Dense subset in product of varieties
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Given two (algebraic) varieties $X,Y$ (not necessarily irreducible) and $D,E$ dense subsets of $X$ and $Y$ respectively, I've read something saying that the product $D times E$ is dense in $X times Y$ (given the Zariski topology of course).
Is it true ? And how do you prove it ?
I've tried to look for it in different books but I couldn't find anything.
Thank you !
algebraic-geometry
asked Dec 12 '11 at 17:21
ng_th
255
add a comment |
up vote
1
down vote
favorite
Given two (algebraic) varieties $X,Y$ (not necessarily irreducible) and $D,E$ dense subsets of $X$ and $Y$ respectively, I've read something saying that the product $D times E$ is dense in $X times Y$ (given the Zariski topology of course).
Is it true ? And how do you prove it ?
I've tried to look for it in different books but I couldn't find anything.
Thank you !
algebraic-geometry
asked Dec 12 '11 at 17:21
ng_th
255
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Given two (algebraic) varieties $X,Y$ (not necessarily irreducible) and $D,E$ dense subsets of $X$ and $Y$ respectively, I've read something saying that the product $D times E$ is dense in $X times Y$ (given the Zariski topology of course).
Is it true ? And how do you prove it ?
I've tried to look for it in different books but I couldn't find anything.
Thank you !
algebraic-geometry
asked Dec 12 '11 at 17:21
ng_th
255
Given two (algebraic) varieties $X,Y$ (not necessarily irreducible) and $D,E$ dense subsets of $X$ and $Y$ respectively, I've read something saying that the product $D times E$ is dense in $X times Y$ (given the Zariski topology of course).
Is it true ? And how do you prove it ?
I've tried to look for it in different books but I couldn't find anything.
Thank you !
algebraic-geometry
algebraic-geometry
asked Dec 12 '11 at 17:21
ng_th
255
asked Dec 12 '11 at 17:21
ng_th
255
asked Dec 12 '11 at 17:21
ng_th
255
asked Dec 12 '11 at 17:21
ng_th
255
asked Dec 12 '11 at 17:21
ng_th
255
255
add a comment |
add a comment |
1 Answer
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Suppose $X,Y$ are varieties over an algebraically closed field $k$.
Claim
If $Dsubset X$ and $ Esubset Y$ are dense subsets, then $Dtimes Esubset Xtimes Y$ is dense too.
Proof
Let $Z$ be the closure of $Dtimes E$ in $Xtimes Y$.
Fix a poind $d_0in D$ and consider the closed subvariety $lbrace d_0rbrace times Y subset Xtimes Y$.
The subset $lbrace d_0rbrace times E subset lbrace d_0rbrace times Y$ has closure $lbrace d_0rbrace times Y ;$ [ because $lbrace d_0rbrace times Y$ is isomorphic to $Y$ and $E$ is dense in $Y$]
so that $lbrace d_0rbrace times Y subset Z $ . We have proved the
Partial Result:
For all $d_0in D $ we have $lbrace d_0rbrace times Y subset Z$
End of proof:
Consider an arbitrary $yin Y$ .
For any $din D$ we know, thanks to the Partial Result, that $(d,y)in Z$. Hence $Dtimes lbrace y rbrace subset Z$.
Since the closure of $Dtimes lbrace y rbrace $ is $Xtimes lbrace y rbrace $ [ because $X times lbrace y rbrace $ is isomorphic to $X$ and $D$ is dense in $X$], we have proved that $Xtimes lbrace y rbrace subset Z$.
Since $yin Y$ was arbitrary, this implies that $Z=Xtimes Y $ i.e. that $Dtimes E$ is dense in $Xtimes Y $
answered Dec 12 '11 at 22:32
Georges Elencwajg
118k7180327
Thank you a lot Georges (and also Qi). I almost had all what was needed to prove the result, but I couldn't put it into good order. I'm glad I learnt something :)
– ng_th
Dec 13 '11 at 9:18
add a comment |
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1 Answer
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Suppose $X,Y$ are varieties over an algebraically closed field $k$.
