Consider the following quadric hypersurfaces in $\mathbb{P}^5$:
$H_1: x_0l_1(x_4,x_5)+q_1(x_1,x_2,x_3,x_4,x_5)=0$,
$H_2: x_0x_5+x_1l_2(x_3,x_4,x_5)+q_2(x_2,x_3,x_4,x_5)=0$,
$H_3: x_3x_5+q_3(x_4,x_5)=0$
where $l_i$: linear equation, $q_j$: quadratic equation.
Then $S=H_1\cap H_2\cap H_3$ is a surface in $\mathbb{P}^5$.
I want to know about the multiplicity of $S$ at $(1,0,0,0,0,0)$.
How can I compute the multiplicity?
Clearly $S$ is singular at $(1,0,0,0,0,0)$. So the multiplicity at $(1,0,0,0,0,0)$ is $\ge 2$.
If $S$ is general (i.e. $l_i$ and $q_j$ are general), then can I find the larger boundary than $2$?