  A planar node set $\mathcal X,$ with $\#\mathcal X=\binom{n+2}{2},$ is called
$GC_n$ set if each node possesses fundamental polynomial in form of a product
of $n$ linear factors. We say that a node uses a line if the line is a factor
of the fundamental polynomial of the node. A line is called $k$-node line if it
passes through exactly $k$-nodes of $\mathcal X.$ At most $n+1$ nodes can be
collinear in any $GC_n$ set and an $(n+1)$-node line is called a maximal line.
The Gasca-Maeztu conjecture (1982) states that every $GC_n$ set has a maximal
line. Until now the conjecture has been proved only for the cases $n \le 5.$
  Here, for a line $\ell$ we introduce and study the concept of $\ell$-lowering
of the set $\mathcal X$ and define so called proper lines. We also provide
refinements of several basic properties of $GC_n$ sets regarding the maximal
lines, $n$-node lines, the used lines, as well as the subset of nodes that use
a given line.