Claim
If $Dsubset X$ and $ Esubset Y$ are dense subsets, then $Dtimes Esubset Xtimes Y$ is dense too.
Proof
Let $Z$ be the closure of $Dtimes E$ in $Xtimes Y$.
Fix a poind $d_0in D$ and consider the closed subvariety $lbrace d_0rbrace times Y subset Xtimes Y$.
The subset $lbrace d_0rbrace times E subset lbrace d_0rbrace times Y$ has closure $lbrace d_0rbrace times Y ;$ [ because $lbrace d_0rbrace times Y$ is isomorphic to $Y$ and $E$ is dense in $Y$]
so that $lbrace d_0rbrace times Y subset Z $ . We have proved the
Partial Result:
For all $d_0in D $ we have $lbrace d_0rbrace times Y subset Z$
End of proof:
Consider an arbitrary $yin Y$ .
For any $din D$ we know, thanks to the Partial Result, that $(d,y)in Z$. Hence $Dtimes lbrace y rbrace subset Z$.
Since the closure of $Dtimes lbrace y rbrace $ is $Xtimes lbrace y rbrace $ [ because $X times lbrace y rbrace $ is isomorphic to $X$ and $D$ is dense in $X$], we have proved that $Xtimes lbrace y rbrace subset Z$.
Since $yin Y$ was arbitrary, this implies that $Z=Xtimes Y $ i.e. that $Dtimes E$ is dense in $Xtimes Y $
answered Dec 12 '11 at 22:32
Georges Elencwajg
118k7180327
Thank you a lot Georges (and also Qi). I almost had all what was needed to prove the result, but I couldn't put it into good order. I'm glad I learnt something :)
– ng_th
Dec 13 '11 at 9:18
add a comment |
up vote
1
down vote
accepted
Suppose $X,Y$ are varieties over an algebraically closed field $k$.
Claim
If $Dsubset X$ and $ Esubset Y$ are dense subsets, then $Dtimes Esubset Xtimes Y$ is dense too.
Proof
Let $Z$ be the closure of $Dtimes E$ in $Xtimes Y$.
Fix a poind $d_0in D$ and consider the closed subvariety $lbrace d_0rbrace times Y subset Xtimes Y$.
The subset $lbrace d_0rbrace times E subset lbrace d_0rbrace times Y$ has closure $lbrace d_0rbrace times Y ;$ [ because $lbrace d_0rbrace times Y$ is isomorphic to $Y$ and $E$ is dense in $Y$]
so that $lbrace d_0rbrace times Y subset Z $ . We have proved the
Partial Result:
For all $d_0in D $ we have $lbrace d_0rbrace times Y subset Z$
End of proof:
Consider an arbitrary $yin Y$ .
For any $din D$ we know, thanks to the Partial Result, that $(d,y)in Z$. Hence $Dtimes lbrace y rbrace subset Z$.
Since the closure of $Dtimes lbrace y rbrace $ is $Xtimes lbrace y rbrace $ [ because $X times lbrace y rbrace $ is isomorphic to $X$ and $D$ is dense in $X$], we have proved that $Xtimes lbrace y rbrace subset Z$.
Since $yin Y$ was arbitrary, this implies that $Z=Xtimes Y $ i.e. that $Dtimes E$ is dense in $Xtimes Y $
answered Dec 12 '11 at 22:32
Georges Elencwajg
118k7180327
Thank you a lot Georges (and also Qi). I almost had all what was needed to prove the result, but I couldn't put it into good order. I'm glad I learnt something :)
– ng_th
Dec 13 '11 at 9:18
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Suppose $X,Y$ are varieties over an algebraically closed field $k$.
Claim
If $Dsubset X$ and $ Esubset Y$ are dense subsets, then $Dtimes Esubset Xtimes Y$ is dense too.
Proof
Let $Z$ be the closure of $Dtimes E$ in $Xtimes Y$.
