Volume 6, Issue 1

It has been proved (see [S. J. Kim, On the existence of Weierstrass gap sequences on trigonal curves. J. Pure Appl. Algebra 63 (1990), 171-180. MR1043748 (91b:14036) Zbl 0712.14019] and [M. Brundu, G. Sacchiero, On the varieties parametrizing trigonal curves with assigned Weierstrass points. Comm. Algebra 26 (1998), 3291-3312. MR1641619 (99g:14040) Zbl 0937.14016]. ) that a Weierstrass point on a trigonal curve, which is not a ramification point for the morphism induced by a trigonal series, has a gap sequence G P of the form G P = (1,2, …, r , r + 1 + α, …, g + α) for some integers r and α satisfying ≤ r ≤ g − 1 and 1 ≤ α ≤ 2 r + 1 − g . Such points will be called one jump Weierstrass points of type ( r ,α). It was further proved [S. J. Kim, On the existence of Weierstrass gap sequences on trigonal curves. J. Pure Appl. Algebra 63 (1990), 171-180. MR1043748 (91b:14036) Zbl 0712.14019] that any one jump numerical sequence of type ( r , α), with r and α in the range above, is the Weierstrass gap sequence of a trigonal curve. Here we prove that the property of having an extremal, in some sense, one jump Weierstrass point characterizes trigonal curves. More precisely, we show that if α belongs to the range α 4 < α ≤ 2 r + 1 − g for a suitable α 4 , any curve with a Weierstrass point of type ( r ,α) is a triple covering of a smooth curve of genus p with , and that there exist examples of such coverings. Therefore when 2 r + 1 − g − α < 3, such a curve is indeed trigonal. As a consequence, any fourgonal curve of genus g ≥ 10 having a one jump Weierstrass point satisfies α ≤ α 4 with few exceptions. Finally, we exhibit examples of fourgonal curves with a Weierstrass point of type ( r , α) with ≤ r ≤ and 1 ≤ α ≤ α 4 .

We characterize the finite projective planes П of order n with a collineation group G acting 2-transitively on a point orbit of length n – 3. Aside from the trivial cases, the plane П must have order 9 or 16 and G is isomorphic to PSL (2,5) or AGL (1,13), respectively.

In this paper we show that there are several other structures that arise from the functions associated with the maximal arcs of Mathon type. So it is shown that maximal arcs of Mathon type are equivalent to additive partial flocks of the quadratic cone in PG(3, q ) and to additive partial q -clans. Further they yield partial ovoids of Q + (5, q ) partial spreads of lines of PG(3, q ), translation k -arcs of PG(2, q ) and m -ovoids of a certain classical generalized quadrangle.

An n -dimensional Riemannian manifold is called k -harmonic for some integer k , 1 ≤ k ≤ n - 1, if the k -th elementary symmetric functions of the principal curvatures of small geodesic spheres are radial functions. We prove that k -harmonic manifolds are necessarily 2-stein and show that locally symmetric manifolds which are k -harmonic for one k , are k -harmonic for all k . We then establish some results relating the harmonic and k -harmonic conditions for the class of non-compact harmonic non-symmetric spaces constructed by Damek and Ricci. We also discuss other notions of k -harmonicity and the problem of their equivalence.

We provide a characterization of the radii minimal projections of polytopes onto j -dimensional subspaces in Euclidean space . Applied to simplices this characterization allows to reduce the computation of an outer radius to a computation in the circumscribing case or to the computation of an outer radius of a lower-dimensional simplex. In the second part of the paper, we use this characterization to determine the sequence of outer ( n – 1)-radii of regular simplices (which are the radii of smallest enclosing cylinders). This settles a question which arose from an error in a paper by Weißbach (1983). In the proof, we first reduce the problem to a constrained optimization problem of symmetric polynomials and then to an optimization problem in a fixed number of variables with additional integer constraints.

Let f and g be elliptic Möbius transformations of respective orders m ≥3, n ≥2, and let s , φ be the hyperbolic distance and angle between their axes. In this paper we prove that for the group 〈 f , g 〉 is discrete, nonelementary and isomorphic to the free product of cyclic groups. Several examples are given showing the effectiveness of the above estimate for cosh s .

The bad locus of a free linear system ℒ on a normal complex projective variety X is defined as the set B( ℒ ) ⊆ X of points that are not contained in any irreducible and reduced member of ℒ. In this paper we provide a geometric description of such locus in terms of the morphism defined by ℒ. In particular, assume that dim X ≥ 2 and ℒ is the complete linear system associated to an ample and spanned line bundle. It is known that in this case B(ℒ) is empty unless X is a surface. Then we prove that, when the latter occurs, B(ℒ) is not empty if and only if ℒ defines a morphism onto a two dimensional cone, in which case B(ℒ) is the inverse image of the vertex of the cone.

CR singularities of real threefolds in ℂ 4 are classified by using holomorphic coordinate changes to transform the quadratic part of the real defining equations into one of a list of normal forms. In the non-degenerate case, it is shown that a real analytic manifold near a CR singular point is formally equivalent to a real algebraic model. Some degenerate cases also have this property.

We say that a Tychonoff space X has computable z -radicals if for all ideals of C ( X ), the smallest z -ideal containing is generated as an ideal by all the s ○ f , where f is in and s is a continuous function ℝ → ℝ with s -1 (0) = {0}. We show that every cozero set of a compact space has computable z -radicals and that a subset X of ℝ n has computable z -radicals if and only if X is locally closed.