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

Hakop Hakopian; Harutyun Kloyan

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arXiv · 1903.10874

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

Hakop Hakopian·Harutyun Kloyan

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Abstract

Let a set of nodes $\mathcal X$ in plain be $n$-independent, i.e., each node has a fundamental polynomial of degree $n.$ Suppose also that $|\mathcal X|= d(n,k-2)+2,$ where $d(n,k-2) = (n+1)+n+\cdots+(n-k+4)$ and $\ k\le n-1.$ 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 $\mathcal X.$ We provide a characterization of the case when there are exactly four such curves. Namely, we prove that then the set $\mathcal X$ 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.

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BibTeX RIS

Hakop Hakopian, Harutyun Kloyan. 2019-03-24. On the dimension of spaces of algebraic curves passing through $n$-independent nodes. https://arxiv.org/abs/1903.10874

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