TITLE: Sections of the conormal bundle
QUESTION [6 upvotes]: Let $X\subset\mathbb{P}^N$ be a quadratic manifold. That is $I(X)$ is generated by quadratic polynomials $Q_1,...,Q_m$.
Let $\mathcal{I}_X$ be the ideal sheaf of $X$ and $\mathcal{I}_X/\mathcal{I}_X^2$ be the conormal bundle of $X$. Is it true that $H^0(X,\mathcal{I}_X/\mathcal{I}_X^2(2))$ has rank $m$ and that it is generated by $dQ_1,...,dQ_m$ ?

REPLY [5 votes]: No. The simplest example is the twisted cubic $C\subset \mathbb{P}^3$. The conormal bundle on $C\cong \mathbb{P}^1$ is $\mathcal{O}_{\mathbb{P}^1}(-5)^2$. Hence $H^0(C,\mathcal{I}_C/\mathcal{I}_C^2(2))$ has dimension 4, while $I(C)$ is generated by 3 quadrics.