Let $\mathcal{F}$ be a family of meromorphic functions on a domain $D$, $k\in\mathbf{N}$ and $\mathcal{H}$ be a normal family of meromorphic functions on $D$ such that 0 is not in $\mathcal{H}$ and $\mathcal{H}$ has no sequence that converges to 0 or $\infty$ spherically locally uniformly on $D$. If for every $f\in\mathcal{F}$, $f(z)\neq 0$, and there exists an $h_{f}\in \mathcal{H}$ such that $f^{(k)}(z)\neq h_{f}(z)$ at every $z\in D$, then the family $\mathcal{F}$ is normal on $D$. This generalizes Gu’s well-known normality criterion. It is interesting that the condition $f(z)\neq 0$ cannot be replaced by that all zeros of $f$ have large multiplicities, at least $k+3$ for instance.

Motivated by a profound observation of A’Campo we investigate the behaviour of the curvature of $f(z,w)=c, f\in \mathbf{C}\{z,w\}, |c|$ small, along infinitesimals. We use the language of infinitesimals as introduced in [2]. Along the way we introduce the important notion of gradient canyon, and prove several theorems in which this notion plays the key role. In this paper we give three such theorems and mention several other facts, to be published elsewhere.

It is known that infinitely many imaginary quadratic fields allow Hermitian lattices which are generated by minimal vectors but have no basis of minimal vectors. In this article we construct systematically such Hermitian lattices over other imaginary quadratic fields. These lattices are binary and unimodular. This construction requires specific non-principal ideals.

For a given finite small binary dihedral group $G\subset\mathrm{GL}(2,\mathbf{C})$ we provide an explicit description of the minimal resolution $Y$ of the singularity $\mathbf{C}^{2}/G$. The minimal resolution $Y$ is known to be either the moduli space of $G$-clusters $G$-Hilb$(\mathbf{C}^{2})$, or the equivalent $\mathcal{M}_{\theta}(Q,R)$, the moduli space of $\theta$-stable quiver representations of the McKay quiver. We use both moduli approaches to give an explicit open cover of $Y$, by assigning to every distinguished $G$-graph $\Gamma$ an open set $U_{\Gamma}\subset\mathcal{M}_{\theta}(Q,R)$, and calculating the explicit equation of $U_{\Gamma}$ using the McKay quiver with relations $(Q,R)$.