singular intersection only comes from tangent?
1
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Let $X$ be a smooth hypersurface in $mathbb {CP}^n$, $H$ be a hyperplane. If $Xcap H$ is singular, is it true that $H$ is a tangent plane at some point of $X$?
Note the converse is always true, if we count the multiplicities.
algebraic-geometry intersection-theory
asked Jan 20 at 18:38
User XUser X
33411
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add a comment |
1
$begingroup$
Let $X$ be a smooth hypersurface in $mathbb {CP}^n$, $H$ be a hyperplane. If $Xcap H$ is singular, is it true that $H$ is a tangent plane at some point of $X$?
Note the converse is always true, if we count the multiplicities.
algebraic-geometry intersection-theory
asked Jan 20 at 18:38
User XUser X
33411
$endgroup$
$begingroup$
Of course, because the tangent space to the intersection is the intersection of tangent spaces.
$endgroup$
– Sasha
Jan 20 at 18:50
$begingroup$
@Sasha Sorry, but I don't understand. What do you mean by "tangent space to the intersection" and why this follows my question?
$endgroup$
– User X
Jan 20 at 22:36
2
$begingroup$
$X cap H$ has dimension $n - 2$. It is singular at point $x in X cap H$ iff the Zariski tangent space to $X cap H$ has dimension greater than $n - 2$. But this space is equal to the intersection of the tangent spaces to $X$ and to $H$ at $x$. Both a hyperplanes in the Zariski tangent space to $mathbb{P}^n$, hence $x$ is singular iff they coincide, i.e., $H$ is tangent to $X$ at $x$.
$endgroup$
– Sasha
Jan 20 at 23:02
$begingroup$
@Sasha Wow... Many thanks!
$endgroup$
– User X
Jan 21 at 10:34
add a comment |
1
1
1
$begingroup$
Let $X$ be a smooth hypersurface in $mathbb {CP}^n$, $H$ be a hyperplane. If $Xcap H$ is singular, is it true that $H$ is a tangent plane at some point of $X$?
Note the converse is always true, if we count the multiplicities.
algebraic-geometry intersection-theory
asked Jan 20 at 18:38
User XUser X
33411
$endgroup$
Let $X$ be a smooth hypersurface in $mathbb {CP}^n$, $H$ be a hyperplane. If $Xcap H$ is singular, is it true that $H$ is a tangent plane at some point of $X$?
Note the converse is always true, if we count the multiplicities.
algebraic-geometry intersection-theory
algebraic-geometry intersection-theory
asked Jan 20 at 18:38
User XUser X
33411
asked Jan 20 at 18:38
User XUser X
33411
asked Jan 20 at 18:38
User XUser X
33411
asked Jan 20 at 18:38
User XUser X
33411
asked Jan 20 at 18:38
User XUser X
33411
33411
$begingroup$
Of course, because the tangent space to the intersection is the intersection of tangent spaces.
$endgroup$
– Sasha
Jan 20 at 18:50
$begingroup$
@Sasha Sorry, but I don't understand. What do you mean by "tangent space to the intersection" and why this follows my question?
$endgroup$
– User X
Jan 20 at 22:36
2
$begingroup$
$X cap H$ has dimension $n - 2$. It is singular at point $x in X cap H$ iff the Zariski tangent space to $X cap H$ has dimension greater than $n - 2$. But this space is equal to the intersection of the tangent spaces to $X$ and to $H$ at $x$. Both a hyperplanes in the Zariski tangent space to $mathbb{P}^n$, hence $x$ is singular iff they coincide, i.e., $H$ is tangent to $X$ at $x$.
$endgroup$
– Sasha
Jan 20 at 23:02
$begingroup$
@Sasha Wow... Many thanks!
$endgroup$
– User X
Jan 21 at 10:34
add a comment |
$begingroup$
Of course, because the tangent space to the intersection is the intersection of tangent spaces.
$endgroup$
– Sasha
Jan 20 at 18:50
$begingroup$
@Sasha Sorry, but I don't understand. What do you mean by "tangent space to the intersection" and why this follows my question?
$endgroup$
– User X
Jan 20 at 22:36
2
$begingroup$
$X cap H$ has dimension $n - 2$. It is singular at point $x in X cap H$ iff the Zariski tangent space to $X cap H$ has dimension greater than $n - 2$. But this space is equal to the intersection of the tangent spaces to $X$ and to $H$ at $x$. Both a hyperplanes in the Zariski tangent space to $mathbb{P}^n$, hence $x$ is singular iff they coincide, i.e., $H$ is tangent to $X$ at $x$.
$endgroup$
– Sasha
Jan 20 at 23:02
$begingroup$
@Sasha Wow... Many thanks!
$endgroup$
– User X
Jan 21 at 10:34
$begingroup$
Of course, because the tangent space to the intersection is the intersection of tangent spaces.
$endgroup$
– Sasha
Jan 20 at 18:50
$begingroup$
Of course, because the tangent space to the intersection is the intersection of tangent spaces.
$endgroup$
– Sasha
Jan 20 at 18:50
$begingroup$
@Sasha Sorry, but I don't understand. What do you mean by "tangent space to the intersection" and why this follows my question?
$endgroup$
– User X
Jan 20 at 22:36
$begingroup$
@Sasha Sorry, but I don't understand. What do you mean by "tangent space to the intersection" and why this follows my question?
$endgroup$
– User X
Jan 20 at 22:36
2
2
$begingroup$
$X cap H$ has dimension $n - 2$. It is singular at point $x in X cap H$ iff the Zariski tangent space to $X cap H$ has dimension greater than $n - 2$. But this space is equal to the intersection of the tangent spaces to $X$ and to $H$ at $x$. Both a hyperplanes in the Zariski tangent space to $mathbb{P}^n$, hence $x$ is singular iff they coincide, i.e., $H$ is tangent to $X$ at $x$.
$endgroup$
– Sasha
Jan 20 at 23:02
$begingroup$
$X cap H$ has dimension $n - 2$. It is singular at point $x in X cap H$ iff the Zariski tangent space to $X cap H$ has dimension greater than $n - 2$. But this space is equal to the intersection of the tangent spaces to $X$ and to $H$ at $x$. Both a hyperplanes in the Zariski tangent space to $mathbb{P}^n$, hence $x$ is singular iff they coincide, i.e., $H$ is tangent to $X$ at $x$.
$endgroup$
– Sasha
Jan 20 at 23:02
$begingroup$
@Sasha Wow... Many thanks!
$endgroup$
– User X
Jan 21 at 10:34
$begingroup$
@Sasha Wow... Many thanks!
$endgroup$
– User X
Jan 21 at 10:34
add a comment |
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$begingroup$
Of course, because the tangent space to the intersection is the intersection of tangent spaces.
$endgroup$
– Sasha
Jan 20 at 18:50
$begingroup$
@Sasha Sorry, but I don't understand. What do you mean by "tangent space to the intersection" and why this follows my question?
$endgroup$
– User X
Jan 20 at 22:36
2
$begingroup$
$X cap H$ has dimension $n - 2$. It is singular at point $x in X cap H$ iff the Zariski tangent space to $X cap H$ has dimension greater than $n - 2$. But this space is equal to the intersection of the tangent spaces to $X$ and to $H$ at $x$. Both a hyperplanes in the Zariski tangent space to $mathbb{P}^n$, hence $x$ is singular iff they coincide, i.e., $H$ is tangent to $X$ at $x$.
$endgroup$
– Sasha
Jan 20 at 23:02
$begingroup$
@Sasha Wow... Many thanks!
$endgroup$
– User X
Jan 21 at 10:34