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Generic hyperplane and transversal intersection
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I'm trying to solve an exercise I.7.7 in Hartshorne's Algebraic Geometry:
7.7. Let $Y$ be a variety of dimension $r$ and degree $d>1$ in $mathbbP^n$. Let $P in Y$ be a nonsingular point. Define $X$ to be
the closure of the union of all lines $overlinePQ$, where $Q in Y,
> Qneq P$.
(a) Show that $X$ is a variety of dimension $r+1$.
(b) Show that $deg X < d$.
I've proved (a); it follows from the fact that there is a dominant rational map from the projective cone over $Y$ to $X$.
However, to prove (b), I'm stuck.
I found that it suffices to prove the following:
Let us consider the set $(mathbbP^n)^*$ of all hyperplanes in
$mathbbP^n$ as the projective $n$-space $mathbbP^n$ with its
Zariski topology.
Let $W$ be the set of all hyperplanes containing $P$. Then W is a
hyperplane in $(mathbbP^n)^*$.
Then, for any (closed) variety $Y subset mathbbP^n$ which contains
$P$ and is nonsingular at $P$, the set
$$ Hin W: mboxFor every irreducible component $Z$ of Ycap H, i(Y,H;Z) = 1. $$
contains a nonempty open subset of $W$.
Now $i(Y,H;Z)$ is the intersection multiplicity of $Y$ and $H$ along $Z$, i.e., the length of the $mathcalO_Z$-module $mathcalO_Z / (I(Y)+I(H))mathcalO_Z$, where $mathcalO_Z$ is the local ring of $mathbbP^n$ at $Z$.
How can I prove this? Thanks.
Edited: I found that if the ideal $I(Y)+I(H)$ is a radical ideal, then $i(Y,H;Z) = 1$ for all $Z$.
algebraic-geometry intersection-theory
asked 2 days ago
Hiro Wat
887
add a comment |Â
up vote
1
down vote
favorite
I'm trying to solve an exercise I.7.7 in Hartshorne's Algebraic Geometry:
7.7. Let $Y$ be a variety of dimension $r$ and degree $d>1$ in $mathbbP^n$. Let $P in Y$ be a nonsingular point. Define $X$ to be
the closure of the union of all lines $overlinePQ$, where $Q in Y,
> Qneq P$.
(a) Show that $X$ is a variety of dimension $r+1$.
(b) Show that $deg X < d$.
I've proved (a); it follows from the fact that there is a dominant rational map from the projective cone over $Y$ to $X$.
However, to prove (b), I'm stuck.
I found that it suffices to prove the following:
Let us consider the set $(mathbbP^n)^*$ of all hyperplanes in
$mathbbP^n$ as the projective $n$-space $mathbbP^n$ with its
Zariski topology.
Let $W$ be the set of all hyperplanes containing $P$. Then W is a
hyperplane in $(mathbbP^n)^*$.
Then, for any (closed) variety $Y subset mathbbP^n$ which contains
$P$ and is nonsingular at $P$, the set
$$ Hin W: mboxFor every irreducible component $Z$ of Ycap H, i(Y,H;Z) = 1. $$
contains a nonempty open subset of $W$.
Now $i(Y,H;Z)$ is the intersection multiplicity of $Y$ and $H$ along $Z$, i.e., the length of the $mathcalO_Z$-module $mathcalO_Z / (I(Y)+I(H))mathcalO_Z$, where $mathcalO_Z$ is the local ring of $mathbbP^n$ at $Z$.
How can I prove this? Thanks.
Edited: I found that if the ideal $I(Y)+I(H)$ is a radical ideal, then $i(Y,H;Z) = 1$ for all $Z$.
algebraic-geometry intersection-theory
asked 2 days ago
Hiro Wat
887
do you know that the degree of a variety can be expressed as the generic number of intersection points that it has with a linear subspace if complementary dimension?
– Jesko Hüttenhain
yesterday
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I'm trying to solve an exercise I.7.7 in Hartshorne's Algebraic Geometry:
7.7. Let $Y$ be a variety of dimension $r$ and degree $d>1$ in $mathbbP^n$. Let $P in Y$ be a nonsingular point. Define $X$ to be
the closure of the union of all lines $overlinePQ$, where $Q in Y,
> Qneq P$.
