2
$\begingroup$

Let $\mathscr I_0=\mathscr J + (x_0x_2,x_1x_2,x_2^2,x_3x_2,x_4x_2)$. where $\mathscr J$ is the ideal sheaf of a rational normal quartic curve in $\mathbb P^3_{x_0,x_1,x_3,x_4}$. Now, construct closed subscheme in $\mathbb{P^4}$ defined by this ideal sheaf.

Then, saturations of above homogeneous ideal is equal to $\mathscr I=\mathscr J + (x_2)$ , because

$(x_0x_2,x_1x_2,x_2^2,x_3x_2,x_4x_2) = \bigoplus_{d \geq 2} (x_2)_d$

(see Hartshorne's ex ii.3.12) So I think the resulted closed subscheme is just a $\mathbb{P^1}$ as a variety

But, I saw a completely different argument in the answer of my previous MO question, which says there is a global nilpotent section comes from the $x_2$. Here is a link: please see the first answer.

Trouble with semicontinuity

It is very persuasive. (he calculated everythings in full details for me) But it contradicts to my own thought and makes me confusing. How can I harmonize those two?

Improve this question
$\endgroup$
3
  • 2
    $\begingroup$ I've been thinking about this for a while, and I'm a little stumped trying to rectify this with Sandor's response. The special fiber should indeed just be the smooth quartic rational curve in $\mathbb{P}^3$, and I don't see where any nilpotents could come from. I think the issue was in fact due to failing to saturate the ideal, so that your description of the ideal didn't actually provide generators for the ideal in the local charts he was working. Playing around in Macaulay, I can't get any of the cohomology groups to jump whatsoever. $\endgroup$
    – Jack Huizenga
    Commented Mar 8, 2012 at 1:21
  • 1
    $\begingroup$ Perhaps this is the right way to fix the exercise: instead of looking at $H^i(t,I_t)$, look at $H^i(t,I_t(1))$. This clearly jumps over $t=0$ as there the curve lies in a plane, while it is nondegenerate for other $t$. $\endgroup$
    – Jack Huizenga
    Commented Mar 8, 2012 at 1:32
  • $\begingroup$ Thanks for your kind check with Macaulay. Donu also mentioned about twisting, and I think so too. Sandor worked on a projective 3 space, whick makes the variable $x_2$ vague. On $\mathbb{P^4}$, saturating works with no problem. Again, Thanks! $\endgroup$
    – Choa
    Commented Mar 8, 2012 at 1:47

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .