Let us fix the base field to be the field of complex numbers (Maybe it's not quite important).
Recall the following definition. Let $X$ be a quasi-projective variety, singular at a point $x$. Let $C_{X,x}$ (resp. $T_{X,x}$) be the tangent cone (resp. Zariski tangent space) of $X$ at $x$. The multiplicity $\textrm{mult}_x(X)$ of $X$ at point $x$ is defined to be the degree of the divisor $\mathbb PC_{X,x}$ in the projective space $\mathbb PT_{X,x}$. (ref. Albarello, Cornalba, Griffiths, Harris, Geometry of algebraic curves, vol I, page 62)
My question is as follows. Let $N$ be a big positive integer. Let $Y\subset \mathbb P^N$ be a smooth cubic hypersurface and let $P\subset Y$ be a fixed general plane. Let $\Theta_P$ be the subscheme of $Gr(3, N+1)$ containing all the planes in $Y$ whose intersection with $P$ contains a line. Are there any effective method to calculate the multiplicity of the variety $\Theta_P$ at the point $[P]\in \Theta_P$ ?
Any comments or suggestions are welcome.
