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Fix integers $2 < d \leq n$.

Suppose that $T$ is a smooth complex curve with marked point $0 \in T$, and $X$ is a closed subscheme of $\mathbb{P}^n_T$, flat over $T$ such that each fiber has Hilbert polynomial $P(x) = dx + 1$. This is the Hilbert polynomial of a degree $d$, arithmetic genus $0$ curve in $\mathbb{P}^n$. Suppose further that the general fiber $X_t$ is a smooth rational curve. If you like, I'm even happy to suppose that this curve spans a $\mathbb{P}^d \subset \mathbb{P}^n$, or even that $d = n$.

If $X_0$ denotes the special fiber with ideal sheaf $I_0$, is there any bound on the smallest number $s$ so that $H^1(\mathbb{P}^n, I_0(s)) = 0$? If you're familiar with the construction of the Hilbert scheme, one must prove that there is an $s$ which works for any closed subscheme of $\mathbb{P}^n$ with a given Hilbert polynomial (i.e. independently of the closed subscheme).

In general, there is an estimate which is known to be sharp (I think) if you consider arbitrary subschemes with a given Hilbert polynomial - the Gotzmann bound. In the case of curves, the bound is on the order of $d^2$, (at the moment though, I'm having trouble finding a reference for this fact). The number that I'm asking for is closely related to the regularity of $X_0$ (so if you know a bound for this number, I would be happy to hear it).

If the special fiber is smooth, then a theorem of Gruson, Lazarsfeld, and Peskine lets us answer this question (their theorem says that if a smooth curve is linearly nondegenerate, then the curve has regularity bounded by degree - codimension + 1. If the curve is contained in a linear subspace, you can compute the regularity in that subspace).

To highlight:

Question: Suppose we have a subscheme which is the limit of smooth rational normal curves. This subscheme could be non-reduced, reducible, contain embedded points, etc. Do we need the full strength of the Gotzmann bound for the regularity in this case or is there a better bound?


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