For the $p$-localized sphere $\mathbb {S}^{2m-1}_{(p)}$ with $p >3$ a prime, we prove that the homotopy nilpotency satisfies $\mbox {nil}\ \mathbb {S}^{2m-1}_{(p)}<\infty$, with respect to any homotopy associative $H$-structure on $\mathbb {S}^{2m-1}_{(p)}$. We also prove that $\mbox {nil}\ \mathbb {S}^{2m-1}_{(p)}= 1$ for all but a finite number of primes $p >3$. Then, for the loop space of the associated $\mathbb {S}^{2m-1}_{(p)}$-projective space $\mathbb {S}^{2m-1}_{(p)}P(n-1)$, with $m,n\ge 2$ and $m\mid p-1$, we derive that $\mbox {nil}\ \Omega (\mathbb {S}^{2m-1}_{(p)}P (n-1))\le 3$.

This paper classifies Poisson structures with the reduced simple normal crossing divisor on a product of Fano varieties of Picard number 1. The characterization of even-dimensional projective spaces from the viewpoint of Poisson structures is given by Lima and Pereira. In this paper, we generalize the characterization of projective spaces to any dimension.

This paper extends a recently proposed singularity analysis method to lower-mobility parallel manipulators having an articulated nacelle. Using screw theory, a twist graph is introduced in order to simplify the constraint analysis of such manipulators. Then, a wrench graph is obtained in order to represent some points at infinity on the Plücker lines of the Jacobian matrix. Using Grassmann–Cayley algebra, the rank deficiency of the Jacobian matrix amounts to the vanishing condition of the superbracket. Accordingly, the parallel singularities are expressed in three different forms involving superbrackets, meet and join operators, and vector cross and dot products, respectively. The approach is explained through the singularity analysis of the H4 robot. All the parallel singularity conditions of this robot are enumerated and the motions associated with these singularities are characterized.

We discuss n4 configurations of n points and n planes in three-dimensional projective space. These have four points on each plane, and four planes through each point. When the last of the 4n incidences between points and planes happens as a consequence of the preceding 4n−1 the configuration is called a ‘theorem’. Using a graph-theoretic search algorithm we find that there are two 84 and one 94 ‘theorems’. One of these 84 ‘theorems’ was already found by Möbius in 1828, while the 94 ‘theorem’ is related to Desargues’ ten-point configuration. We prove these ‘theorems’ by various methods, and connect them with other questions, such as forbidden minors in graph theory, and sets of electrons that are energy minimal.

If a mapping of several complex variables into projective space is holomorphic in each pair of variables, then it is globally holomorphic.

We distribute the points and lines of PG(2, 2n+1) according to a special structure that we call the daisy structure. This distribution is intimately related to a special block design which turns out to be isomorphic to PG(n, 2).

We show a blocking set of 3q points in PG(2, 2n+1)that intersects each line in at least two points and we apply this to find a lower bound for the heterochromatic number of the projective plane.

Let ${{\mathbf{P}}^{n}}$ be the $n$-dimensional projective space over some algebraically closed field $k$ of characteristic 0. For an integer $t\,\ge \,3$ consider the invertible sheaf $O\left( t \right)$ on ${{\mathbf{P}}^{n}}$ (Serre twist of the structure sheaf). Let $N\,=\,\left( \underset{n}{\mathop{t+n}}\, \right)$, the dimension of the space of global sections of $O\left( t \right)$, and let $k$ be an integer satisfying $0\,\le \,k\,\le \,N\,-\,\left( 2n\,+\,2 \right)$. Let ${{P}_{1}},\ldots ,{{P}_{k}}$ be general points on ${{\mathbf{P}}^{n}}$ and let $\pi :\,X\,\to \,{{\mathbf{P}}^{n}}$ be the blowing-up of ${{\mathbf{P}}^{n}}$ at those points. Let ${{E}_{i}}\,=\,{{\pi }^{-1}}\left( {{P}_{i}} \right)$ with $1\,\le \,i\,\le \,k$ be the exceptional divisor. Then $M={{\pi }^{*}}\left( O(t) \right)\,\otimes \,{{O}_{X}}\left( -{{E}_{1}}-\cdots -{{E}_{k}} \right)$ is a very ample invertible sheaf on $X$.

We proove an analogue of the Beilinson-Bernstein Theorem, in the case of the projective space, for the ring of arithmetical differential operators introduced by P. Berthelot.