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Let $\mathcal{D} \approx \mathbb{P}^{\delta_d} $ be the space of nonzero homogeneous degree $d$ polynomials in three variables upto scaling, where $\delta_d = \frac{d(d+3)}{2} $ (basically degree $d$ curves in $\mathbb{P}^2$). Suppose $\mathcal{V}$ is a smooth sub variety of $\mathcal{D}$, not necessarily closed that has the following property: a generic element $[f] \in \mathcal{V}$ has a singularity only at the point $[1,0,0]$. Suppose $k$ is the dimension of $\mathcal{V}$. Define the following fibered products: $$ \overline{\mathcal{C}}(k):= \{ ([f], p_1, \ldots, p_k) \in \overline{\mathcal{V}} \times (\mathbb{P}^2)^k: f(p_1) =0, \ldots, f(p_k) =0 \} $$

$$ \mathcal{C}(k):= \{ ([f], p_1, \ldots, p_k) \in \mathcal{V} \times (\mathbb{P}^2)^k: f(p_1) =0, ~\nabla f|_{p_1} \neq 0, \ldots, f(p_k) =0, ~\nabla f|_{p_k} \neq 0 \} $$

Here $\overline{\mathcal{V}}$ denotes the closure of $\mathcal{V}$ in $\mathcal{D}$. I have two questions:

1) Is it true that $\mathcal{C}(k)$ is dense in $ \overline{\mathcal{C}}(k) $ (I apologize for the notation). And why?

2) Consider the projection map $$ \pi: \overline{\mathcal{C}}(k) \rightarrow (\mathbb{P}^2)^k $$ Is it true that the image $$ \pi( \overline{\mathcal{C}}(k) - \mathcal{C}(k) ) \subset (\mathbb{P}^2)^k$$ is "small" inside $(\mathbb{P}^2)^k$, ie it is contained in a union of subavrieties of dimension $2k-1$ or less?

The specific example I have in mind is when $\mathcal{V} = \mathcal{A}$, where $$\mathcal{A}:= \{ [f] \in \mathcal{D}: f([1,0,0]) =0, ~~\nabla f|_{[1,0,0]} =0, ~~det \nabla^2 f|_{[1,0,0]} \neq 0 \} $$

ie the space of curves having a simple node at $[1,0,0]$. It can be shown that a generic element of $\mathcal{A}$ has only one singular point (namely $[1,0,0]$).

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1) It is dense because, given a curve with a finite set of points, one can modify the curve's equation slightly so that it is smooth except for $[1,0,0]$, and then modify each of the points slightly so they still lie on the curve.

More precisely, given a point which might not lie in the closure, since the space of curves smooth away from $[1:0:0]$ is dense in the space of all curves, we can view the curve with the marked points as a special fiber to a family of curves over a DVR whose generic fiber is smooth. Then we lift the marked points, possibly having to make an integral extension to the DVR, just by fixing one coordinate and then adjoining a solution to the equation. Then we look at the image of the generic point in $C$ and deduce that the questionable point lies in the closure of that point.

2) Clearly not if $d$ is sufficiently large relative to $k$. Clearly so if $k$ is sufficiently large relative to $d$.

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