For each odd prime p, we see $BP^{odd}(Bp_+^{1+4})=0$ where $p_+^{1+4}$ is the extraspecial p-group of order p⁵ and of exponent p.

We give a method for characterizing base points of polar curves on a normal surface of type zⁿ = f(x,y), after a resolution of singularities by Jung's method.

Let W_g → W_z be a ramified p-sheeted covering of Riemann surfaces of genus g and z, (z > 0) where p is an odd prime. Assume that the Galois group is either dihedral or cyclic. Assume, moreover, that the covering is full; that is, there us an integral divisor E, of degree 2r on W_z which lifts to be canonical on W_g. Then g = rp + 1, where r ≥ 1. Clearly, W_g admits 2^2z half-canonical linear series of dimension at least r − z arising from divisors on W_z whose double is E. Theorem 1 Of these 2^2z half-canonical linear series u_z (= 2^z−1 (2^z−1)) have dimension at least r − z + 1. Theorem 2 Let W_g (g = 3r + 1, r ≥ 3) admit four half canonical linear series, three of dimension r − 1, and one of dimension r, whose sum is bi-canonical, where the half-canonical linear series of dimension r is unique. Then W_g is a full elliptic-trigonal Riemann surface. (This characterizes the cases z = 1, p = 3, g ≥ 10)

We show the existence of special curves which are ramified coverings of irrational curves with surjective Wahl maps. We also show the failure of a key property of Gaussian maps in positive characteristic, i.e. the Gherardelli's lower bound for the rank of a Gaussian map.

Let f be a transcendental meromorphic function and R be a rational function, $R\not\equiv 0$, and let k be a positive integer. In this paper, we obtain some results concerning the zeros of f^(k)−R, which generalize and improve related results of Wang-Fang and Bergweiler-Pang.

The object of this paper is to give an estimation of the Łojasiewicz exponent of the gradient of a holomorphic function under Kouchnirenko's nondegeneracy condition, using information from the Newton polyhedron.

A compact Riemann surface of genus g ≥ 2 is said to be extremal if it admits an extremal disk, a disk of the maximal radius determined by g. If g = 2 or g ≥ 4, it is known that how many extremal disks an extremal surface of genus g can admit. In the present paper we deal with the case of g = 3. Considering the side-pairing patterns of the fundamental polygons, we show that extremal surfaces of genus 3 admit at most two extremal disks and that 16 surfaces admit exactly two. Also we describe the group of automorphisms and hyperelliptic surfaces.

A new class of semi-Riemannian and lightlike manifolds (including globally null) is constructed by using a hypersurface of an orientable Riemannian manifold, endowed with the second fundamental form instead of a metric induced from the ambient space. We show the existence (or non-existence) of harmonic tensor fields and harmonic maps and extend to the semi-Riemannian and lightlike case a result of Chen-Nagano [4]. Then we deal with general lightlike submanifolds immersed in a semi-Riemannian manifold and propose a definition of minimal lightlike submanifolds, which generalize the one given in [7] in the Minkowski space ${\bf R}^4_1$. Several examples are given throughout.

Building on ideas of R. Mizner, [17] - [18], and C. Laurent-Thiébaut, [14], we study the CR geometry of real orientable hypersurfaces of a Sasakian manifold. These are shown to be CR manifolds of CR codimension two and to possess a canonical connection D (parallelizing the maximally complex distribution) similar to the Tanaka-Webster connection (cf. [21]) in pseudohermitian geometry. Examples arise as circle subbundles $S^1 \to N \stackrel{\pi}{\rightarrow} M$, of the Hopf fibration, over a real hypersurface M in the complex projective space. Exploiting the relationship between the second fundamental forms of the immersions N → S²ⁿ⁺¹ and M → CPⁿ and a horizontal lifting technique we prove a CR extension theorem for CR functions on N. Under suitable assumptions [$\mathrm{Ric}_D(Z,\overline{Z})+2g(Z,(I-a)\overline{Z})\geq 0$, $Z \in T_{1,0}(N)$, where a is the Weingarten operator of the immersion N → S²ⁿ⁺¹] on the Ricci curvature Ric_D of D, we show that the first Kohn-Rossi cohomology group of M vanishes. We show that whenever $\mathrm{Ric}_D(Z,\overline{W})-2g(Z,\overline{W})=(\mu \circ \pi)g(Z,\overline{W})$ for some $\mu \in C^\infty (M)$, M is a pseudo-Einstein manifold.

We show that there exist hyperbolic once-punctured torus bundles with 2-generator fundamental groups which have invariant trace fields of arbitrarily high degree. We also answer a question of Bowditch on stabilisers of Markoff triples.

We classify 3-dimensional null 2-type submanifolds of the Euclidean space E⁵ with parallel normalized mean curvature vector under certain hypothesis.

In this paper, we study the symplectic volume of the moduli space of polygons by using Witten's formula. We propose to use this volume as a measure for the flexibility of a polygon with fixed side-lengths. The main result of our is that among all the polygons with fixed perimeter in S³ or E³ the regular one is the most flexible and that among all the spherical polygons the regular one with side-length π/2 is the most flexible.