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I'm in trouble with the exercise problem iii.12.3 in Hartshorne's AG.

Let $X_{1}$ be a rational normal curve in $\mathbb{P^4}$ ( = image of 4th veronese embedding of $\mathbb{P^1}$).

$X_{0}$ be a rational quartic curve in $\mathbb{P^3}$ with parametrization $[t^4, t^3u, tu^3, u^4]$.

Construct flat family $X$ using projection $\pi: \mathbb{P^4}\to \mathbb{P^3}$ parametrized by $\mathbb{A^1}$,with the given fibers $X_{1}$ and $X_{0}$ for $t=1$ and $t=0$.

Let $\mathcal{I} \subset$ $\mathcal{O_{\mathbb{P^4}\times\mathbb{A^1}}}$ be a total idael sheaf of $X$.

Show that the functions $h^0(t,\mathcal I_{t})$ and $h^1(t,\mathcal{I}_{t})$ are jump at $t=0$.

Still, my calculation doesn't lead to such answer. Rather, it seems to be that they are constant functions. I think such result is more acceptable because all fibers of $X$ are just $\mathbb{P^1}$.

What's wrong with me? I'll appriciate any comments.

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Also posted at MathOverflow – user16299 Mar 5 '12 at 8:52

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