# Some Results on Seshadri constants on Surfaces of general type

## TL;DR

This paper investigates Seshadri constants on surfaces of general type, providing classifications for certain multi-point constants and proving rationality of global constants on product surfaces.

## Contribution

It offers new classifications of multi-point Seshadri constants and establishes the rationality of global Seshadri constants on product surfaces of curves.

## Key findings

- Classified possible values of multi-point Seshadri constants between 0 and 1/r.
- Proved global Seshadri constants are rational on surfaces of the form C×C.
- Extended understanding of Seshadri constants on surfaces of general type.

## Abstract

We prove two new results for Seshadri constants on surfaces of general type. Let $X$ be a surface of general type. In the first part, inspired by \cite{B-S}, we list the possible values for the multi-point Seshadri constant $\varepsilon(K_X,x_1,x_2,...,x_r)$ when it lies between $0$ and $1/r$, where $K_X$ is the canonical line bundle on $X$. In the second part, we assume $X$ of the form $C \times C$, where $C$ is a general smooth curve of genus $g \geq 2$. Given such $X$ and an ample line bundle $L$ on $X$ with some conditions on it, we show that the global Seshadri constant of $L$ is a rational number.