Let $f: Y=Bl_0^\omega(\mathbb{C}^3)\to \mathbb{C}^3$ be a weighted blow up of $\mathbb{C}^3$ with weights $w(x,y,z)=(1,1,2)$. Then $Y$ and the exceptional divisor $E\cong \mathbb{P}(1,1,2)$ are singular at the same point. Locally, the germ of singularity in $Y$ is isomorphic to a cone over Veronese surface, living in $\mathbb{C}^6$, and $E$, as a Weil divisor, can be thought of as a cone over the image of a quadratic curve mapped in $\mathbb{P}^5$ by the Veronese embedding $v: \mathbb{P}^2\to\mathbb{P}^5$  .

Blow up $\mathbb{C}^6$ at the vertex of the cone, $g:Bl_0\mathbb{C}^6\to \mathbb{C}^6$singularities in $Y$ and $E$ should both be resolved. And we should have some $\mathbb{Q}$-linear equivalence: $$g^*E\sim_{\mathbb{Q}} \tilde{E}+dE'$$ where $E'$ is the exceptional divisor of $g$ and $\tilde{E}$ is the strict transform of $E$ under $g$.

My questions is: How to compute $d$?

What I have tried is to write, locally, $Y$ as $\mathbb{C}^3/\mathbb{Z}_2$ and realized as a cone over Veronese surface, so can be $\textrm{Spec}~\mathbb{C}[x^2, y^2, z^2, xy, yz, xz]$, and $E$ can be cut by $z=0$.

But I am confused about how to define multiplicity of $E$ at the vertex, as a divisor in $Y$. I don't know how to get $d$ by similar argument as the Proposition 3.6, Chapter V in Hartshorne's book.

$\DeclareMathOperator\Bl{Bl}$Let $f: Y=\Bl_0^\omega(\mathbb{C}^3)\to \mathbb{C}^3$ be a weighted blow up of $\mathbb{C}^3$ with weights $w(x,y,z)=(1,1,2)$. Then $Y$ and the exceptional divisor $E\cong \mathbb{P}(1,1,2)$ are singular at the same point. Locally, the germ of singularity in $Y$ is isomorphic to a cone over Veronese surface, living in $\mathbb{C}^6$, and $E$, as a Weil divisor, can be thought of as a cone over the image of a quadratic curve mapped in $\mathbb{P}^5$ by the Veronese embedding $v: \mathbb{P}^2\to\mathbb{P}^5$.

Blow up $\mathbb{C}^6$ at the vertex of the cone, $g:Bl_0\mathbb{C}^6\to \mathbb{C}^6$ singularities in $Y$ and $E$ should both be resolved. And we should have some $\mathbb{Q}$-linear equivalence: $$g^*E\sim_{\mathbb{Q}} \tilde{E}+dE'$$ where $E'$ is the exceptional divisor of $g$ and $\tilde{E}$ is the strict transform of $E$ under $g$.

My questions is: How to compute $d$?

What I have tried is to write, locally, $Y$ as $\mathbb{C}^3/\mathbb{Z}_2$ and realized as a cone over Veronese surface, so can be $\operatorname{Spec} \mathbb{C}[x^2, y^2, z^2, xy, yz, xz]$, and $E$ can be cut by $z=0$.

But I am confused about how to define multiplicity of $E$ at the vertex, as a divisor in $Y$. I don't know how to get $d$ by similar argument as the Proposition 3.6, Chapter V in Hartshorne's book.

Let $f: Y=Bl_0^\omega(\mathbb{C}^3)\to \mathbb{C}^3$ be a weighted blow up of $\mathbb{C}^3$ with weights $w(x,y,z)=(1,1,2)$. Then $Y$ and the exceptional divisor $E\cong \mathbb{P}(1,1,2)$ are singular at the same point. If I consider the log resolution $g$ of this singularity with respect to the pair $(Y,E)$, how to compute $g^*E$?

I think locally, the germ of singularity in $Y$ is isomorphic to a cone over Veronese surface, living in $\mathbb{C}^6$, and $E$ is the cone over a quadratic curve.

$g$ could be the blow up of $\mathbb{C}^6$ at 0, but I don't know what the number $d$ is in the following:

 $$g^*E\sim_{\mathbb{Q}} \tilde{E}+dE'$$ where $E'$ is the exceptional divisor of $g$.

Let $f: Y=Bl_0^\omega(\mathbb{C}^3)\to \mathbb{C}^3$ be a weighted blow up of $\mathbb{C}^3$ with weights $w(x,y,z)=(1,1,2)$. Then $Y$ and the exceptional divisor $E\cong \mathbb{P}(1,1,2)$ are singular at the same point. Locally, the germ of singularity in $Y$ is isomorphic to a cone over Veronese surface, living in $\mathbb{C}^6$, and $E$, as a Weil divisor, can be thought of as a cone over the image of a quadratic curve mapped in $\mathbb{P}^5$ by the Veronese embedding $v: \mathbb{P}^2\to\mathbb{P}^5$ .

Blow up $\mathbb{C}^6$ at the vertex of the cone, $g:Bl_0\mathbb{C}^6\to \mathbb{C}^6$singularities in $Y$ and $E$ should both be resolved. And we should have some $\mathbb{Q}$-linear equivalence:  $$g^*E\sim_{\mathbb{Q}} \tilde{E}+dE'$$ where $E'$ is the exceptional divisor of $g$ and $\tilde{E}$ is the strict transform of $E$ under $g$.

My questions is: How to compute $d$?

What I have tried is to write, locally, $Y$ as $\mathbb{C}^3/\mathbb{Z}_2$ and realized as a cone over Veronese surface, so can be $\textrm{Spec}~\mathbb{C}[x^2, y^2, z^2, xy, yz, xz]$, and $E$ can be cut by $z=0$.

But I am confused about how to define multiplicity of $E$ at the vertex, as a divisor in $Y$. I don't know how to get $d$ by similar argument as the Proposition 3.6, Chapter V in Hartshorne's book.