Flat but not very flat families
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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.
algebraic-geometry
asked Dec 15 '18 at 5:00
hyyyyyhyyyyy
265
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add a comment |
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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.
algebraic-geometry
asked Dec 15 '18 at 5:00
hyyyyyhyyyyy
265
$endgroup$
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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?
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– hyyyyy
Dec 22 '18 at 2:23
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Yes. That's correct. Maybe i mistakenly copied from the wrong tab. I deleted my comment.
$endgroup$
– random123
Dec 22 '18 at 4:21
add a comment |
$begingroup$
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.
algebraic-geometry
asked Dec 15 '18 at 5:00
hyyyyyhyyyyy
265
$endgroup$
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.
algebraic-geometry
algebraic-geometry
asked Dec 15 '18 at 5:00
hyyyyyhyyyyy
265
asked Dec 15 '18 at 5:00
hyyyyyhyyyyy
265
asked Dec 15 '18 at 5:00
hyyyyyhyyyyy
265
asked Dec 15 '18 at 5:00
hyyyyyhyyyyy
265
asked Dec 15 '18 at 5:00
hyyyyyhyyyyy
265
265
$begingroup$
@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?
$endgroup$
– hyyyyy
Dec 22 '18 at 2:23
$begingroup$
Yes. That's correct. Maybe i mistakenly copied from the wrong tab. I deleted my comment.
$endgroup$
– random123
Dec 22 '18 at 4:21
add a comment |
$begingroup$
@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?
$endgroup$
– hyyyyy
Dec 22 '18 at 2:23
$begingroup$
Yes. That's correct. Maybe i mistakenly copied from the wrong tab. I deleted my comment.
$endgroup$
– random123
Dec 22 '18 at 4:21
$begingroup$
@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?
$endgroup$
– hyyyyy
Dec 22 '18 at 2:23
$begingroup$
@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?
$endgroup$
– hyyyyy
Dec 22 '18 at 2:23
$begingroup$
Yes. That's correct. Maybe i mistakenly copied from the wrong tab. I deleted my comment.
$endgroup$
– random123
Dec 22 '18 at 4:21
$begingroup$
Yes. That's correct. Maybe i mistakenly copied from the wrong tab. I deleted my comment.
$endgroup$
– random123
Dec 22 '18 at 4:21
add a comment |
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$begingroup$
@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?
$endgroup$
– hyyyyy
Dec 22 '18 at 2:23
$begingroup$
Yes. That's correct. Maybe i mistakenly copied from the wrong tab. I deleted my comment.
$endgroup$
– random123
Dec 22 '18 at 4:21