Singular locus of dual hypersurfaces
$begingroup$
Everything is over field $mathbb C$. Let $X$ be a hypersurface of degree $d$ in $mathbb P^n$. We know that if $X$ is smooth, then its dual $X^vee$ is still a hypersurface in $(mathbb P^n)^vee$, but not necessarily smooth. I want to know that, for a general choose of $X$, can we compute the dimension of the singular locus $X^vee_{sing}$? Where can I find a discussion of this?
For some reason, I believe in general it is of codimension $1$ In $X^vee$, or it is empty. But I don’t know how to prove this.
Thanks in advance.
algebraic-geometry complex-geometry intersection-theory
asked Jan 7 at 22:48
User XUser X
34111
$endgroup$
add a comment |
$begingroup$
Everything is over field $mathbb C$. Let $X$ be a hypersurface of degree $d$ in $mathbb P^n$. We know that if $X$ is smooth, then its dual $X^vee$ is still a hypersurface in $(mathbb P^n)^vee$, but not necessarily smooth. I want to know that, for a general choose of $X$, can we compute the dimension of the singular locus $X^vee_{sing}$? Where can I find a discussion of this?
For some reason, I believe in general it is of codimension $1$ In $X^vee$, or it is empty. But I don’t know how to prove this.
Thanks in advance.
algebraic-geometry complex-geometry intersection-theory
asked Jan 7 at 22:48
User XUser X
34111
$endgroup$
add a comment |
$begingroup$
Everything is over field $mathbb C$. Let $X$ be a hypersurface of degree $d$ in $mathbb P^n$. We know that if $X$ is smooth, then its dual $X^vee$ is still a hypersurface in $(mathbb P^n)^vee$, but not necessarily smooth. I want to know that, for a general choose of $X$, can we compute the dimension of the singular locus $X^vee_{sing}$? Where can I find a discussion of this?
For some reason, I believe in general it is of codimension $1$ In $X^vee$, or it is empty. But I don’t know how to prove this.
Thanks in advance.
algebraic-geometry complex-geometry intersection-theory
asked Jan 7 at 22:48
User XUser X
34111
$endgroup$
Everything is over field $mathbb C$. Let $X$ be a hypersurface of degree $d$ in $mathbb P^n$. We know that if $X$ is smooth, then its dual $X^vee$ is still a hypersurface in $(mathbb P^n)^vee$, but not necessarily smooth. I want to know that, for a general choose of $X$, can we compute the dimension of the singular locus $X^vee_{sing}$? Where can I find a discussion of this?
For some reason, I believe in general it is of codimension $1$ In $X^vee$, or it is empty. But I don’t know how to prove this.
Thanks in advance.
algebraic-geometry complex-geometry intersection-theory
algebraic-geometry complex-geometry intersection-theory
asked Jan 7 at 22:48
User XUser X
34111
asked Jan 7 at 22:48
User XUser X
34111
edited Jan 9 at 17:05
User X
asked Jan 7 at 22:48
User XUser X
34111
asked Jan 7 at 22:48
User XUser X
34111
asked Jan 7 at 22:48
User XUser X
34111
34111
add a comment |
add a comment |
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