Let $X$ be a smooth, projective variety over $k$ and $Y$ a smooth closed subvariety of codimension r. Let $\mathcal{N}_{Y/X}$ be the rank r normal bundle. Is it possible to determine when $\mathrm{det}\mathcal{N}_{Y/X}$ is ample?

As far as I know for $X$ a projective space this is true, but what is for example for Hirzebruch surfaces or $\mathbb{P}^1\times\mathbb{P}^1$?