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I'm doing an exercise in Hartshorne's Algebraic Geometry, Ex 9.5 in Chapter III, whose part (a) states the following:

Given an example to show that if $\{X_t\}$ is a flat family of closed subschemes of $\mathbb{P}^n$, then the projective cone $\{C(X_t)\}$ need not be a flat family in $\mathbb{P}^n$.

Since we can determine flatness by Hilbert polynomials. I want to use the dimension formula $dim((S_t/I_t)[x])_d = \sum_{i = 0}^{d}dim(S_t/I_t)_i$, to construct a family with same Hilbert polynomial but have different dimension in lower degrees.

This turned out to be some special example of flat but not very flat families related to the remaining part of this exercise, but I failed in finding such counter examples.

hyyyyy
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  • @random123 Thanks for comment. I think you are refering to S. Wang and J. Zhao's article examples and counter examples, but maybe your link is mistakened? – hyyyyy Dec 22 '18 at 02:23
  • Yes. That's correct. Maybe i mistakenly copied from the wrong tab. I deleted my comment. – random123 Dec 22 '18 at 04:21

1 Answers1

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Hint: compute the Hilbert function of three points in $\Bbb P^2$. There are two options, depending on some geometric information. Can you arrange a flat family which exploits this?

I'll put the Hilbert polynomial behind a spoiler block so you can make a go of computing it yourself before reading what it is.

The Hilbert function of three points in $\Bbb P^2$ is $$\begin{cases} 0 & n<1 \\ 2 & n=1 \\ 3 & n>1\end{cases}$$ if they're collinear, or $$\begin{cases} 0 & n<1 \\ 3 & n\geq 1\end{cases}$$ if they're not collinear.

A strategy for proving this:

Prove this by looking at the evaluation map $k[x,y,z]_d\to k^3$.

Do you see how use this to solve the problem?

KReiser
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  • Hi, I conjecture that your Ansatz suggests to consider a flat family like $X_t$ defined by $(1:0:0), (0:1:0)$, and $(1:1:t)$ in $\mathbb{P}^2$. (that's what I found und the idea sound quite similar to yours, because the intention of this choice was also to exploit that these points are only on a line together at $t = 0$. – user267839 May 01 '21 at 10:39
  • But one issue confuses me. We can regard this family as closed subscheme $X_t \subset \mathbb{P}^2 \times \mathbb{A}^1$ and doesn't it's flatness contradicts to Hartshorne's Thm 9.9 Chapt. III? Is says that all fibers of a flat family have same Hilbert polynomial. On the other hand by construction and your observation on collinearity this would imply that $X_0$ and $X_1$ have different Hilbert polynomials, thus by Thm 9.9 not flat.
    On the other hand it is obviuosly flat since isomorphic to $X_1 \times \mathbb{A}^1$. Could you resolve my confusion? Otherwise I'm sure if I understand your     – user267839 May 01 '21 at 10:45
  • motivation to make use of the property of Hilbert function of three points in oorder to constrctuct a flat family $X_t$ with desired properties. Could you elaborate it a bit more? – user267839 May 01 '21 at 10:47
  • I think I see your argument now. If $M(X)= \oplus_d M(X)d$ is the graded complex associated to subvariety $X \subset P^2$ and $M(CX)=M(CX)_d$ the gc associated to the cone $CX \subset P^3$, then $\dim M(CX)_d= \sum{i=0}^d \dim M(X)_i$. Therefore the Hilbert polynomial of $CX_0$ differs always from HP of $CX_t$, $t \neq 0$ since asymptotically $\dim M(CX_0)_d$ and $\dim M(CX_t)$ differ for $d$ big. That's the argument, I think. – user267839 May 01 '21 at 20:09
  • (If one prefers not to interact with another user, one simply does not do it: it's that easy! ;). I have deleted unnecessary comments.) – Pedro May 03 '21 at 17:56