On the dimension of spaces of algebraic curves passing through $n$-independent nodes

Let the set of nodes $X$ in the plain be $n$-independent, i.e. each node has a fundamental polynomial of degree $n$. Suppose also that $|X|= (n+1)+n+\cdots+(n-k+4)+2$ and $3\leq k\leq n-1$. In this paper we prove that there can be at most 4 linearly independent curves of degree less than or equal to $k$ passing through all the nodes of $X$. We provide a characterization of the case when there are exactly four such curves. Namely, we prove that then the set $X$ has a very special construction: all its nodes but two belong to a (maximal) curve of degree $k-2$. At the end, an important application to the Gasca–Maeztu conjecture is provided.