In this paper, we define the totally ramified value number (TRVN) of the Gauss map of a complete minimal surface, and prove that there exist algebraic minimal surfaces that have the TRVN equal to $2.5$.

We study the mapping class groups of compact orientable surfaces from the viewpoint of measure equivalence. In this paper, we announce some classification result of the mapping class groups in terms of measure equivalence and the result that there exist various kinds of discrete groups which are not measure equivalent to the mapping class groups. As a byproduct of the proof, it turns out that the mapping class group is exact.

Let $L$ be a very ample line bundle on a smooth complex projective variety $X$ of dimension $\geq 6$. We classify the polarized manifolds $(X, L)$ such that there exists a smooth member $A$ of $|L|$ endowed with a branched covering of degree four $\pi \colon A \rightarrow \mathbf{P}^{n}$. The cases of $\deg \pi =2$ and $3$ are already studied by Lanteri-Palleschi-Sommese. Recently the case of $\deg \pi =5$ is studied by Amitani.