qualitative questions about the global properties of solutions surface (i.e., leaves of a foliation) on 3-manifolds. It is now known that foliations admitting a closed leaf of suitably high genus abound on all closed or orientable 3-manifolds that are not rational homology spheres (S. Goodman, Proc. Nat. Acad. Sci. U.S.A.71 (1974), 4414–4415), and this leads to natural questions about the “positions” of such leaves relative to the rest of the foliation. One such question, suggested by Goodman's theorem on closed transversals (S. Goodman, ibid.), is considered here.     

Note on irreducible triangulations of surfaces

In this paper, we shall show that an irreducible triangulation of a closed surface F² has at most cg vertices, where g stands for a genus of F² and c a constant. © 1995, John Wiley & Sons, Inc.     

Degree-regular triangulations of torus and Klein bottle

A triangulation of a connected closed surface is called weakly regular if the action of its automorphism group on its vertices is transitive. A triangulation of a connected closed surface is called degree-regular if each of its vertices have the same degree. Clearly, a weakly regular triangulation is degree-regular. In [8], Lutz has classified all the weakly regular triangulations on at most 15 vertices. In [5], Datta and Nilakantan have classified all the degree-regular triangulations of closed surfaces on at most 11 vertices. In this article, we have proved that any degree-regular triangulation of the torus is weakly regular. We have shown that there exists ann-vertex degree-regular triangulation of the Klein bottle if and only if n is a composite number ≥ 9. We have constructed two distinctn-vertex weakly regular triangulations of the torus for eachn ≥ 12 and a (4m + 2)-vertex weakly regular triangulation of the Klein bottle for eachm ≥ 2. For 12 ≤n ≤ 15, we have classified all then-vertex degree-regular triangulations of the torus and the Klein bottle. There are exactly 19 such triangulations, 12 of which are triangulations of the torus and remaining 7 are triangulations of the Klein bottle. Among the last 7, only one is weakly regular.