We study the splitting of Fermat Jacobians of prime degree $\ell$ over an algebraic closure of a finite field of characteristic $p$ not equal to $\ell$. We prove that their decomposition is determined by the residue degree of $p$ in the cyclotomic field of the $\ell$-th roots of unity. We provide a numerical criterion that allows to compute the absolutely simple subvarieties and their multiplicity in the Fermat Jacobian.

This paper presents an approach to injectivity theorems via the Mountain Pass Lemma and raises an open question. The main result of this paper (Theorem~1.1) is proved by means of the Mountain Pass Lemma and states that if the eigenvalues of $F' (\x)F' (\x)^{T}$ are uniformly bounded away from zero for $\x \in \hbox{\Bbbvii R}^{n}$, where $F \colon \hbox{\Bbbvii R}^n \rightarrow \hbox{\Bbbvii R}^n$ is a class $\cC^{1}$ map, then $F$ is injective. This was discovered in a joint attempt by the authors to prove a stronger result conjectured by the first author: Namely, that a sufficient condition for injectivity of class $\cC^{1}$ maps $F$ of $\hbox{\Bbbvii R}^n$ into itself is that all the eigenvalues of $F'(\x)$ are bounded away from zero on $\hbox{\Bbbvii R}^n$. This is stated as Conjecture~2.1. If true, it would imply (via {\it Reduction-of-Degree}) {\it injectivity of polynomial maps} $F \colon \hbox{\Bbbvii R}^n \rightarrow \hbox{\Bbbvii R}^n$ {\it satisfying the hypothesis}, $\det F'(\x) \equiv 1$, of the celebrated Jacobian Conjecture (JC) of Ott-Heinrich Keller. The paper ends with several examples to illustrate a variety of cases and known counterexamples to some natural questions.

Let $(X,L)$ be a polarized manifold over the complex number field with $\dim X=n$. In this paper, we consider a conjecture of M.~C.~Beltrametti and A.~J.~Sommese and we obtain that this conjecture is true if $n=3$ and $h^{0}(L)\geq 2$, or $\dim \Bs |L|\leq 0$ for any $n\geq 3$. Moreover we can generalize the result of Sommese.

Let $\Bbb G_{p,q}(\Bbb F)$ be the Grassmann space of all $q$-dimensional $\Bbb F$-vector subspaces of $\Bbb F^{p}$, where $\Bbb F$ stands for $\Bbb R$, $\Bbb C$ or $\Bbb H$ (the quaternions). Here $\Bbb G_{p,q}(\Bbb F)$ is regarded as a real algebraic variety. The paper investigates which ${\cal C}^\infty$ maps from a nonsingular real algebraic variety $X$ into $\Bbb G_{p,q}(\Bbb F)$ can be approximated, in the ${\cal C}^\infty$ compact-open topology, by real algebraic morphisms.

Using naive algebraic geometric methods a new proof of the following celebrated theorem of Magnus is given: Let $G$ be a group with a presentation having $n$ generators and $m$ relations. If $G$ also has a presentation on $n-m$ generators, then $G$ is free of rank $n-m$.