Given a real vector space V of finite dimension, together with a particular homogeneous field of bivectors that we call a field of projective forces, we define a law of dynamics such that the position of the particle is a ray i.e. a half-line drawn from the origin of V. The impulsion is a bivector whose support is a 2-plane containing the ray. Throwing the particle with a given initial impulsio...

Abstract For a three-dimensional quantum polynomial algebra $A=\mathcal {A}(E,\sigma )$ , Artin, Tate, and Van den Bergh showed that A is finite over its center if only $|\sigma |<\infty $ . Moreover, Artin $E\neq \mathbb P^{2}$ then has fat point module, which plays an important role in noncommutative algebraic geometry; however, the converse not true general. In this paper, we will show mo...

We present a method for calculating the zeta function of a smooth projective variety over a finite field which proceeds by induction on the dimension. Specifically, we outline an algorithm which reduces the problem of calculating a numerical approximation for the action of Frobenius on the middle-dimensional rigid cohomology of a smooth variety, to that of performing the same calculation for a ...

0.1. Notations and Conventions. During this note, we will fix a base field k (e.g. the complex numbers C). The multiplicative group of the field k will be denoted by Gm. All the vector spaces considered will be finite dimensional vector spaces over the field k. For a given vector space V of dimension≥ 1, we will denote the dual vector space, the projective space parameterizing 1-dimensional sub...

In this chapter we will relate the topology of smooth projective varieties over the complex numbers with counting points over finite fields, via the Weil conjectures. If X is a variety defined over a finite field Fq, one can count its points over the various finite extensions of Fq; denote Nm = |X(Fmq )| (for instance, if X ⊂ AnFq is affine, given by equations f1, . . . , fk, then Nm = |{x ∈ Fm...

In this note, we prove that the projective normality of (P(V )/G,L), the celebrated theorem of Erdös-Ginzburg-Ziv and normality of an affine semigroup are all equivalent, where V is a finite dimensional representation of a finite cyclic group G over C and L is the descent of the line bundle O(1)⊗|G|.