Let $X\subset\mathbb{P}^N$ be an irreducible complex variety. Fix an integer $a\geq 2$ and call $P\_a$ the following property: given $x\_1,\dots,x\_a\in X$ general points there exists an irreducible curve $C\subset X$ such that $x\_1,\dots,x\_a\in C$, $C$ spans a projective space of dimension $2a-1$ and $\deg(C) > 2a-1$.


For instance, $P\_2$ means that two general points of $X$ can be connected by a curve of degree at least $4$ lying in $\mathbb{P}^3$.


I would like to ask whether there is some natural condition one could put on $X$ (especially when $X$ is rational) for $P\_a$ to hold?


Thank you.



Varieties connected by curves in projective spaces of small dimension
I am writing an answer to the question in the comments. Let $a$ be a positive integer. Let $X\subset \mathbb{P}^N$ be a linearly nondegenerate, smooth, projective variety. Let $\alpha \in \text{Hom}(\text{Pic}(X)/\text{Pic}^0(X),\mathbb{Z})$ be the class of a rational curve in $X$. Denote by $a$ the $\mathcal{O}\_{\mathbb{P}^N}(1)|\_X$-degree of $\alpha$. Assume that for $a$ general points of $X$ there exists an irreducible curve of degree $\alpha$ in $X$ that contains those points. This implies the inequality, $$\langle c\_1(T\_X),\alpha \rangle \geq a(\text{dim}(X)-1) - (\text{dim}(X)-3).$$ For smooth complete intersections in $\mathbb{P}^N$ of type $(d\_1,\dots,d\_c)$, this is equivalent to the inequality, $$\frac{N-3-c}{a} + c+2 \geq d\_1 + \dots + d\_c,$$ and this inequality is also sufficient if the characteristic is $0$ and the complete intersection is general in moduli.


Inside the space of irreducible, genus-$0$ stable maps of class $2\alpha$ containing $a$ general points, consider the locus of those whose linear span has dimension at most $2a-1$. By the usual theory of the Thom-Porteous formula, the codimension of every irreducible component of this locus is at most $N+1-2a$. On the other hand, the locus of double covers of curves of class $\alpha$ is in this locus and has codimension $\langle c\_1(T\_X),\alpha \rangle - 2$. Thus, whenever the following inequality holds, $$\langle c\_1(T\_X),\alpha \rangle \geq N+3-2a,$$ then the double covers deform / generize to irreducible, genus-$0$ stable maps that are not double covers (hence are birational to their image) and whose linear span has dimension at most $2a-1$.


In particular, if $X$ is a smooth, complete intersection in $\mathbb{P}^N$ of type $(d\_1,\dots,d\_c)$, the inequality above holds if $$a(N+3-(d\_1+\dots+d\_c)) \geq N+3.$$ For $a=1$, of course this is correct: linearly degenerate "conics" must be double covers of lines. For $a=2$, this recovers the inequality in the comments: $d\_1+\dots+d\_c$ is no greater than $(N+3)/2$.


