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?

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?

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 $\mathbb{P^1}$

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?

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?