Irrational Seshadri constants from dihedral orbits

Abstract: We construct an irrational one-point Seshadri constant on the blow-up $X_9$ of $\mathbb{P}<sup>2$ at nine very general points. The divisor $L=9H-3(E_1+\cdots+E_5)-2(E_6+\cdots+E_9)$ is ample and satisfies $\varepsilon(L;x)=2\sqrt{5}$ at a very general point $x\in X_9$. To prove this, we establish that the Seshadri constant of $$\mathcal{O}_{\mathbb{P}<sup>1\times\mathbb{P}<sup>1}(1,1)$$ at ten very general points is $1/\sqrt{5}$, completing the reflection approach proposed by Dionne and Roth for the ten-point case. We also prove that the same equality holds at a very general free orbit of a fixed dihedral group of order ten. This yields an irrational one-point Seshadri constant on the singular quotient surface. A plane model of its minimal resolution and a deformation of the blow-up centers then give the result on $X_9$.