On a result concerning algebraic curves passing through $n$-independent nodes

TL;DR

This paper investigates the maximum number of low-degree algebraic curves passing through a specific set of plane nodes, providing a characterization of when this maximum is achieved and extending previous results in algebraic geometry.

Contribution

It establishes an upper bound on the number of low-degree curves passing through an $n$-independent node set and characterizes the structure of the node set when this bound is tight.

Findings

At most three linearly independent curves of degree ≤ n-1 pass through the nodes.

Characterization of node sets with exactly three such curves.

Nodes either lie on a curve of degree n-2 or all but three lie on a curve of degree n-3.

Abstract

Let a set of nodes $\mathcal X$ in the plane be $n$-independent, i.e., each node has a fundamental polynomial of degree $n.$ Assume that\\ $\#\mathcal X=d(n,n-3)+3= (n+1)+n+\cdots+5+3.$ In this paper we prove that there are at most three linearly independent curves of degree less than or equal to $n-1$ that pass through all the nodes of $\mathcal X.$ We provide a characterization of the case when there are exactly three such curves. Namely, we prove that then the set $\mathcal X$ has a special construction: either all its nodes belong to a curve of degree $n-2,$ or all its nodes but three belong to a (maximal) curve of degree $n-3.$ This result complements a result established recently by H. Kloyan, D. Voskanyan, and H. H. Note that the proofs of the two results are completely different.