We pose the problem to determine explicit defining equations of various elliptic fibrations on a given $K3$ surface, and study the case of the Kummer surfaces of the product of two elliptic curves.

Let $S = G / G'$ be a rational homogeneous space defined by a complex simple Lie group $G$ and a maximal parabolic subgroup $G'$. For a base point $s \in S$, let $\mathcal{C}_s \subset \mathbb{P}T_s (S)$ be the variety of minimal rational tangents at $s$. In the study of rigidity of rational homogeneous spaces, the following question naturally arises. Let $X$ be a Fano manifold of Picard number 1 such that the variety of minimal rational tangents at a general point $x \in X$, $\mathcal{C}_x \subset \mathbb{P}T_x (X)$, is isomorphic to $\mathcal{C}_s \subset \mathbb{P}T_s (S)$. Is $X$ biholomorphic to $S$? An affirmative answer has been given by Mok when $S$ is a Hermitian symmetric space or a homogeneous contact manifold. Extending Mok's method further and combining it with the theory of differential systems on $S$, we will give an affirmative answer when $G'$ is associated to a long simple root.

In this paper we introduce a maximal divisorial set in the arc space of a variety. The generalized Nash problem is reduced to a translation problem of the inclusion of two maximal divisorial sets. We study this problem and show a counter example to the most natural expectation even for a non-singular variety.

We give a detailed proof of the following main result of [Isk1]. There are two non-conjugate embeddings of $S_3 \times \mathbb{Z}_2$ into the Cremona group given by the linear action of this group on the plane and the action on the two-dimensional torus respectively.

We compute the Poincaré polynomial of hyperplane arrangements associated with a class of signed complete graphs. We also make a factorization of the Poincaré polynomial over the integers.

We investigate unibranched singularities of dual varieties of even-dimensional smooth projective varieties in characteristic 2.

The Castelnuovo-Weil lattice of a curve refers to the ring of correspondence of the curve or the endomorphism ring of its Jacobian variety, viewed as a lattice with respect to the natural trace form. The main subject is the minimal norm $\mu (C)$ of this lattice, and we discuss the question as to whether $\mu (C) = 2g$ holds true.