General hypersurfaces containing a general member of HilbP^n

Question

Let W be a subvariety of some Hilbert scheme of $P^n$, $n \geq 3$. Assume the general member $C_w \subset P^n$ of W is pure positive dimensional, not necessarily reduced (e.g., defined by the square of the ideal of a (variable) smooth curve in $P^3$).

Let $V_d={ (w , F_d) | F_d \; \text{is a hypersurface of degree d containing} \; C_w}$.

Then for all $d>>0$ the map $(w,F_d) \rightarrow F_d$ is generically injective.

In other words, for all $d>>0$, if w is general in W and $F_d$ is general in the space of hypersurfaces of degree d containing $C_w$ then w is the only member of W such that $F_d$ contains $C_w$.

Answer 1

I believe that this is not true. For every integer $g$ that is sufficiently positive, I believe that Henry Pinkham has proved that projective cones in $\mathbb{P}^g$ over canonically embedded, genus $g$ curves in $\mathbb{P}^{g-1}$ are "rigid" in the sense that the only (embedded) deformations arise as cones over deformations of the curve in $\mathbb{P}^{g-1}$. (Ed. Actually, that might be due to Mumford, but I am having trouble tracking down a reference.) Assuming this, let $\mathcal{I}\subset \mathcal{O}_{\mathbb{P}^g}$ be the ideal sheaf of such a cone.

Denote by $\mathfrak{m}$ the maximal ideal sheaf in $\mathcal{O}_{\mathbb{P}^g}$ of the vertex of the cone. Consider the ideal sheaves $\mathcal{J}$ such that $$\mathfrak{m}\cdot \mathcal{I} \subset \mathcal{J} \subset \mathcal{I}$$ and such that the quotient $\mathcal{I}/\mathcal{J}$ has length $1$, i.e., the quotient equals the skyscraper sheaf of the vertex. By Max Noether's Theorem, the dimension of the locus of such ideals (for fixed ideal $\mathcal{I}$) equals $(g^2-5g+4)/2$. On the other hand, the locus parameterizing ideal sheaves $\mathcal{G}\subset \mathcal{I}$ of colength $1$ whose cokernel has support disjoint from the vertex has dimension $g$. If $g\geq 8$, it appears that the locus of ideal sheaves $\mathcal{J}$ as above forms an irreducible component of the Hilbert scheme (allowing also the cone to vary).

That is bad news. For each such ideal sheaf $\mathcal{I}$ and ideal sheaf $\mathcal{J}$, consider any other ideal sheaf $\mathcal{J}'\subset \mathcal{I}$ of colength $1$ containing $\mathfrak{m}\cdot \mathcal{I}$. Among polynomials $F_d$ of degree $d$ that are contained in $\mathcal{J}\cdot \mathcal{O}(d)$, it is only one more condition to be contained in $\mathcal{J}\cap \mathcal{J}'\cdot \mathcal{O}(d)$. Thus, varying $\mathcal{J}'$ in a $1$-parameter family, this collection of hyperplanes in $H^0(\mathbb{P}^g,\mathcal{J}\cdot \mathcal{O}(d))$ should sweep out the entire vector space. Therefore, it appears to me that the question above has a negative answer.