Simple cyclic covers of the plane and Seshadri constants of some general hypersurfaces in weighted projective space

TL;DR

This paper investigates the Seshadri constants of general hypersurfaces in weighted projective spaces, showing they approach the square root of the degree as the weight parameter increases.

Contribution

It provides bounds for Seshadri constants on certain hypersurfaces and demonstrates their convergence to an irrational limit as weights grow.

Findings

Seshadri constants are bounded between \\sqrt{d} - d/m and \\sqrt{d}.

Constants approach \\sqrt{d} as m increases.

Results apply to general hypersurfaces in weighted projective spaces.

Abstract

Let $X$ be a general hypersurface of degree $md$ in the weighted projective space with weights $1,1,1,m$ for some for $d\geq 2$ and $m\geq 3$. We prove that the Seshadri constant of the ample generator of the N\'eron-Severi space at a general point $x\in X$ lies in the interval $\left[\sqrt{d}- \frac d m, \sqrt{d}\right]$ and thus approaches the possibly irrational number $\sqrt d$ as $m$ grows.