Assume that for some $n \in \mathbb{N}$ I have a (possibly singular) irreducible, non-degenerate complex surface $X_n \subset \mathbb{P}^{N}$ with the following properties:

 1. for all $p \in X_n$ there exists a $1$-dimensional family $\mathcal{C}_p$ of rational normal curves of degree $n$ containing $p$; 
 2. any two points $p, \, q \in X_n$ are joined by at least one rational normal curve of degree $n$ $C_{pq} \in \mathcal{C}_p \cap \mathcal{C}_q$.

>**Q1.** Is it possible to completely classify those surfaces $X_n$ with the above properties?

An example is the $n^{\mathrm{th}}$ Veronese surface, namely $\mathbb{P}^2$ embedded in $\mathbb{P}^{n(n+3)/2}$ by $|\mathcal{O}_{\mathbb{P}^2}(n)|$. Are there more? Maybe singular examples?

In case **Q1** turns out to be hopeless, let me ask

>**Q2.** Is it possible to explicitly bound $N$ from above in function of $n$, i.e. finding an explicit numerical function $\varphi$ such that $N \leq \varphi(n)?$ For instance, is it true that $N \leq n(n+3)/2$, i.e. that the Veronese surface provides the example with the highest codimension?

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**Remark.** If one replace "any" in the second property by "sufficiently general" then there are other examples: for instance we can consider the quadric surface in $\mathbb{P}^3$, in which every two points, not on the same ruling, are joined by a smooth conic (see potentially dense's and J. Starr's comments below).