Question on an example about flatness in Hartshorne

Question

I have been having trouble understanding some statements regarding flatness in Hartshorne - in particular relating to some of the examples in the text. Any help would be appreciated!

Here is the issue:

• In example III.9.8.4 Hartshorne discusses an example of a family of twisted cubics arising from the projection of $\mathbb{P}^3$ to $\mathbb{P}^2$ from a point. The result is that the flat limit of the twisted cubic is not only singular, but it has an embedded point at the singular point.

• In example III.9.10.1 he explains why if one takes the reduced induced structure on the fibers, then the family is not flat.

My question is: If one takes the flat family $Y\to\mathbb{A}^1$ from the first bullet, and uses the canonical map $Y_{red}\to Y$, then by composing one gets a family $Y_{red}\to \mathbb{A}^1$, which should be flat by proposition III.9.7 (which states that you have flatness over a smooth curve if every associated point of $Y$ maps to the generic point of the curve). Now, I expected $Y_{red}$ to be the family in the second bullet, but since it is not flat, this cannot be the case. What is going on?

Thanks!

Answer 1

Enrique, I think what's happening is that $Y$ does not have an embedded point, only its special fiber does. Taking $Y_{\mathrm red}$ does not change anything.

Answer 2

Sandor is right. Indeed, in Hartshorne's example the total family $Y$ is defined by the ideal

$$I=(a^2(x+1)-z^2, ax(x+1)-yz, xz-ay, y^2-x^2(x+1)) \subset k[a, x,y,z],$$ whereas the central fibre (corresponding to $a=0$) is defined by

$$I_0 = ( z^2, yz, xz, y^2-x^2(x+1)) \subset k[x,y,z].$$

The following Macauley2 script shows that

• $I$ is a radical ideal, hence $Y$ is a reduced affine scheme, that is $Y=Y_{red}$.

• $I_0$ is not radical. In fact, the central fiber $Y_0$ has an embedded point at the node $(0,0,0)$ corresponding to the nilpotent element $z$.

  i1 : k=ZZ/32003;

  i2 : S=k[a,x,y,z];

  i3 : I=ideal (a^2*(x+1)-z^2, a*x*(x+1)-y*z, x*z-a*y, y^2-x^2*(x+1));

  o3 : Ideal of S

  i4 : I==radical I

  o4 = true

  i5 : T=k[x,y,z];

  i6 : I0=ideal (z^2, y*z, x*z, y^2-x^2*(x+1));

  o6 : Ideal of T

  i7 : I0 == radical I0

  o7 = false