normal bundle

《Algebraic Geometry》 · 第 182 页 · Chapter II，Section 8，Definition

Let $Y$ be a nonsingular subvariety of a nonsingular variety $X$ over $k$. The locally free sheaf $\mathcal J/\mathcal J^2$ we call the conormal sheaf of $Y$ in $X$. Its dual $\mathcal N_{Y/X}=\mathcal Hom_{\mathcal O_Y}(\mathcal J/\mathcal J^2,\mathcal O_Y)$ is called the normal sheaf of $Y$ in $X$. It is locally free of rank $r=\operatorname{codim}(Y,X)$.

《Algebraic Geometry》 · 第 433 页 · Appendix A，Example 4.1.3

Let $N=N_{X/\mathbb P^4}$ be the normal bundle of $X$ in $\mathbb P^4$.