I'm reading a material that states:
Definition: Let F be a foliation on a analytical manifold N. A normal crossing divisor on N is a collection of submanifolds $E$ of $N$ such that for every point $p\in E$ there is a coordinate system for $N$ $(x_1,...,x_n)$ such that $$ E=(\Pi _{i\in A}x_i=0)\text{ locally in $p$.} $$ for $A\subset\{1,...,n\}$. Can somebody explain what does this product means, geometrically? Does this mean that at least one of the coordinates is set as zero?
Does this also means that $E$ is a algebraic manifold? In this case, an irreductible component of $E$ is the zero locus of the irreductible components of $E$?