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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 $C_w$.

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    $\begingroup$ What is the question? $\endgroup$ – Sasha Apr 19 '18 at 13:20
  • $\begingroup$ @Sasha. The question is whether, for $d$ sufficiently positive, for a sufficiently general closed point $w$ of $W$ parameterizing a closed subscheme $\text{Zero}(\mathcal{I}_w)$ of $\mathbb{P}^n$, for a general member $F_d$ of the linear series $\mathcal{I}_w\cdot \mathcal{O}(d)$, is $w$ the unique point $v$ of $W$ such that $F_d$ is in the linear series $\mathcal{I}_v\cdot \mathcal{O}(d)$. This can fail for small values of $d$, e.g., if $d$ equals $1$ and $W$ parameterizes closed subschemes that do not span $\mathbb{P}^n$. $\endgroup$ – Jason Starr Apr 19 '18 at 15:21
  • $\begingroup$ Stated as above, the question is whether, for every sufficiently general $w$ in $W$, for every $v$ in $W\setminus\{w\}$, does the quotient sheaf $\mathcal{I}_w/(\mathcal{I}_v\cap \mathcal{I}_w)$ have positive-dimensional support. If so, then the Hilbert function of this quotient increases as a function of $d$. Together with the usual upper semicontinuity arguments, there exists an integer $d_0$ such that for every integer $d\geq d_0$, the Hilbert function in degree $d$ is strictly greater than the dimension of $W$, and the result follows. $\endgroup$ – Jason Starr Apr 19 '18 at 17:06
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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.

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