We prove some basic theorems concerning lemniscate configurations in an Euclidean space of dimension $ n geq 3$. Lemniscates are defined as follows. Given m points $w_j $ in $mathbb R^n$, consider the function $F(x)$ which is the product of the dista
nces $ |x-w_j|$: the singular level sets of the function $F$ are called lemniscates. We show via complex analysis that the critical points of $F$ have Hessian of positivity at least $(n-1)$. This implies that, if $F$ is a Morse function, then $F$ has only local minima and saddle points with negativity 1. The critical points lie in the convex span of the points $|w_j| $ (these are absolute minima): but we made also the discovery that $F$ can also have other local minima, and indeed arbitrarily many. We discuss several explicit examples. We finally prove in the appendix that all critical points are isolated.
The first main purpose of this paper is to contribute to the existing knowledge about the complex projective surfaces $S$ of general type with $p_g(S) = 0$ and their moduli spaces, constructing 19 new families of such surfaces with hitherto unknown f
undamental groups. We also provide a table containing all the known such surfaces with K^2 <=7. Our second main purpose is to describe in greater generality the fundamental groups of smooth projective varieties which occur as the minimal resolutions of the quotient of a product of curves by the action of a finite group. We classify, in the two dimensional case, all the surfaces with q=p_g = 0 obtained as the minimal resolution of such a quotient, having rational double points as singularities. We show that all these surfaces give evidence to the Bloch conjecture.
