Let $V\subset \mathbb{A}^n$ be a variety defined by equations of degree $\leq D$, or, what is the same, an intersection of hypersurfaces of degree $\leq D$. Let $V^+_0$ be an irreducible component of $V$ of maximal dimension $d$. Is there a variety $W$ defined by equations of degree $O(n D)$ such that $V\cap W$ has $V^+_0$ as its only top-dimensional irreducible component?

(Additional questions/remarks: (a) Do Gröbner bases give an answer (presumably "yes")? (b) We know that asking for $W$ to be defined by equations of degree $\leq D$ is too much: see How does taking intersections and irreducible components affect the degree?)

UPDATE: $O(n D)$ is not in general enough. Can we do $O(D^2)$? Or $O(n D^2)$?

Let $V\subset \mathbb{A}^n$ be a variety defined by equations of degree $\leq D$, or, what is the same, an intersection of hypersurfaces of degree $\leq D$. Let $V^+_0$ be an irreducible component of $V$ of maximal dimension $d$. Is there a variety $W$ defined by equations of degree $O(n D)$ such that $V\cap W$ has $V^+_0$ as its only top-dimensional irreducible component?

(Additional questions/remarks: (a) Do Gröbner bases give an answer (presumably "yes")? (b) We know that asking for $W$ to be defined by equations of degree $\leq D$ is too much: see How does taking intersections and irreducible components affect the degree?)

UPDATE: $O(n D)$ does not always hold. Can we do $O(D^2)$? Or $O(n D^2)$?

Let $V\subset \mathbb{A}^n$ be a variety defined by equations of degree $\leq D$, or, what is the same, an intersection of hypersurfaces of degree $\leq D$. Let $V^+_0$ be an irreducible component of $V$ of maximal dimension $d$. Is there a variety $W$ defined by equations of degree $O(n D)$ such that $V\cap W$ has $V^+_0$ as its only top-dimensional irreducible component?

(Additional questions/remarks: (a) Do Gröbner bases give an answer (presumably "yes")? (b) We know that asking for $W$ to be defined by equations of degree $\leq D$ is too much: see How does taking intersections and irreducible components affect the degree?)

Let $V\subset \mathbb{A}^n$ be a variety defined by equations of degree $\leq D$, or, what is the same, an intersection of hypersurfaces of degree $\leq D$. Let $V^+_0$ be an irreducible component of $V$ of maximal dimension $d$. Is there a variety $W$ defined by equations of degree $O(n D)$ such that $V\cap W$ has $V^+_0$ as its only top-dimensional irreducible component?

(Additional questions/remarks: (a) Do Gröbner bases give an answer (presumably "yes")? (b) We know that asking for $W$ to be defined by equations of degree $\leq D$ is too much: see How does taking intersections and irreducible components affect the degree?)

UPDATE: $O(n D)$ is not in general enough. Can we do $O(D^2)$? Or $O(n D^2)$?