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Assume that for some $n \in \mathbb{N}$ I have a (possibly singular) irreducible, linearly normal 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?

${}$

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).

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?

${}$

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).

Assume that for some $n \in \mathbb{N}$ I have a (possibly singular) irreducible, linearly normal 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?

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).

Assume that for some $n \in \mathbb{N}$ I have a (possibly singular) irreducible, linearly normal 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?

${}$

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).

Assume that for some $n \in \mathbb{N}$ I have a (possibly singular) irreducible, linearly normal 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 $|\mathcal{O}_{\mathbb{P}^2}(n)|$, but I suspect that in general this is not the only one: for instance, in the case $n=2$ (where through any point passes a $1$-dimensional family of smooth conics and any two points are joined by at least one smooth conic) I know two examples: a smooth quadric surface in $\mathbb{P}^3$ and the Veronese surface in $\mathbb{P}^5$. 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?

Assume that for some $n \in \mathbb{N}$ I have a (possibly singular) irreducible, linearly normal 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?

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).