Normal sheaf (${\mathcal{N}}_{Z/X}$)

For a closed immersion $i:Z\hookrightarrow X$ with ideal sheaf $\mathcal{I}$, the normal sheaf is ${\mathcal{N}}_{Z/X}=\mathcal{H}o{m}_{{\mathcal{O}}_Z}(\mathcal{I}/{\mathcal{I}}^2,{\mathcal{O}}_Z)$. It is the dual of the conormal sheaf. For a regular embedding it is a locally free sheaf; in the smooth case it agrees with the normal bundle. Without regularity it need not be locally free, but its global sections still classify first-order embedded deformations.

If a projective complete intersection $Z\subseteq {\mathbf{P}}^n$ is cut out by a homogeneous regular sequence of degrees $d_1,\dots ,d_c$, then $\mathcal{I}/{\mathcal{I}}^2\cong {⨁}_j{\mathcal{O}}_Z(-d_j)$ and ${\mathcal{N}}_{Z/{\mathbf{P}}^n}\cong {⨁}_j{\mathcal{O}}_Z(d_j)$. The classes of the defining equations form the conormal basis: the first syzygies of a regular sequence are the Koszul complex relations, whose coefficients vanish modulo the defining ideal. Thus the generator classes have no residual linear relations. Dualizing gives the stated normal sheaf.