Relations between homogeneous polynomials












6












$begingroup$


Let $F_1,ldots ,F_ellin mathbb{C}[X_1,ldots , X_n]$ be general homogeneous polynomials of degree $d$. For $p<d$, it seems likely that there is no nontrivial relation $sum F_iG_i=0$ with $G_1,ldots ,G_ell$ homogeneous of degree $p$. Is this known? If yes, what would be a reference (or a proof)?










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$endgroup$












  • $begingroup$
    I think this question needs some clarification. I think you may mean something along the lines of: let $X$ be the moduli space of $n$-tuples of homogeneous polynomials of degree $d$; within $X$, there is a subspace $Y$ consisting of homogeneous polynomials that satisfy a nontrivial relation $sum F_i G_i = 0$ for some $G_i$ homogeneous of degree $p < d$ (i.e. the union of the subspaces determined by each tuple $G$). Is it true that $Y subsetneq X$?
    $endgroup$
    – user44191
    7 hours ago












  • $begingroup$
    Sure. Is that different from what I wrote?
    $endgroup$
    – abx
    7 hours ago










  • $begingroup$
    I found what you wrote ambiguous or underdetermined, e.g. with what you mean by "general"; I thought what I wrote made it more explicitly clear (though it may be overkill as written).
    $endgroup$
    – user44191
    7 hours ago










  • $begingroup$
    This seem trivially false if I haven't misunderstood the question. Take $n = 1, p = 0, d = 1, ell = 2$. Then $F_1 = ax, F_2 = bx$ for some $a, b$, so $b F_1 - a F_2 = 0$. I think both $n, ell$ play some role.
    $endgroup$
    – user44191
    7 hours ago












  • $begingroup$
    Oh, right. $ellleq n$ would be fine for me, though this can be certainly weakened.
    $endgroup$
    – abx
    7 hours ago
















6












$begingroup$


Let $F_1,ldots ,F_ellin mathbb{C}[X_1,ldots , X_n]$ be general homogeneous polynomials of degree $d$. For $p<d$, it seems likely that there is no nontrivial relation $sum F_iG_i=0$ with $G_1,ldots ,G_ell$ homogeneous of degree $p$. Is this known? If yes, what would be a reference (or a proof)?










share|cite|improve this question











$endgroup$












  • $begingroup$
    I think this question needs some clarification. I think you may mean something along the lines of: let $X$ be the moduli space of $n$-tuples of homogeneous polynomials of degree $d$; within $X$, there is a subspace $Y$ consisting of homogeneous polynomials that satisfy a nontrivial relation $sum F_i G_i = 0$ for some $G_i$ homogeneous of degree $p < d$ (i.e. the union of the subspaces determined by each tuple $G$). Is it true that $Y subsetneq X$?
    $endgroup$
    – user44191
    7 hours ago












  • $begingroup$
    Sure. Is that different from what I wrote?
    $endgroup$
    – abx
    7 hours ago










  • $begingroup$
    I found what you wrote ambiguous or underdetermined, e.g. with what you mean by "general"; I thought what I wrote made it more explicitly clear (though it may be overkill as written).
    $endgroup$
    – user44191
    7 hours ago










  • $begingroup$
    This seem trivially false if I haven't misunderstood the question. Take $n = 1, p = 0, d = 1, ell = 2$. Then $F_1 = ax, F_2 = bx$ for some $a, b$, so $b F_1 - a F_2 = 0$. I think both $n, ell$ play some role.
    $endgroup$
    – user44191
    7 hours ago












  • $begingroup$
    Oh, right. $ellleq n$ would be fine for me, though this can be certainly weakened.
    $endgroup$
    – abx
    7 hours ago














6












6








6


1



$begingroup$


Let $F_1,ldots ,F_ellin mathbb{C}[X_1,ldots , X_n]$ be general homogeneous polynomials of degree $d$. For $p<d$, it seems likely that there is no nontrivial relation $sum F_iG_i=0$ with $G_1,ldots ,G_ell$ homogeneous of degree $p$. Is this known? If yes, what would be a reference (or a proof)?










