Dimension of singular curves of a given degree Announcing the arrival of Valued Associate...
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Dimension of singular curves of a given degree
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Consider plane curves of degree $d$ in $mathbb{P}^2$. These are elements of the linear system $|O(d)|$.
What is the dimension of singular curves of degree $d$. Can this be computed? Is there some reference for this?
Thanks in advance!
algebraic-geometry
asked Mar 26 at 10:56
user52991user52991
351310
$endgroup$
add a comment |
$begingroup$
Consider plane curves of degree $d$ in $mathbb{P}^2$. These are elements of the linear system $|O(d)|$.
What is the dimension of singular curves of degree $d$. Can this be computed? Is there some reference for this?
Thanks in advance!
algebraic-geometry
asked Mar 26 at 10:56
user52991user52991
351310
$endgroup$
$begingroup$
The locus of singular hypersurfaces of degree $d$ in $mathbb{P}^n$ is a divisor (called the discriminant) in the moduli space of hypersurfaces of degree $d$ in $mathbb{P}^n$. So, in your case, the dimension is $dim |mathcal{O}(d)| -1$.
$endgroup$
– Harry
Mar 26 at 11:15
$begingroup$
@ Harry, can you direct me to some reference?
$endgroup$
– user52991
Mar 26 at 11:17
$begingroup$
The following paper is classical: ems-ph.org/journals/…
$endgroup$
– Harry
Apr 5 at 7:51
add a comment |
$begingroup$
Consider plane curves of degree $d$ in $mathbb{P}^2$. These are elements of the linear system $|O(d)|$.
What is the dimension of singular curves of degree $d$. Can this be computed? Is there some reference for this?
Thanks in advance!
algebraic-geometry
asked Mar 26 at 10:56
user52991user52991
351310
$endgroup$
Consider plane curves of degree $d$ in $mathbb{P}^2$. These are elements of the linear system $|O(d)|$.
What is the dimension of singular curves of degree $d$. Can this be computed? Is there some reference for this?
Thanks in advance!
algebraic-geometry
algebraic-geometry
asked Mar 26 at 10:56
user52991user52991
351310
asked Mar 26 at 10:56
user52991user52991
351310
asked Mar 26 at 10:56
user52991user52991
351310
asked Mar 26 at 10:56
user52991user52991
351310
asked Mar 26 at 10:56
user52991user52991
351310
351310
$begingroup$
The locus of singular hypersurfaces of degree $d$ in $mathbb{P}^n$ is a divisor (called the discriminant) in the moduli space of hypersurfaces of degree $d$ in $mathbb{P}^n$. So, in your case, the dimension is $dim |mathcal{O}(d)| -1$.
$endgroup$
– Harry
Mar 26 at 11:15
$begingroup$
@ Harry, can you direct me to some reference?
$endgroup$
– user52991
Mar 26 at 11:17
$begingroup$
The following paper is classical: ems-ph.org/journals/…
$endgroup$
– Harry
Apr 5 at 7:51
add a comment |
$begingroup$
The locus of singular hypersurfaces of degree $d$ in $mathbb{P}^n$ is a divisor (called the discriminant) in the moduli space of hypersurfaces of degree $d$ in $mathbb{P}^n$. So, in your case, the dimension is $dim |mathcal{O}(d)| -1$.
$endgroup$
– Harry
Mar 26 at 11:15
$begingroup$
@ Harry, can you direct me to some reference?
$endgroup$
– user52991
Mar 26 at 11:17
$begingroup$
The following paper is classical: ems-ph.org/journals/…
$endgroup$
– Harry
Apr 5 at 7:51
$begingroup$
The locus of singular hypersurfaces of degree $d$ in $mathbb{P}^n$ is a divisor (called the discriminant) in the moduli space of hypersurfaces of degree $d$ in $mathbb{P}^n$. So, in your case, the dimension is $dim |mathcal{O}(d)| -1$.
$endgroup$
– Harry
Mar 26 at 11:15
$begingroup$
The locus of singular hypersurfaces of degree $d$ in $mathbb{P}^n$ is a divisor (called the discriminant) in the moduli space of hypersurfaces of degree $d$ in $mathbb{P}^n$. So, in your case, the dimension is $dim |mathcal{O}(d)| -1$.
$endgroup$
– Harry
Mar 26 at 11:15
$begingroup$
@ Harry, can you direct me to some reference?
$endgroup$
– user52991
Mar 26 at 11:17
$begingroup$
@ Harry, can you direct me to some reference?
$endgroup$
– user52991
Mar 26 at 11:17
$begingroup$
The following paper is classical: ems-ph.org/journals/…
$endgroup$
– Harry
Apr 5 at 7:51
$begingroup$
The following paper is classical: ems-ph.org/journals/…
$endgroup$
– Harry
Apr 5 at 7:51
add a comment |
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$begingroup$
The locus of singular hypersurfaces of degree $d$ in $mathbb{P}^n$ is a divisor (called the discriminant) in the moduli space of hypersurfaces of degree $d$ in $mathbb{P}^n$. So, in your case, the dimension is $dim |mathcal{O}(d)| -1$.
$endgroup$
– Harry
Mar 26 at 11:15
$begingroup$
@ Harry, can you direct me to some reference?
$endgroup$
– user52991
Mar 26 at 11:17
$begingroup$
The following paper is classical: ems-ph.org/journals/…
$endgroup$
– Harry
Apr 5 at 7:51