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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).


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 $X = \bigcup_{i=1}^d \ell_i$.


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.

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  • $\begingroup$ I would bet this is feasible (by degenerations, again) but I suspect it takes some work. It could be that they are not "maximal rank" curves, but one can compute their regularity, and work out low degree cases. Are you interested in some particular application? $\endgroup$ – quim May 23 '13 at 20:01
  • $\begingroup$ quim: Yes, actually I am, but since the application would constitute an important point in my thesis if it works, I can't exactly ask about it here. I will say, though, that showing these curves are "maximal rank" would be strictly weaker than solving my application. So if you (or anyone) sees a counterexample to this (for general chains of lines), please let me know. $\endgroup$ – Charles Staats May 23 '13 at 21:08
  • $\begingroup$ Ironic that I should see this now, when contemplating this question for a week before posting, but I think I have a solution. I'll post it later if it actually holds up--I haven't completely run it through its paces yet. $\endgroup$ – Charles Staats May 24 '13 at 1:37
  • $\begingroup$ Okay, the method I found does give many cases, but does not completely solve the problem--yet. $\endgroup$ – Charles Staats May 25 '13 at 3:29
  • $\begingroup$ Update: The first case I cannot handle by my current methods is $n=3$, $d=11$, $k=5$. In other words: Is a general chain of 11 lines in $\mathbb P^3$ contained in any quintic hypersurface? The maximal rank conjecture would predict "no"; currently, I can show that it is contained in at most a pencil of quintics. $\endgroup$ – Charles Staats May 26 '13 at 0:44
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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.

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