share|cite|improve this question











$endgroup$




Let $F_1,ldots ,F_ellin mathbb{C}[X_1,ldots , X_n]$ be general homogeneous polynomials of degree $d$. For $p<d$, it seems likely that there is no nontrivial relation $sum F_iG_i=0$ with $G_1,ldots ,G_ell$ homogeneous of degree $p$. Is this known? If yes, what would be a reference (or a proof)?







ag.algebraic-geometry ac.commutative-algebra






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 8 hours ago







abx

















asked 8 hours ago









abxabx

23.8k34885




23.8k34885












  • $begingroup$
    I think this question needs some clarification. I think you may mean something along the lines of: let $X$ be the moduli space of $n$-tuples of homogeneous polynomials of degree $d$; within $X$, there is a subspace $Y$ consisting of homogeneous polynomials that satisfy a nontrivial relation $sum F_i G_i = 0$ for some $G_i$ homogeneous of degree $p < d$ (i.e. the union of the subspaces determined by each tuple $G$). Is it true that $Y subsetneq X$?
    $endgroup$
    – user44191
    7 hours ago












  • $begingroup$
    Sure. Is that different from what I wrote?
    $endgroup$
    – abx
    7 hours ago










  • $begingroup$
    I found what you wrote ambiguous or underdetermined, e.g. with what you mean by "general"; I thought what I wrote made it more explicitly clear (though it may be overkill as written).
    $endgroup$
    – user44191
    7 hours ago










  • $begingroup$
    This seem trivially false if I haven't misunderstood the question. Take $n = 1, p = 0, d = 1, ell = 2$. Then $F_1 = ax, F_2 = bx$ for some $a, b$, so $b F_1 - a F_2 = 0$. I think both $n, ell$ play some role.
    $endgroup$
    – user44191
    7 hours ago












  • $begingroup$
    Oh, right. $ellleq n$ would be fine for me, though this can be certainly weakened.
    $endgroup$
    – abx
    7 hours ago


















  • $begingroup$
    I think this question needs some clarification. I think you may mean something along the lines of: let $X$ be the moduli space of $n$-tuples of homogeneous polynomials of degree $d$; within $X$, there is a subspace $Y$ consisting of homogeneous polynomials that satisfy a nontrivial relation $sum F_i G_i = 0$ for some $G_i$ homogeneous of degree $p < d$ (i.e. the union of the subspaces determined by each tuple $G$). Is it true that $Y subsetneq X$?
    $endgroup$
    – user44191
    7 hours ago












  • $begingroup$
    Sure. Is that different from what I wrote?
    $endgroup$
    – abx
    7 hours ago










  • $begingroup$
    I found what you wrote ambiguous or underdetermined, e.g. with what you mean by "general"; I thought what I wrote made it more explicitly clear (though it may be overkill as written).
    $endgroup$
    – user44191
    7 hours ago










  • $begingroup$
    This seem trivially false if I haven't misunderstood the question. Take $n = 1, p = 0, d = 1, ell = 2$. Then $F_1 = ax, F_2 = bx$ for some $a, b$, so $b F_1 - a F_2 = 0$. I think both $n, ell$ play some role.
    $endgroup$
    – user44191
    7 hours ago












  • $begingroup$
    Oh, right. $ellleq n$ would be fine for me, though this can be certainly weakened.
    $endgroup$
    – abx
    7 hours ago
















$begingroup$
I think this question needs some clarification. I think you may mean something along the lines of: let $X$ be the moduli space of $n$-tuples of homogeneous polynomials of degree $d$; within $X$, there is a subspace $Y$ consisting of homogeneous polynomials that satisfy a nontrivial relation $sum F_i G_i = 0$ for some $G_i$ homogeneous of degree $p < d$ (i.e. the union of the subspaces determined by each tuple $G$). Is it true that $Y subsetneq X$?
$endgroup$
– user44191
7 hours ago






