Hypersurfaces containing a general chain of lines

Question

Let $X$ be a general chain of $d$ lines in $\mathbb P^n$, where $n \geq 3$. Let $I$ be the homogeneous ideal of polynomials vanishing on $X$. What is the Hilbert function $$P(k) = \dim I_k$$ of $X$? Note that it is equivalent to let $P(k)$ be one more than the dimension of the space of degree $k$ hypersurfaces containing $X$.

In particular, an answer would be implied if the Maximal Rank Conjecture holds for $X$, i.e., if for all $k$, the morphism $$H^0(\mathbb P^n, \mathscr O(k)) \to H^0(X, \mathscr O(k))$$ is either injective or surjective (i.e., of maximal rank given its domain and codomain).

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By a "general chain of lines," I mean the following: choose $d+1$ general points $p_0, \dotsc, p_d \in \mathbb P^n$; let $\ell_i$ be the line spanned by $p_{i-1}$ and $p_i$; and let $X = \bigcup_{i=1}^d \ell_i$.

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Background: Many results of this nature are claimed when $X$ is a general (smooth) curve of a specified genus and degree; see, for instance, papers by Ballico and Ellia including the phrase "maximal rank conjecture" in the title, as well as papers by Hartshorne and Hirschowitz (an number of which are in French). The arguments typically proceed by some form of degeneration. However, since the degenerate varieties involved are not the main subjects of inquiry, it can be hard to recognize what exactly is known about degenerate varieties.

Answer 1

The result closer to your question I know is the following:

Let $X\subset\mathbb{P}^{N}$ be a variety set theoretically defined by homogeneous polynomials $G_{i}$ of degree $d_{i}$, for $i = 1,..,m$, and let $l\geq 2$ be an integer. If $$\sum_{i=1}^{m}d_{i}\leq \frac{N(l-1)+m}{l}$$ then $X$ is rationally chain connected by chains of lines of length at most $l$. In particular if $X$ is smooth and the above inequality is satisfied then $X$ is rationally connected by rational curves of degree at most $l$.

As corollaries we get:

• Let $X\subset\mathbb{P}^{N}$ be a hypersurface of degree $d\leq N-1$. Then $X$ is rationally chain connected by a chain of lines of length at most $N-1$.
• Let $X\subset\mathbb{P}^{N}$ be a scheme theoretical complete intersection of codimension $c$. If $\deg(X)\leq\frac{N(l-1)+c}{l}$ then $X$ is rationally chain connected by chains of lines of length at most $l$.
• Let $X\subseteq\mathbb{P}^{N}$ be a smooth complete intersection of codimension $c$ defined by homogeneous polynomials $G_{i}$ of degree $d_{i}$, for $i = 1,..,c$, such that $\sum_{i=1}^{c}d_{i}\leq N-1$. Then $$length(X) = \lceil \frac{N-c}{N-\sum_{i=1}^{c}d_{i}}\rceil,$$ where $\lceil k\rceil$ is the smallest integer greater or equal than $k$.

You can find all of this here: http://arxiv.org/abs/1106.0124.