Mixing
What is the codimension of the set of non-node singularites?
$begingroup$
Let $F$ be a given homogeneous polynomial in $mathbb C[x_0,ldots,x_n]$. It defines some hypersurface $X=Z(F)$ in $mathbb {P}^n$. Let
$$U:={xin X:text{$x$ is a non-node singular point in $X$}}.$$
My question is:
What is the codimension of $U$ in $X$?
In general, Let $Ssubset X$ be the closed subset of singular points, so $S$ has codimension at least $1$. I think $Usubset S$ should also be a closed subset, and for $n$ large it should be proper, so $Usubset X$ should has codimension at least $2$. It would be good if we can tell for some given $X=Z(F)$ whether the bound is obtained. Is there some known criterion like this?
Thanks in advance.
algebraic-geometry reference-request complex-geometry
asked Jan 21 at 20:23
AkatsukiAkatsuki
1,1241725
$endgroup$
add a comment |
$begingroup$
Let $F$ be a given homogeneous polynomial in $mathbb C[x_0,ldots,x_n]$. It defines some hypersurface $X=Z(F)$ in $mathbb {P}^n$. Let
$$U:={xin X:text{$x$ is a non-node singular point in $X$}}.$$
My question is:
What is the codimension of $U$ in $X$?
In general, Let $Ssubset X$ be the closed subset of singular points, so $S$ has codimension at least $1$. I think $Usubset S$ should also be a closed subset, and for $n$ large it should be proper, so $Usubset X$ should has codimension at least $2$. It would be good if we can tell for some given $X=Z(F)$ whether the bound is obtained. Is there some known criterion like this?
Thanks in advance.
algebraic-geometry reference-request complex-geometry
asked Jan 21 at 20:23
AkatsukiAkatsuki
1,1241725
$endgroup$
$begingroup$
Can you tell me your definition of node?
$endgroup$
– Mohan
Jan 21 at 23:13
$begingroup$
@Mohan Maybe I should use the term “ordinary double point”, i.e. the tangent cone is non-degenerate.
$endgroup$
– Akatsuki
Jan 22 at 14:21
add a comment |
$begingroup$
Let $F$ be a given homogeneous polynomial in $mathbb C[x_0,ldots,x_n]$. It defines some hypersurface $X=Z(F)$ in $mathbb {P}^n$. Let
$$U:={xin X:text{$x$ is a non-node singular point in $X$}}.$$
My question is:
What is the codimension of $U$ in $X$?
In general, Let $Ssubset X$ be the closed subset of singular points, so $S$ has codimension at least $1$. I think $Usubset S$ should also be a closed subset, and for $n$ large it should be proper, so $Usubset X$ should has codimension at least $2$. It would be good if we can tell for some given $X=Z(F)$ whether the bound is obtained. Is there some known criterion like this?
Thanks in advance.
algebraic-geometry reference-request complex-geometry
asked Jan 21 at 20:23
AkatsukiAkatsuki
1,1241725
$endgroup$
Let $F$ be a given homogeneous polynomial in $mathbb C[x_0,ldots,x_n]$. It defines some hypersurface $X=Z(F)$ in $mathbb {P}^n$. Let
$$U:={xin X:text{$x$ is a non-node singular point in $X$}}.$$
My question is:
What is the codimension of $U$ in $X$?
In general, Let $Ssubset X$ be the closed subset of singular points, so $S$ has codimension at least $1$. I think $Usubset S$ should also be a closed subset, and for $n$ large it should be proper, so $Usubset X$ should has codimension at least $2$. It would be good if we can tell for some given $X=Z(F)$ whether the bound is obtained. Is there some known criterion like this?
Thanks in advance.
algebraic-geometry reference-request complex-geometry
algebraic-geometry reference-request complex-geometry
asked Jan 21 at 20:23
AkatsukiAkatsuki
1,1241725
asked Jan 21 at 20:23
AkatsukiAkatsuki
1,1241725
asked Jan 21 at 20:23
AkatsukiAkatsuki
1,1241725
asked Jan 21 at 20:23
AkatsukiAkatsuki
1,1241725
asked Jan 21 at 20:23
AkatsukiAkatsuki
1,1241725
1,1241725
$begingroup$
Can you tell me your definition of node?
$endgroup$
– Mohan
Jan 21 at 23:13
$begingroup$
@Mohan Maybe I should use the term “ordinary double point”, i.e. the tangent cone is non-degenerate.
$endgroup$
– Akatsuki
Jan 22 at 14:21
add a comment |
$begingroup$
Can you tell me your definition of node?
$endgroup$
– Mohan
Jan 21 at 23:13
$begingroup$
@Mohan Maybe I should use the term “ordinary double point”, i.e. the tangent cone is non-degenerate.
$endgroup$
– Akatsuki
Jan 22 at 14:21
$begingroup$
Can you tell me your definition of node?
$endgroup$
– Mohan
Jan 21 at 23:13
$begingroup$
Can you tell me your definition of node?
$endgroup$
– Mohan
Jan 21 at 23:13
$begingroup$
@Mohan Maybe I should use the term “ordinary double point”, i.e. the tangent cone is non-degenerate.
$endgroup$
– Akatsuki
Jan 22 at 14:21
$begingroup$
@Mohan Maybe I should use the term “ordinary double point”, i.e. the tangent cone is non-degenerate.
$endgroup$
– Akatsuki
Jan 22 at 14:21
add a comment |
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$begingroup$
Can you tell me your definition of node?
$endgroup$
– Mohan
Jan 21 at 23:13
$begingroup$
@Mohan Maybe I should use the term “ordinary double point”, i.e. the tangent cone is non-degenerate.
$endgroup$
– Akatsuki
Jan 22 at 14:21