$begingroup$
I think this question needs some clarification. I think you may mean something along the lines of: let $X$ be the moduli space of $n$-tuples of homogeneous polynomials of degree $d$; within $X$, there is a subspace $Y$ consisting of homogeneous polynomials that satisfy a nontrivial relation $sum F_i G_i = 0$ for some $G_i$ homogeneous of degree $p < d$ (i.e. the union of the subspaces determined by each tuple $G$). Is it true that $Y subsetneq X$?
$endgroup$
– user44191
7 hours ago














$begingroup$
Sure. Is that different from what I wrote?
$endgroup$
– abx
7 hours ago




$begingroup$
Sure. Is that different from what I wrote?
$endgroup$
– abx
7 hours ago












$begingroup$
I found what you wrote ambiguous or underdetermined, e.g. with what you mean by "general"; I thought what I wrote made it more explicitly clear (though it may be overkill as written).
$endgroup$
– user44191
7 hours ago




$begingroup$
I found what you wrote ambiguous or underdetermined, e.g. with what you mean by "general"; I thought what I wrote made it more explicitly clear (though it may be overkill as written).
$endgroup$
– user44191
7 hours ago












$begingroup$
This seem trivially false if I haven't misunderstood the question. Take $n = 1, p = 0, d = 1, ell = 2$. Then $F_1 = ax, F_2 = bx$ for some $a, b$, so $b F_1 - a F_2 = 0$. I think both $n, ell$ play some role.
$endgroup$
– user44191
7 hours ago






$begingroup$
This seem trivially false if I haven't misunderstood the question. Take $n = 1, p = 0, d = 1, ell = 2$. Then $F_1 = ax, F_2 = bx$ for some $a, b$, so $b F_1 - a F_2 = 0$. I think both $n, ell$ play some role.
$endgroup$
– user44191
7 hours ago














$begingroup$
Oh, right. $ellleq n$ would be fine for me, though this can be certainly weakened.
$endgroup$
– abx
7 hours ago




$begingroup$
Oh, right. $ellleq n$ would be fine for me, though this can be certainly weakened.
$endgroup$
– abx
7 hours ago










1 Answer
1






active

oldest

votes


















6












$begingroup$

I think this can be controlled as follows. Let $Z subset mathbb{P}^{n-1}$ be the complete intersection defined by $F_i$. Then there is a Koszul resolution
$$
dots to mathcal{O}(-2d)^{binom{ell}{2}} to mathcal{O}(-d)^ell to mathcal{O} to mathcal{O}_Z to 0.
$$

Twisting it by $mathcal{O}(d+p)$ we obtain
$$
dots to mathcal{O}(p-d)^{binom{ell}{2}} to mathcal{O}(p)^ell to mathcal{O}(d+p) to mathcal{O}_Z(d+p) to 0.
$$

Your question is equivalent to injectivity of the induced map
$$
H^0(mathcal{O}(p)^ell) to H^0(mathcal{O}(d+p)).
$$

If $n ge ell$ the cohomology spectral sequence of the twisted Koszul complex proves this. Is that enough for your purposes?






share|cite|improve this answer









$endgroup$









  • 1




    $begingroup$
    Thanks, but could you explain why the Koszul complex gives that?
    $endgroup$
    – abx
    7 hours ago










  • $begingroup$
    Let $E$ the cokernel of $ldots rightarrow mathcal{O}(p-d)^{binom{l}{2}}$. This truncated comples has length $l-1$. Hence, the dimensions of cohomology groups which play a role in order to compute $H^0(E)$ are less than $l-1$.
    $endgroup$
    – Libli
    6 hours ago












  • $begingroup$
    Hence, if you assume that $l leq n$ and noting that all terms of this truncated complex are acyclic in degree strictly less than $n$, you get that $H^0(E)= 0$ and the injectivity claimed by Sasha.
    $endgroup$
    – Libli
    6 hours ago













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1 Answer
1






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1 Answer
1






active

oldest

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active

oldest

votes






active

oldest

votes









6












$begingroup$

I think this can be controlled as follows. Let $Z subset mathbb{P}^{n-1}$ be the complete intersection defined by $F_i$. Then there is a Koszul resolution
$$
dots to mathcal{O}(-2d)^{binom{ell}{2}} to mathcal{O}(-d)^ell to mathcal{O} to mathcal{O}_Z to 0.
$$

