Some line and conic arrangements and their Waldschmidt constants

• Main Authors: Huynh, Dinh Tuan, Linh, Tran N. K, Long, Le Ngoc
• Format: Journal Article
• Language: English
• Published: 07-10-2024
• Subjects: Mathematics - Algebraic Geometry; Mathematics - Combinatorics
• DOI: 10.48550/arxiv.2410.05029

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.