I would like a reference to a proof or a proof of the following fact.
Let $ X $ be a nonsingular variety over an algebraically closed field $ k $ and $ Y $ a nonsingular subvariety of $ X $ of codimension $ 1 $ and with ideal sheaf $ \mathcal{I} = \mathcal{O}_X(-Y) $. Note that the sheaf $ \mathcal{I} / \mathcal{I}^2 $ on $ X $ is supported on $ Y $ which we call by the same. Then we have $ \mathcal{I} / \mathcal{I}^2 = \mathcal{I}|_Y = \mathcal{O}_Y(-Y) $, or in other words, the normal sheaf is given by $ \mathcal{O}_Y(Y) $.