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Continuity of intersection points of projective algebraic varieties
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Suppose that $X subset mathbbCP^n$ is a (possibly singular) algebraic curve. Let $H_tau subset mathbbCP^n$ be a family of hyperplanes, with $tau in (-1,1)$.
By Bezout's theorem, the count of intersection points (with multiplicity) is independent of $tau$. (Let us assume that $H_tau$ does not contain any irreducible components of $X$, and that $X$ is reduced. This justifies the application of Bezout's theorem).
Question: I would like to say that the intersection points vary continuously in $mathbbCP^n$, with respect to the Euclidean topology. Is this statement true, and if so, where can it be found in the literature?
complex-analysis algebraic-geometry
asked Aug 6 at 18:51
user142700
1,190515
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Suppose that $X subset mathbbCP^n$ is a (possibly singular) algebraic curve. Let $H_tau subset mathbbCP^n$ be a family of hyperplanes, with $tau in (-1,1)$.
By Bezout's theorem, the count of intersection points (with multiplicity) is independent of $tau$. (Let us assume that $H_tau$ does not contain any irreducible components of $X$, and that $X$ is reduced. This justifies the application of Bezout's theorem).
Question: I would like to say that the intersection points vary continuously in $mathbbCP^n$, with respect to the Euclidean topology. Is this statement true, and if so, where can it be found in the literature?
complex-analysis algebraic-geometry
asked Aug 6 at 18:51
user142700
1,190515
1
what is the role of $tau$ here? You might need some kind of "generic" requirement. As in, take two lines through the origin, and some angle $tau$ parametrizing their slope in $mathbb CP^1$. Fixing one $H_0$, we can see that no matter what happens, as $tau$ passes through this, there will be infinitely many intersections when they are the same line. The same issue arises in higher dimensions, although they need not be the same hyperplane, right?
– Andres Mejia
Aug 6 at 19:01
Of course, are you saying euclidian topology on $(-1,1)$?
– Andres Mejia
Aug 6 at 19:03
@AndresMejia I am indeed using the Euclidean topology. Regarding the "genericity assumption", I agree with you that one ought to add more hypotheses. I will update the question to reflect this. Thanks!
– user142700
Aug 6 at 20:04
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Suppose that $X subset mathbbCP^n$ is a (possibly singular) algebraic curve. Let $H_tau subset mathbbCP^n$ be a family of hyperplanes, with $tau in (-1,1)$.
By Bezout's theorem, the count of intersection points (with multiplicity) is independent of $tau$. (Let us assume that $H_tau$ does not contain any irreducible components of $X$, and that $X$ is reduced. This justifies the application of Bezout's theorem).
Question: I would like to say that the intersection points vary continuously in $mathbbCP^n$, with respect to the Euclidean topology. Is this statement true, and if so, where can it be found in the literature?
complex-analysis algebraic-geometry
asked Aug 6 at 18:51
user142700
1,190515
Suppose that $X subset mathbbCP^n$ is a (possibly singular) algebraic curve. Let $H_tau subset mathbbCP^n$ be a family of hyperplanes, with $tau in (-1,1)$.
By Bezout's theorem, the count of intersection points (with multiplicity) is independent of $tau$. (Let us assume that $H_tau$ does not contain any irreducible components of $X$, and that $X$ is reduced. This justifies the application of Bezout's theorem).
Question: I would like to say that the intersection points vary continuously in $mathbbCP^n$, with respect to the Euclidean topology. Is this statement true, and if so, where can it be found in the literature?
complex-analysis algebraic-geometry
asked Aug 6 at 18:51
user142700
1,190515
edited Aug 6 at 20:06
asked Aug 6 at 18:51
user142700
1,190515
asked Aug 6 at 18:51
user142700
1,190515
asked Aug 6 at 18:51
user142700
1,190515
1,190515
1
what is the role of $tau$ here? You might need some kind of "generic" requirement. As in, take two lines through the origin, and some angle $tau$ parametrizing their slope in $mathbb CP^1$. Fixing one $H_0$, we can see that no matter what happens, as $tau$ passes through this, there will be infinitely many intersections when they are the same line. The same issue arises in higher dimensions, although they need not be the same hyperplane, right?
– Andres Mejia
Aug 6 at 19:01
Of course, are you saying euclidian topology on $(-1,1)$?
– Andres Mejia
Aug 6 at 19:03
@AndresMejia I am indeed using the Euclidean topology. Regarding the "genericity assumption", I agree with you that one ought to add more hypotheses. I will update the question to reflect this. Thanks!
– user142700
Aug 6 at 20:04
add a comment |Â
1
what is the role of $tau$ here? You might need some kind of "generic" requirement. As in, take two lines through the origin, and some angle $tau$ parametrizing their slope in $mathbb CP^1$. Fixing one $H_0$, we can see that no matter what happens, as $tau$ passes through this, there will be infinitely many intersections when they are the same line. The same issue arises in higher dimensions, although they need not be the same hyperplane, right?
– Andres Mejia
Aug 6 at 19:01
Of course, are you saying euclidian topology on $(-1,1)$?
– Andres Mejia
Aug 6 at 19:03
@AndresMejia I am indeed using the Euclidean topology. Regarding the "genericity assumption", I agree with you that one ought to add more hypotheses. I will update the question to reflect this. Thanks!
– user142700
Aug 6 at 20:04
1
1
what is the role of $tau$ here? You might need some kind of "generic" requirement. As in, take two lines through the origin, and some angle $tau$ parametrizing their slope in $mathbb CP^1$. Fixing one $H_0$, we can see that no matter what happens, as $tau$ passes through this, there will be infinitely many intersections when they are the same line. The same issue arises in higher dimensions, although they need not be the same hyperplane, right?
– Andres Mejia
Aug 6 at 19:01
what is the role of $tau$ here? You might need some kind of "generic" requirement. As in, take two lines through the origin, and some angle $tau$ parametrizing their slope in $mathbb CP^1$. Fixing one $H_0$, we can see that no matter what happens, as $tau$ passes through this, there will be infinitely many intersections when they are the same line. The same issue arises in higher dimensions, although they need not be the same hyperplane, right?
– Andres Mejia
Aug 6 at 19:01
Of course, are you saying euclidian topology on $(-1,1)$?
– Andres Mejia
Aug 6 at 19:03
Of course, are you saying euclidian topology on $(-1,1)$?
– Andres Mejia
Aug 6 at 19:03
@AndresMejia I am indeed using the Euclidean topology. Regarding the "genericity assumption", I agree with you that one ought to add more hypotheses. I will update the question to reflect this. Thanks!
– user142700
Aug 6 at 20:04
@AndresMejia I am indeed using the Euclidean topology. Regarding the "genericity assumption", I agree with you that one ought to add more hypotheses. I will update the question to reflect this. Thanks!
– user142700
Aug 6 at 20:04
add a comment |Â
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what is the role of $tau$ here? You might need some kind of "generic" requirement. As in, take two lines through the origin, and some angle $tau$ parametrizing their slope in $mathbb CP^1$. Fixing one $H_0$, we can see that no matter what happens, as $tau$ passes through this, there will be infinitely many intersections when they are the same line. The same issue arises in higher dimensions, although they need not be the same hyperplane, right?
– Andres Mejia
Aug 6 at 19:01
Of course, are you saying euclidian topology on $(-1,1)$?
– Andres Mejia
Aug 6 at 19:03
@AndresMejia I am indeed using the Euclidean topology. Regarding the "genericity assumption", I agree with you that one ought to add more hypotheses. I will update the question to reflect this. Thanks!
– user142700
Aug 6 at 20:04