We introduce the different focal loci (focal points, planes and hyperplanes) of \break (n-1)-dimensional families (congruences) of lines in ${\Bbb P}^{n}$ and study their invariants, geometry and the relation among them. We also study some particular congruences whose focal loci have special behavior, namely $(n-1)$-secant lines to an $(n-2)$-fold and $(n-1)$-tangent lines to a hypersurface. In case $n=4$ we also give, under some smoothness assumptions, a classification result for these congruences.

In Part I we showed that the $L^q$-spectrum of the 3-fold convolution of the Cantor measure has a non-differentiable point at a $q_0 < 0$ [LW], therefore the standard multifractal formalism does not hold. In this Part II, we prove a modified multifractal formalism for the measure.

Let $X$ be a complex irreducible smooth projective curve of genus at least two and $M(r,d)$ a moduli space of stable parabolic vector bundles over $X$ of rank $r$ and degree $d$ with a fixed parabolic structure. For any parabolic bundle $E_*\in M(r,d)$ and a subbundle $F\, \subset\, E$ of rank $r'$ and fixed induced parabolic structure, set $s^{par}(E_*,F_*)\, :=\, dr'-\text{deg}(F)r$, where $F_*$ is $F$ equipped with the induced parabolic structure. If $E_*$ has a subbundle of rank $r'$ with the fixed induced parabolic structure, then let $s^{par}_{r'}(E_*)$ be the minimum of $s^{par}(E_*,F_*)$ taken over all such subbundles $F$. We investigate the strata of $M(r,d)$ defined by values of $s^{par}_{r'}(E_*)$.

Let $X$ be a Fano manifold of pseudoindex $i_X$ whose Picard number is at least two and let $R$ be an extremal ray of $X$ with exceptional locus $\operatorname{Exc}(R)$. We prove an inequality which bounds the length of $R$ in terms of $i_X$ and of the dimension of $\operatorname{Exc}(R)$ and we investigate the border cases.

In particular we classify Fano manifolds $X$ of pseudoindex $i_X$ obtained blowing up a smooth variety $Y$ along a smooth subvariety $T$ such that $\dim T < i_X$.

In this paper, we study the fundamental groups of closed manifolds of positive sectional curvature which admit compatible local isometric torus $T^k$-actions. We explore relations between basic properties of an isometric $T^k$-action and the structure of the fundamental group of $M$. Using these relations, we prove several results on the fundamental groups.