Does the rational normal curve embedding extend as a mapping from the "bulk" to some bigger ambient space?

Question

The complex projective line $\mathbb{P}^1(\mathbb{C})$ can be identified with the sphere at infinity of hyperbolic $3$-space, modeled say by the Poincare open $3$-ball in $\mathbb{R}^3$ (the sphere at infinity would then corresponding to the unit $2$-sphere in the Poincare model).

There is a nice famous map, the embedding of $\mathbb{P}^1(\mathbb{C})$ as a rational normal curve of degree $d$ inside $\mathbb{P}^d(\mathbb{C})$. In homogeneous coordinates, it maps $v \in \mathbb{C}^2 \setminus \{ \mathbf{0} \}$ to $\operatorname{Sym}^d(v) \in \operatorname{Sym}^d(\mathbb{C}^2) \setminus \{\mathbf{0}\}$. Let us denote the induced embedding by $\nu_d$, so that $$ \nu_d: \mathbb{P}^1(\mathbb{C}) \to \mathbb{P}^d(\mathbb{C}).$$

My question is this. Can one extend $\nu_d$, to some other map, say $\tilde{\nu}_d$, from $\overline{B(0,1)}$ (thought of as the closure of the Poincare open ball, so as to also include the sphere at infinity), to some real analytic manifold $M$ containing $\mathbb{P}^d$ as some kind of boundary "at infinity" in some sense, and such that one recovers $\nu_d$ when restricting to the sphere at infinity in the domain of $\tilde{\nu}_d$?

As such, I think the answer is most probably yes, since I did not impose many restrictions (and since my question is somewhat vague). However, I am interested in some natural construction, possibly involving some Grassmannian as $M$ (I think there may be a related construction in an article by Claude LeBrun).

I will think about this soon. I think it is an interesting question though.

Answer 1

I think one way to approach this "geometric dream", so to speak, is as follows. Using the so-called stellar isomorphism, $\mathbb{P}^d(\mathbb{C})$, thought of as $\mathbb{P}(\operatorname{Sym}^d(\mathbb{C}^2))$, is the symmetric product of $d$ copies of $\mathbb{P}^1(\mathbb{C})$. On the other hand, the rational curve inside $\mathbb{P}^d(\mathbb{C})$ corresponds to the case where the $d$ points on $\mathbb{P}^1(\mathbb{C})$ all coincide.

So I think one may take $M$ to be the unordered configuration space of $d$ distinct points in the open ball $B(0,1)$. Then $\mathbb{P}^d(\mathbb{C})$ corresponds to the subset of the boundary at infinity of $M$ where all $d$ points lie on the sphere at infinity (note that they are no longer necessarily distinct).

The map $\tilde{\nu}: \overline{B(0,1)} \to \overline{M}$ which maps a point, say $p \in \overline{B(0,1)}$ to $p$ with multiplicity $d$, which can be thought of as a point in $\overline{M}$, has the property that it gives $\nu_d$ upon restricting the map to the sphere at infinity.

I believe this is essentially the kind of map I wanted to construct.