Lemma
Let \(\mathcal L\) be an invertible \(\mathcal O_S\)-module and \(\mathcal E\) a locally free \(\mathcal O_S\)-module. Then there is an \(S\)-isomorphism \[\varphi:X=\Proj(\Sym(\mathcal E))\xrightarrow{\sim}\Proj(\Sym(\mathcal E\otimes \mathcal L)) = Y\] such that \(i^*\mathcal O_Y(n) = \mathcal O_X(n)\otimes \pi^*(\mathcal L^{\otimes n})\) for all \(n\in \mathbb Z\). In particular, \[\Proj(\Sym(\mathcal L))\cong \Proj(\Sym(\mathcal O_S)) \cong S.\]
Proof
This follows immediately from the more general fact giving an isomorphism between Proj of a graded \(\mathcal O_S\)-algebra and its “twist” buy increasing tensor powers of an invertible sheaf. In particular, we have an isomorphism \(\Sym^n(\mathcal E\otimes \mathcal L) \xrightarrow{\sim} \Sym^n (\mathcal E)\otimes (\mathcal L)^{\otimes n}\) given locally on sections by \[(e_1\otimes \ell_1)\otimes...\otimes(e_n\otimes \ell_n) \mapsto (e_1...e_n)\otimes(\ell_1\otimes...\otimes \ell_n).\] The result then follows from the aforementioned proposition.