Let $X$ be a projective variety in $\mathbb{P}^n.$ I'm trying to prove the following fact that the total space of the line bundle $\mathcal{O}_X(-1)$ is isomorphic to the blow-up of the affine cone $\widehat{X}$ at the origin in $\mathbb{A}^{n+1}$.
The statement is quite obvious to me visually, but I'm getting stuck proving it in the language of schemes.
I can prove that $X\simeq \text{Proj}(\bigoplus _{n\geq 0} H^0(X,\mathcal{O}_X(n)))$, so $\widehat{X}\simeq \text{Spec}(\bigoplus _{n\geq 0} H^0(X,\mathcal{O}_X(n)))$, and hence $Bl_0{\widehat{X}}=\text{Proj}(\bigoplus I^n)$, where $I$ should be the ideal generated by the images of $x_0,\cdots,x_n$ in $H^0(X,\mathcal{O}_X(1))$. On the other hand, the underlying space of $\mathcal{O}_X(-1)$ is given by $|\mathcal{O}_X(-1)| = \text{Spec}(Sym^{\cdot}\mathcal{O}_X(1))$, where $Sym^{\cdot}$ denotes the symmetric algebra. But at this point, I'm stuck. Any help would be appreciated.
$\DeclareMathOperator{\Spec}{Spec}$ $\DeclareMathOperator{\Proj}{Proj}$ $\DeclareMathOperator{\Bl}{Bl}$ $\DeclareMathOperator{\deg}{deg}$ Firstly, the cone $\widehat{X}$ cannot be written as $\Spec\bigoplus_{n\geq 0}H^0(X,\mathcal{O}_X(n))$. For example, let $X=\Proj k[x,y,z]/(x^3-yz^2)$. Then $\bigoplus_{n\geq 0}H^0(X,\mathcal{O}_X(n))\simeq k[s,t,u,v]/(su-t^2,tv-u^2,sv-tu)$, but $$ \widehat{X}=\Spec k[x,y,z]/(x^3-yz^2)\not\simeq \Spec k[s,t,u,v]/(su-t^2,tv-u^2,sv-tu) $$ (compare the dimension of the tangent space at the unique singular point). The point is that $\Proj A\simeq \Proj B$ does not imply $\Spec A\simeq \Spec B$.
I will give an elementary proof. Note that the blow-up $\Bl_0\widehat{X}$ can be identified with the closure of $\widehat{X}\setminus\{0\}$ in the blow-up $\Bl_0\mathbb{A}^{n+1}$ of $\mathbb{A}^{n+1}$ (see Example 7.15.1 of "Algebraic Geometry" by Hartshorne). One can easily check that $\Bl_0\mathbb{A}^{n+1}$ is isomorphic to $\mathbb{V}(\mathcal{O}_{\mathbb{P}^n}(-1))$, where $\mathbb{V}(\mathcal{E})$ denotes the total space of a vector bundle $\mathcal{E}$. Let $\pi\colon \mathbb{V}(\mathcal{O}_{\mathbb{P}^n}(-1))\to \mathbb{P}^n$ be the canonical projection. An easy computation shows that $\widehat{X}\setminus \{0\}\subset \Bl_0 \mathbb{A}^{n+1}$ corresponds to $\pi^{-1}(X)\setminus E\subset \mathbb{V}(\mathcal{O}_{\mathbb{P}^n}(-1))$, where $E$ is the image of the zero-section. Since $\pi^{-1}(X)\setminus E$ is dense in $\pi^{-1}(X)$, we get $$ \Bl_0\widehat{X}\simeq \pi^{-1}(X)=\mathbb{V}(\mathcal{O}_X(-1)). $$