Fix a poind $d_0in D$ and consider the closed subvariety $lbrace d_0rbrace times Y subset Xtimes Y$.
The subset $lbrace d_0rbrace times E subset lbrace d_0rbrace times Y$ has closure $lbrace d_0rbrace times Y ;$ [ because $lbrace d_0rbrace times Y$ is isomorphic to $Y$ and $E$ is dense in $Y$]
so that $lbrace d_0rbrace times Y subset Z $ . We have proved the
Partial Result:
For all $d_0in D $ we have $lbrace d_0rbrace times Y subset Z$
End of proof:
Consider an arbitrary $yin Y$ .
For any $din D$ we know, thanks to the Partial Result, that $(d,y)in Z$. Hence $Dtimes lbrace y rbrace subset Z$.
Since the closure of $Dtimes lbrace y rbrace $ is $Xtimes lbrace y rbrace $ [ because $X times lbrace y rbrace $ is isomorphic to $X$ and $D$ is dense in $X$], we have proved that $Xtimes lbrace y rbrace subset Z$.
Since $yin Y$ was arbitrary, this implies that $Z=Xtimes Y $ i.e. that $Dtimes E$ is dense in $Xtimes Y $
answered Dec 12 '11 at 22:32
Georges Elencwajg
118k7180327
Suppose $X,Y$ are varieties over an algebraically closed field $k$.
Claim
If $Dsubset X$ and $ Esubset Y$ are dense subsets, then $Dtimes Esubset Xtimes Y$ is dense too.
Proof
Let $Z$ be the closure of $Dtimes E$ in $Xtimes Y$.
Fix a poind $d_0in D$ and consider the closed subvariety $lbrace d_0rbrace times Y subset Xtimes Y$.
The subset $lbrace d_0rbrace times E subset lbrace d_0rbrace times Y$ has closure $lbrace d_0rbrace times Y ;$ [ because $lbrace d_0rbrace times Y$ is isomorphic to $Y$ and $E$ is dense in $Y$]
so that $lbrace d_0rbrace times Y subset Z $ . We have proved the
Partial Result:
For all $d_0in D $ we have $lbrace d_0rbrace times Y subset Z$
End of proof:
Consider an arbitrary $yin Y$ .
For any $din D$ we know, thanks to the Partial Result, that $(d,y)in Z$. Hence $Dtimes lbrace y rbrace subset Z$.
Since the closure of $Dtimes lbrace y rbrace $ is $Xtimes lbrace y rbrace $ [ because $X times lbrace y rbrace $ is isomorphic to $X$ and $D$ is dense in $X$], we have proved that $Xtimes lbrace y rbrace subset Z$.
Since $yin Y$ was arbitrary, this implies that $Z=Xtimes Y $ i.e. that $Dtimes E$ is dense in $Xtimes Y $
answered Dec 12 '11 at 22:32
Georges Elencwajg
118k7180327
answered Dec 12 '11 at 22:32
Georges Elencwajg
118k7180327
answered Dec 12 '11 at 22:32
Georges Elencwajg
118k7180327
answered Dec 12 '11 at 22:32
Georges Elencwajg
118k7180327
118k7180327
Thank you a lot Georges (and also Qi). I almost had all what was needed to prove the result, but I couldn't put it into good order. I'm glad I learnt something :)
– ng_th
Dec 13 '11 at 9:18
add a comment |
Thank you a lot Georges (and also Qi). I almost had all what was needed to prove the result, but I couldn't put it into good order. I'm glad I learnt something :)
– ng_th
Dec 13 '11 at 9:18
Thank you a lot Georges (and also Qi). I almost had all what was needed to prove the result, but I couldn't put it into good order. I'm glad I learnt something :)
– ng_th
Dec 13 '11 at 9:18
Thank you a lot Georges (and also Qi). I almost had all what was needed to prove the result, but I couldn't put it into good order. I'm glad I learnt something :)
– ng_th
Dec 13 '11 at 9:18
add a comment |
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