= Normal sheaf of a projective complete intersection
If a <projective complete intersection> $Z\subseteq\mathbf P^n$ is cut out by a homogeneous <regular sequence> of degrees $d_1,\ldots,d_c$, then $\mathcal I/\mathcal I^2\cong\bigoplus_j\mathcal O_Z(-d_j)$ and $\mathcal N_{Z/\mathbf P^n}\cong\bigoplus_j\mathcal O_Z(d_j)$. 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>.
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