Twisting it by $mathcal{O}(d+p)$ we obtain
$$
dots to mathcal{O}(p-d)^{binom{ell}{2}} to mathcal{O}(p)^ell to mathcal{O}(d+p) to mathcal{O}_Z(d+p) to 0.
$$

Your question is equivalent to injectivity of the induced map
$$
H^0(mathcal{O}(p)^ell) to H^0(mathcal{O}(d+p)).
$$

If $n ge ell$ the cohomology spectral sequence of the twisted Koszul complex proves this. Is that enough for your purposes?






share|cite|improve this answer









$endgroup$









  • 1




    $begingroup$
    Thanks, but could you explain why the Koszul complex gives that?
    $endgroup$
    – abx
    7 hours ago










  • $begingroup$
    Let $E$ the cokernel of $ldots rightarrow mathcal{O}(p-d)^{binom{l}{2}}$. This truncated comples has length $l-1$. Hence, the dimensions of cohomology groups which play a role in order to compute $H^0(E)$ are less than $l-1$.
    $endgroup$
    – Libli
    6 hours ago












  • $begingroup$
    Hence, if you assume that $l leq n$ and noting that all terms of this truncated complex are acyclic in degree strictly less than $n$, you get that $H^0(E)= 0$ and the injectivity claimed by Sasha.
    $endgroup$
    – Libli
    6 hours ago


















6












$begingroup$

I think this can be controlled as follows. Let $Z subset mathbb{P}^{n-1}$ be the complete intersection defined by $F_i$. Then there is a Koszul resolution
$$
dots to mathcal{O}(-2d)^{binom{ell}{2}} to mathcal{O}(-d)^ell to mathcal{O} to mathcal{O}_Z to 0.
$$

Twisting it by $mathcal{O}(d+p)$ we obtain
$$
dots to mathcal{O}(p-d)^{binom{ell}{2}} to mathcal{O}(p)^ell to mathcal{O}(d+p) to mathcal{O}_Z(d+p) to 0.
$$

Your question is equivalent to injectivity of the induced map
$$
H^0(mathcal{O}(p)^ell) to H^0(mathcal{O}(d+p)).
$$

If $n ge ell$ the cohomology spectral sequence of the twisted Koszul complex proves this. Is that enough for your purposes?






share|cite|improve this answer









$endgroup$









  • 1




    $begingroup$
    Thanks, but could you explain why the Koszul complex gives that?
    $endgroup$
    – abx
    7 hours ago










  • $begingroup$
    Let $E$ the cokernel of $ldots rightarrow mathcal{O}(p-d)^{binom{l}{2}}$. This truncated comples has length $l-1$. Hence, the dimensions of cohomology groups which play a role in order to compute $H^0(E)$ are less than $l-1$.
    $endgroup$
    – Libli
    6 hours ago












  • $begingroup$
    Hence, if you assume that $l leq n$ and noting that all terms of this truncated complex are acyclic in degree strictly less than $n$, you get that $H^0(E)= 0$ and the injectivity claimed by Sasha.
    $endgroup$
    – Libli
    6 hours ago
















6












6








6





$begingroup$

I think this can be controlled as follows. Let $Z subset mathbb{P}^{n-1}$ be the complete intersection defined by $F_i$. Then there is a Koszul resolution
$$
dots to mathcal{O}(-2d)^{binom{ell}{2}} to mathcal{O}(-d)^ell to mathcal{O} to mathcal{O}_Z to 0.
$$

Twisting it by $mathcal{O}(d+p)$ we obtain
$$
dots to mathcal{O}(p-d)^{binom{ell}{2}} to mathcal{O}(p)^ell to mathcal{O}(d+p) to mathcal{O}_Z(d+p) to 0.
$$

Your question is equivalent to injectivity of the induced map
$$
H^0(mathcal{O}(p)^ell) to H^0(mathcal{O}(d+p)).
$$

