Dihedral reflections and an infinite series of irrational Seshadri constants

Abstract

Laface and Ugaglia recently constructed an irrational one-point Seshadri constant on the blow-up of $\mathbb{P}^2$ at nine very general points by combining a dihedral orbit on $\mathbb{P}^1\times\mathbb{P}^1$, a sequence of de Jonquières transformations, and a reflection argument along a $(-4)$-curve with balanced normal bundle. We show that the same mechanism extends uniformly to every odd integer $n\geq 5$. For $n=2k+1$ we prove $$ \varepsilon\bigl(\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(n-4,1);p_1,\ldots,p_{2n}\bigr)=\sqrt{\frac{n-4}{n}} $$ at $2n$ very general points, and already at a very general free orbit of a fixed dihedral group of order $2n$. For every odd $n\geq 7$ this produces an explicit ample line bundle on the blow-up of $\mathbb{P}^2$ at $k+7=(n+13)/2$ very general points whose one-point Seshadri constant at a very general point equals $$ 2\sqrt{n(n-4)}. $$ More precisely, after one quadratic transformation we obtain the ample divisor $$ L_n=(3n-4)H-nE_1-(n-2)(E_2+\cdots+E_5)-4(E_6+\cdots+E_{k+5})-2(E_{k+6}+E_{k+7}), $$ with $L_n^2=4n(n-4)$ and $\varepsilon(L_n;x)=\sqrt{L_n^2}$. We also isolate an abstract balanced-reflection principle underlying the construction: a nef class on a special fiber can be reflected across a rational curve of square $-2a$ whenever the curve has normal bundle $\mathcal{O}_{\mathbb{P}^1}(-a)^{\oplus 2}$ in the total space, and the reflected class is nef on very general fibers.