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Dimension of singular curves of a given degree

0

$begingroup$

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

share|cite|improve this question

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 | 

0

$begingroup$

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

share|cite|improve this question

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

share|cite|improve this question

asked Mar 26 at 10:56

user52991user52991

351310

$endgroup$

share|cite|improve this question

asked Mar 26 at 10:56

user52991user52991

351310

share|cite|improve this question

asked Mar 26 at 10:56

user52991user52991

351310

asked Mar 26 at 10:56

user52991user52991

351310

asked Mar 26 at 10:56

user52991user52991

351310