  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 $X^5_{r,0}$ ($r \leq 5$).
