Mixing
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An inequality about intersection multiplicity [Hartshorne Ex.I.5.4]
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Let $Y,Zsubset mathbb{A}^2$ defined by the polynomials $f$ and $g$. The intersection multiplicity of $Y$ and $Z$ at $p$ is defined as the length of the module $mathcal{O}_{mathbb{A}^2,P}/(f,g)$, denotes by $(Y Z)_p$. For simplicity,let us assume $P=(0,0)$. This length is finite (an argument can be found on this website), but Hartshorne also wants us to prove that it satisfies an inequality $$(YZ)_pgeq mu_P(f)mu_P(g),$$ where $mu_P(f)$ is the the smallest integer $r$ such that the homogenous degree $r$ part of $f$ is non-zero.
I am quite frankly unable to prove this, or have any idea where this could come from.
algebraic-geometry
asked Feb 1 at 8:47
user09127user09127
2166
$endgroup$
add a comment |
$begingroup$
Let $Y,Zsubset mathbb{A}^2$ defined by the polynomials $f$ and $g$. The intersection multiplicity of $Y$ and $Z$ at $p$ is defined as the length of the module $mathcal{O}_{mathbb{A}^2,P}/(f,g)$, denotes by $(Y Z)_p$. For simplicity,let us assume $P=(0,0)$. This length is finite (an argument can be found on this website), but Hartshorne also wants us to prove that it satisfies an inequality $$(YZ)_pgeq mu_P(f)mu_P(g),$$ where $mu_P(f)$ is the the smallest integer $r$ such that the homogenous degree $r$ part of $f$ is non-zero.
I am quite frankly unable to prove this, or have any idea where this could come from.
algebraic-geometry
asked Feb 1 at 8:47
user09127user09127
2166
$endgroup$
$begingroup$
Consider the basis ${1,x,y,x^2,xy,y^2,...}$ of $R[x,y]$ as an $R$ module. Count the terms left over after you mod out by $f$ and $g$.
$endgroup$
– KReiser
Feb 1 at 20:40
$begingroup$
This seems believable, but I have trouble figuring out the details in general. Could you elaborate?
$endgroup$
– user09127
Feb 2 at 11:08
add a comment |
$begingroup$
Let $Y,Zsubset mathbb{A}^2$ defined by the polynomials $f$ and $g$. The intersection multiplicity of $Y$ and $Z$ at $p$ is defined as the length of the module $mathcal{O}_{mathbb{A}^2,P}/(f,g)$, denotes by $(Y Z)_p$. For simplicity,let us assume $P=(0,0)$. This length is finite (an argument can be found on this website), but Hartshorne also wants us to prove that it satisfies an inequality $$(YZ)_pgeq mu_P(f)mu_P(g),$$ where $mu_P(f)$ is the the smallest integer $r$ such that the homogenous degree $r$ part of $f$ is non-zero.
I am quite frankly unable to prove this, or have any idea where this could come from.
algebraic-geometry
asked Feb 1 at 8:47
user09127user09127
2166
$endgroup$
Let $Y,Zsubset mathbb{A}^2$ defined by the polynomials $f$ and $g$. The intersection multiplicity of $Y$ and $Z$ at $p$ is defined as the length of the module $mathcal{O}_{mathbb{A}^2,P}/(f,g)$, denotes by $(Y Z)_p$. For simplicity,let us assume $P=(0,0)$. This length is finite (an argument can be found on this website), but Hartshorne also wants us to prove that it satisfies an inequality $$(YZ)_pgeq mu_P(f)mu_P(g),$$ where $mu_P(f)$ is the the smallest integer $r$ such that the homogenous degree $r$ part of $f$ is non-zero.
I am quite frankly unable to prove this, or have any idea where this could come from.
algebraic-geometry
algebraic-geometry
asked Feb 1 at 8:47
user09127user09127
2166
asked Feb 1 at 8:47
user09127user09127
2166
asked Feb 1 at 8:47
user09127user09127
2166
asked Feb 1 at 8:47
user09127user09127
2166
asked Feb 1 at 8:47
user09127user09127
2166
2166
$begingroup$
Consider the basis ${1,x,y,x^2,xy,y^2,...}$ of $R[x,y]$ as an $R$ module. Count the terms left over after you mod out by $f$ and $g$.
$endgroup$
– KReiser
Feb 1 at 20:40
$begingroup$
This seems believable, but I have trouble figuring out the details in general. Could you elaborate?
$endgroup$
– user09127
Feb 2 at 11:08
add a comment |
$begingroup$
Consider the basis ${1,x,y,x^2,xy,y^2,...}$ of $R[x,y]$ as an $R$ module. Count the terms left over after you mod out by $f$ and $g$.
$endgroup$
– KReiser
Feb 1 at 20:40
$begingroup$
This seems believable, but I have trouble figuring out the details in general. Could you elaborate?
$endgroup$
– user09127
Feb 2 at 11:08
$begingroup$
Consider the basis ${1,x,y,x^2,xy,y^2,...}$ of $R[x,y]$ as an $R$ module. Count the terms left over after you mod out by $f$ and $g$.
$endgroup$
– KReiser
Feb 1 at 20:40
$begingroup$
Consider the basis ${1,x,y,x^2,xy,y^2,...}$ of $R[x,y]$ as an $R$ module. Count the terms left over after you mod out by $f$ and $g$.
$endgroup$
– KReiser
Feb 1 at 20:40
$begingroup$
This seems believable, but I have trouble figuring out the details in general. Could you elaborate?
$endgroup$
– user09127
Feb 2 at 11:08
$begingroup$
This seems believable, but I have trouble figuring out the details in general. Could you elaborate?
$endgroup$
– user09127
Feb 2 at 11:08
add a comment |
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$begingroup$
Consider the basis ${1,x,y,x^2,xy,y^2,...}$ of $R[x,y]$ as an $R$ module. Count the terms left over after you mod out by $f$ and $g$.
$endgroup$
– KReiser
Feb 1 at 20:40
$begingroup$
This seems believable, but I have trouble figuring out the details in general. Could you elaborate?
$endgroup$
– user09127
Feb 2 at 11:08