The Tango bundle $T$ over $\p ^5$ is proved to be the pull--back of the twisted Cayley bundle $C(1)$ via a map $f \colon \p ^5 \rightarrow Q_5$ existing only in characteristic 2. The Frobenius morphism $\varphi$ factorizes via such $f$.

It is known that if $X$ is an $n$--dimensional normal variety, and $D$ a nef and big Cartier divisor on it such that the associated map $\varphi_D$ is generically finite then $D^n\geq 2(h^0(X,\Oc_X(D))-n)$. We study the case in which the equality holds for $n=3$ and $D=K_X$ is the canonical divisor. \par We also produce a bound for the admissible degree of the canonical map of a threefold, when it is supposed to be generically finite.

In this work we will consider compact submanifold $M^{n}$ immersed in the Euclidean sphere $S^{n+p}$ with parallel mean curvature vector and we introduce a Schr\"{o}dinger operator $L=-\Delta+V$, where $\Delta$ stands for the Laplacian whereas $V$ is some potential on $M^{n}$ which depends on $n,p$ and $h$ that are respectively, the dimension, codimension and mean curvature vector of $M^{n}$. We will present a gap estimate for the first eigenvalue $\mu_{1}$ of $L$, by showing that either $\mu_{1}=0$ or $\mu_{1}\leq-n(1+H^{2})$. As a consequence we obtain new characterizations of spheres, Clifford tori and Veronese surfaces that extend a work due to Wu \cite{wu} for minimal submanifolds.

Let $\p^{19}$ be the parametrizing space of cubic surfaces in $\p^3$. Let $M$ be the moduli space of non-singular cubic surfaces and $\overline{M}$ be a suitable compactification. We study components of boundaries of the relative subspaces of non-singular cubic surfaces with star points in $\p^{19}$ and in $\overline{M}$.

Two invariants, called the $c_1$-sectional geometric genus and the $\co(1)$-sectional geometric genus, are introduced for generalized polarized manifolds, i.e., pairs of a projective manifold and an ample vector bundle on it. They are generalizations of the $c_1$-sectional genus and the $\co(1)$-sectional genus introduced by T.~Fujita for generalized polarized manifolds. They are also generalizations of the sectional geometric genus introduced by Y.~Fukuma for polarized manifolds. We classify generalized polarized manifolds of the smallest $c_1$ or $\co(1)$-sectional geometric genus under the spannedness condition for ample vector bundles.

In the study of extremal surfaces it is important to see the side-pairings of the generic fundamental polygons. For the regular 30-gon in the hyperbolic plane it is known that there are 1726 side-pairing patterns to make a compact surface of genus three. If we consider the mirror images of these patterns, then there are essentially 927 patterns. In the present paper we give the 927 side-pairing patterns completely.