We consider the variety $\Sigma_{m}$ := {($p$, $X$) : $X$ is a degree $n$ + 1 hypersurface over $\mathbb{C}$ with mult$_{p}(X) \geq m$} $\subseteq$ $\mathbb{P}$$^{n}$ $\times$ $\mathbb{P}$$^{N}$, where $N$ = $\binom{2n + 1}{n}$ and $1 \leq m \leq n - 1$. My question is can we find a projective fano variety $Y_{m}$ such that every pair $(p, X) \in \Sigma_{m}$ corresponds precisely to a member $D_{m}$ in the linear series $|-K_{Y}|$ $?$
Moreover, can we find a natural morphism sends $Y_{m}$ to $\mathbb{P}^{n}$, $D_{m}$ to $X$ $?$ For example, if we consider $m = n - 1$, we can let $Y_{n - 1}$ be the blow up of $\mathbb{P}^{n}$ along the point $p$ and $D_{m}$ is the strict transform of $X$. However, for remaining cases, l can't find such models.
If l can find such $Y_{m}$, l can show the GIT slope $t$ stability of $(p, X)$ by checking K stability of $(Y_{m}, cD_{m})$, for the positive constant $c$.
Any answers are welcome.
