Let $K_m$ be a finite commutative semi-local ring of characteristic $m$, and let $MQ$ be the generalized dicyclic group. Descriptions are given of the simple and projective $K_{m} MQ$-modules.

In this paper, we generalize Zariski’s discriminant criterion under which an analytic family of plane curve singularities are equivalent. Furthermore, we give a necessary condition of local irreducibility of plane curves with singularities.

We give an alternative proof of the mod $p$ vanishing theorem by F. Fang of Seiberg-Witten invariants under a cyclic group action of prime order, and generalize it to the case when $b_1 \geq 1$. Although we also use the finite dimensional approximation of the monopole map as well as Fang, our method is rather geometric. Furthermore, non-trivial examples of mod $p$ vanishing are given.

For a number field $K$, we show that any $S$-arithmetic subgroup of $SL_2(K)$ contains a subgroup of finite index generated by three elements if card $(S) \geq 2$.

An explicit integral basis is given for infinitely many cyclic quintic fields.

A construction is introduced for modifying hyperkähler manifolds with tri-Hamiltonian circle action, that in favourable situations increases the second Betti number by one. This is based on the symplectic cut construction of Lerman. In 4 or 8 dimensions the construction may be interpreted as adding a D6-brane. A number of examples are given and a generalisation to three-Sasaki geometry discussed.