If $n ge ell$ the cohomology spectral sequence of the twisted Koszul complex proves this. Is that enough for your purposes?






share|cite|improve this answer









$endgroup$



I think this can be controlled as follows. Let $Z subset mathbb{P}^{n-1}$ be the complete intersection defined by $F_i$. Then there is a Koszul resolution
$$
dots to mathcal{O}(-2d)^{binom{ell}{2}} to mathcal{O}(-d)^ell to mathcal{O} to mathcal{O}_Z to 0.
$$

Twisting it by $mathcal{O}(d+p)$ we obtain
$$
dots to mathcal{O}(p-d)^{binom{ell}{2}} to mathcal{O}(p)^ell to mathcal{O}(d+p) to mathcal{O}_Z(d+p) to 0.
$$

Your question is equivalent to injectivity of the induced map
$$
H^0(mathcal{O}(p)^ell) to H^0(mathcal{O}(d+p)).
$$

If $n ge ell$ the cohomology spectral sequence of the twisted Koszul complex proves this. Is that enough for your purposes?







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered 8 hours ago









SashaSasha

21k22755




21k22755








  • 1




    $begingroup$
    Thanks, but could you explain why the Koszul complex gives that?
    $endgroup$
    – abx
    7 hours ago










  • $begingroup$
    Let $E$ the cokernel of $ldots rightarrow mathcal{O}(p-d)^{binom{l}{2}}$. This truncated comples has length $l-1$. Hence, the dimensions of cohomology groups which play a role in order to compute $H^0(E)$ are less than $l-1$.
    $endgroup$
    – Libli
    6 hours ago












  • $begingroup$
    Hence, if you assume that $l leq n$ and noting that all terms of this truncated complex are acyclic in degree strictly less than $n$, you get that $H^0(E)= 0$ and the injectivity claimed by Sasha.
    $endgroup$
    – Libli
    6 hours ago
















  • 1




    $begingroup$
    Thanks, but could you explain why the Koszul complex gives that?
    $endgroup$
    – abx
    7 hours ago










  • $begingroup$
    Let $E$ the cokernel of $ldots rightarrow mathcal{O}(p-d)^{binom{l}{2}}$. This truncated comples has length $l-1$. Hence, the dimensions of cohomology groups which play a role in order to compute $H^0(E)$ are less than $l-1$.
    $endgroup$
    – Libli
    6 hours ago












  • $begingroup$
    Hence, if you assume that $l leq n$ and noting that all terms of this truncated complex are acyclic in degree strictly less than $n$, you get that $H^0(E)= 0$ and the injectivity claimed by Sasha.
    $endgroup$
    – Libli
    6 hours ago










1




1




$begingroup$
Thanks, but could you explain why the Koszul complex gives that?
$endgroup$
– abx
7 hours ago




$begingroup$
Thanks, but could you explain why the Koszul complex gives that?
$endgroup$
– abx
7 hours ago












$begingroup$
Let $E$ the cokernel of $ldots rightarrow mathcal{O}(p-d)^{binom{l}{2}}$. This truncated comples has length $l-1$. Hence, the dimensions of cohomology groups which play a role in order to compute $H^0(E)$ are less than $l-1$.
$endgroup$
– Libli
6 hours ago






$begingroup$
Let $E$ the cokernel of $ldots rightarrow mathcal{O}(p-d)^{binom{l}{2}}$. This truncated comples has length $l-1$. Hence, the dimensions of cohomology groups which play a role in order to compute $H^0(E)$ are less than $l-1$.
$endgroup$
– Libli
6 hours ago














$begingroup$
Hence, if you assume that $l leq n$ and noting that all terms of this truncated complex are acyclic in degree strictly less than $n$, you get that $H^0(E)= 0$ and the injectivity claimed by Sasha.
$endgroup$
– Libli
6 hours ago






$begingroup$
Hence, if you assume that $l leq n$ and noting that all terms of this truncated complex are acyclic in degree strictly less than $n$, you get that $H^0(E)= 0$ and the injectivity claimed by Sasha.
$endgroup$
– Libli
6 hours ago




















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