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arXiv · 2410.05029

Some line and conic arrangements and their Waldschmidt constants

Dinh Tuan Huynh·Tran N. K. Linh·Le Ngoc Long
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Abstract

We study the Waldschmidt constant of some configurations in the projective plane. In the first part, we show that the Waldschmidt constant of a set $\mathbb{X}$ of $n$ points where at least $n-3$ points among them lie on a line is either equal to $1, \frac{2n-3}{n-1}, 2, \frac{16}{7}, \frac{7}{3}, \frac{17}{7},$ or $\frac{5}{2}$. Together with the Hilbert polynomials, this gives a complete geometric characterization for $\mathbb{X}$. Next, we study some specific configurations whose Waldschmidt constants are bounded from above by $\frac{5}{2}$. Under this condition, we describe all configurations of $n$ points with $n-1$ points among them lying on an irreducible conic, and we also study some specific configurations of $9$ points.

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Dinh Tuan Huynh, Tran N. K. Linh, Le Ngoc Long. 2024-10-07. Some line and conic arrangements and their Waldschmidt constants. https://arxiv.org/abs/2410.05029

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