(a) Show that $X$ is a variety of dimension $r+1$.
(b) Show that $deg X < d$.
I've proved (a); it follows from the fact that there is a dominant rational map from the projective cone over $Y$ to $X$.
However, to prove (b), I'm stuck.
I found that it suffices to prove the following:
Let us consider the set $(mathbbP^n)^*$ of all hyperplanes in
$mathbbP^n$ as the projective $n$-space $mathbbP^n$ with its
Zariski topology.
Let $W$ be the set of all hyperplanes containing $P$. Then W is a
hyperplane in $(mathbbP^n)^*$.
Then, for any (closed) variety $Y subset mathbbP^n$ which contains
$P$ and is nonsingular at $P$, the set
$$ Hin W: mboxFor every irreducible component $Z$ of Ycap H, i(Y,H;Z) = 1. $$
contains a nonempty open subset of $W$.
Now $i(Y,H;Z)$ is the intersection multiplicity of $Y$ and $H$ along $Z$, i.e., the length of the $mathcalO_Z$-module $mathcalO_Z / (I(Y)+I(H))mathcalO_Z$, where $mathcalO_Z$ is the local ring of $mathbbP^n$ at $Z$.
How can I prove this? Thanks.
Edited: I found that if the ideal $I(Y)+I(H)$ is a radical ideal, then $i(Y,H;Z) = 1$ for all $Z$.
algebraic-geometry intersection-theory
asked 2 days ago
Hiro Wat
887
I'm trying to solve an exercise I.7.7 in Hartshorne's Algebraic Geometry:
7.7. Let $Y$ be a variety of dimension $r$ and degree $d>1$ in $mathbbP^n$. Let $P in Y$ be a nonsingular point. Define $X$ to be
the closure of the union of all lines $overlinePQ$, where $Q in Y,
> Qneq P$.
(a) Show that $X$ is a variety of dimension $r+1$.
(b) Show that $deg X < d$.
I've proved (a); it follows from the fact that there is a dominant rational map from the projective cone over $Y$ to $X$.
However, to prove (b), I'm stuck.
I found that it suffices to prove the following:
Let us consider the set $(mathbbP^n)^*$ of all hyperplanes in
$mathbbP^n$ as the projective $n$-space $mathbbP^n$ with its
Zariski topology.
Let $W$ be the set of all hyperplanes containing $P$. Then W is a
hyperplane in $(mathbbP^n)^*$.
Then, for any (closed) variety $Y subset mathbbP^n$ which contains
$P$ and is nonsingular at $P$, the set
$$ Hin W: mboxFor every irreducible component $Z$ of Ycap H, i(Y,H;Z) = 1. $$
contains a nonempty open subset of $W$.
Now $i(Y,H;Z)$ is the intersection multiplicity of $Y$ and $H$ along $Z$, i.e., the length of the $mathcalO_Z$-module $mathcalO_Z / (I(Y)+I(H))mathcalO_Z$, where $mathcalO_Z$ is the local ring of $mathbbP^n$ at $Z$.
How can I prove this? Thanks.
Edited: I found that if the ideal $I(Y)+I(H)$ is a radical ideal, then $i(Y,H;Z) = 1$ for all $Z$.
algebraic-geometry intersection-theory
asked 2 days ago
Hiro Wat
887
edited 2 days ago
asked 2 days ago
Hiro Wat
887
asked 2 days ago
Hiro Wat
887
asked 2 days ago
Hiro Wat
887
887
do you know that the degree of a variety can be expressed as the generic number of intersection points that it has with a linear subspace if complementary dimension?
– Jesko Hüttenhain
yesterday
add a comment |Â
do you know that the degree of a variety can be expressed as the generic number of intersection points that it has with a linear subspace if complementary dimension?
– Jesko Hüttenhain
yesterday
do you know that the degree of a variety can be expressed as the generic number of intersection points that it has with a linear subspace if complementary dimension?
– Jesko Hüttenhain
yesterday
do you know that the degree of a variety can be expressed as the generic number of intersection points that it has with a linear subspace if complementary dimension?
– Jesko Hüttenhain
yesterday
add a comment |Â
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do you know that the degree of a variety can be expressed as the generic number of intersection points that it has with a linear subspace if complementary dimension?
– Jesko Hüttenhain
yesterday