Seshadri constants on some blow-ups of projective spaces

TL;DR

This paper investigates Seshadri constants and nef cones on blow-ups of projective spaces along lines and points, providing explicit calculations and restrictions for various dimensions and configurations.

Contribution

It computes the nef cone of certain blow-ups of $P^3$ and analyzes Seshadri constants for specific line bundles on these and higher-dimensional varieties.

Findings

Nef cone of $X^3_{r,0}$ for $r\, extleq 6$ computed

Seshadri constants of ample line bundles on $X^n_{r,0}$ for specified dimensions and $r$ values analyzed

Restrictions on $s$ for $l$-very ample line bundles on $X^n_{0,s}$ established

Abstract

Let $X^n_{r,s}$ denote the blow-up of $\mathbb{P}^n$ along $r$ general lines and $s$ general points. In this paper, we focus on $l$-very ample line bundles on $X^n_{0,s}$ and investigate their Seshadri constants with some restrictions on $s$. Additionally, we compute the nef cone of $X^3_{r,0}$ for $r\leq 6$ and study the Seshadri constants of some ample line bundles on it. We also examine the Seshadri constants of some ample line bundles on $X^4_{r,0}$ ($r\leq 7$) and $X^5_{r,0}$ ($r \leq 5$).