Volume 6, Issue 3

Here we investigate the canonical Gaussian map for higher multiple coverings of curves, the case of double coverings being completely understood thanks to previous work by Duflot. In particular, we prove that every smooth curve can be covered with degree not too high by a smooth curve having arbitrarily large genus and surjective canonical Gaussian map. As a consequence, we recover an asymptotic version of a recent result by Ciliberto and Lopez and we address the Arbarello subvariety in the moduli space of curves.

We investigate 4-dimensional (topological locally compact connected) elation Laguerre planes that admit solvable automorphism groups and show that there is, up to isomorphism, precisely one such plane of group dimension 10. This result completes the classification of all 4-dimensional elation Laguerre planes of group dimension 10.

The following result concerning completely regular ovals is proved: Let Π be a projective plane of even order and let 𝒪 be a completely regular oval with nucleus N . Then Π is ( N,N )-transitive. Combining this result with previous results [A. Maschietti, Symplectic translation planes and line ovals. Adv. Geom. 3 (2003), 123–143. MR1967995 (2004c:51008) Zbl 1030.51002] one obtains: A projective plane of even order admits a completely regular oval if and only if the plane is dual to a symplectic translation plane.

Consider the mapping F associating to each point x of a convex surface the set of all points at maximal intrinsic distance from x . We provide a large class of surfaces on which F is single-valued and involutive. Moreover, we show that there are point-symmetric surfaces of revolution with F single-valued but not involutive.

We prove that the elation groups of a certain infinite family of Roman generalized quadrangles are not isomorphic to those of associated flock generalized quadrangles.

In this paper we study the possibility of defining a similarity structure on the torus and the Klein bottle using the combinatorial data of a triangulation. Given a choice of moduli for the triangles of a triangulation of a surface, the problem is to decide whether such moduli are compatible with a global similarity structure on the surface. We study this problem under two dierent viewpoints. From one side we look at the combinatorial data of triangulations, and we develop an algorithmic method, which allows us to reduce the general problem to a simpler one, which is easily solved. From the other side we study the problem more algebraically, looking at the properties of the holonomy, and we give a complete characterization of the choices of moduli defining global similarity structures on the torus (or on the Klein bottle).

A Euclidean t -design, as introduced by Neumaier and Seidel (1988), is a finite set 𝒳 ⊂ ℝ n with a weight function w : 𝒳 → ℝ + for which holds for every polynomial ƒ of total degree at most t ; here R is the set of norms of the points in 𝒳, W r is the total weight of all elements of 𝒳 with norm r , S r is the n -dimensional sphere of radius r centered at the origin, and is the average of ƒ over S r . Neumaier and Seidel (1988), as well as Delsarte and Seidel (1989), also proved a Fisher-type inequality |𝒳| ≥ N ( n , | R |, t ) (assuming that the design is antipodal if t is odd). For fixed n and | R | we have N ( n , | R |, t ) = O ( t n −1 ). This paper contains two main results. First, we provide a recursive construction for Euclidean t -designs in ℝ n . Namely, we show how to use certain Gauss–Jacobi quadrature formulae to ‘lift’ a Euclidean t -design in ℝ n −1 to a Euclidean t -design in ℝ n , preserving both the norm spectrum R and the weight sum W r for each r ∈ R . For fixed n and | R | this construction yields a design of size O ( t n −1 ); however, the coefficient of t n −1 here is significantly greater than it is in N ( n , | R |, t ). A Euclidean design with exactly N ( n , | R |, t ) points is called tight; in both of the above mentioned papers it was conjectured that a tight Euclidean design with t ≥ 4 must be a spherical design, that is, | R | = 1 and w is constant on 𝒳. Bannai and Bannai (2003) disproved this conjecture by exhibiting an example for the parameters, ( n , | R |, t ) = (2,2,4). Here we construct tight Euclidean designs for n = 2 and every t and | R | with | R | ≤ .

A riemannian manifold M with associated Jacobi operators R X (R X Y = R( Y, X ) X ), X in TM , is said to be k -stein, k ≥ 1, if there exists a function μ k on M such that tr(R k X ) = μ k | X | 2 k for all X in TM . We study the k -stein condition on Lie groups of Iwasawa type and in particular in those which are Carnot spaces. We show that a Carnot space which is k -stein for some k > 1 is necessarily a Damek–Ricci space; Damek–Ricci spaces are Einstein and 2-stein and they are not k -stein for any k ≥ 3, unless they are symmetric. Moreover, we show that a harmonic Lie group of Iwasawa type which is 3-stein is a symmetric space of noncompact type and rank one.

We give a classification of smooth surfaces S in the Grassmannian 𝔾(1,3) of lines in ℙ 3 such that the restriction to S of the universal quotient bundle is not simple.

Let ( X , d X ) be a geodesically complete Hadamard space endowed with a Borel-measure μ . Assume that there exists a group Γ of isometries of X which acts totally discontinuously and cocompactly on X and preserves μ. We show that the topological entropy of the geodesic flow on the space of (parametrized) geodesics of the compact quotient Γ\ X is equal to the volume entropy of μ (if X satisfies a certain local uniformity condition). This extends a result of Manning for riemannian manifolds of nonpositive curvature to the singular case. The result in particular holds for Bruhat–Tits buildings, for which we also compute the entropy explicitly.