Can two projective surfaces intersect in points only?
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Let $S_1,S_2subset mathbb P^n$ be two algebraic surfaces (we may assume that $n=3$). Is it possible that $S_1cap S_2=x_1,dots,x_N$ a finite set of points?
I can imagine the surfaces two be disjoint, if they are parallel; to intersect along a curve, which should be the general situation; and to intersect along a surface, which occurs if $S_1=S_2$. Thus it seems like everything else is possible and that's why I am curious whether a point intersection could also be possible.
geometry algebraic-geometry surfaces intersection-theory
asked Sep 7 at 7:53
James
29613
add a comment |Â
up vote
0
down vote
favorite
Let $S_1,S_2subset mathbb P^n$ be two algebraic surfaces (we may assume that $n=3$). Is it possible that $S_1cap S_2=x_1,dots,x_N$ a finite set of points?
I can imagine the surfaces two be disjoint, if they are parallel; to intersect along a curve, which should be the general situation; and to intersect along a surface, which occurs if $S_1=S_2$. Thus it seems like everything else is possible and that's why I am curious whether a point intersection could also be possible.
geometry algebraic-geometry surfaces intersection-theory
asked Sep 7 at 7:53
James
29613
Do you means "surfaces" (subvarieties of dimension two) or "hypersurfaces" (subvarieties defined by a single equation). Two hypersurfaces in $Bbb P^n$ meet in a set of dimension at least $n-2$.
– Lord Shark the Unknown
Sep 7 at 9:29
I mean surfaces, i.e. two-dimensional subvarieties.
– James
Sep 7 at 11:49
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $S_1,S_2subset mathbb P^n$ be two algebraic surfaces (we may assume that $n=3$). Is it possible that $S_1cap S_2=x_1,dots,x_N$ a finite set of points?
I can imagine the surfaces two be disjoint, if they are parallel; to intersect along a curve, which should be the general situation; and to intersect along a surface, which occurs if $S_1=S_2$. Thus it seems like everything else is possible and that's why I am curious whether a point intersection could also be possible.
geometry algebraic-geometry surfaces intersection-theory
asked Sep 7 at 7:53
James
29613
Let $S_1,S_2subset mathbb P^n$ be two algebraic surfaces (we may assume that $n=3$). Is it possible that $S_1cap S_2=x_1,dots,x_N$ a finite set of points?
I can imagine the surfaces two be disjoint, if they are parallel; to intersect along a curve, which should be the general situation; and to intersect along a surface, which occurs if $S_1=S_2$. Thus it seems like everything else is possible and that's why I am curious whether a point intersection could also be possible.
geometry algebraic-geometry surfaces intersection-theory
geometry algebraic-geometry surfaces intersection-theory
asked Sep 7 at 7:53
James
29613
asked Sep 7 at 7:53
James
29613
asked Sep 7 at 7:53
James
29613
asked Sep 7 at 7:53
James
29613
asked Sep 7 at 7:53
James
29613
29613
Do you means "surfaces" (subvarieties of dimension two) or "hypersurfaces" (subvarieties defined by a single equation). Two hypersurfaces in $Bbb P^n$ meet in a set of dimension at least $n-2$.
– Lord Shark the Unknown
Sep 7 at 9:29
I mean surfaces, i.e. two-dimensional subvarieties.
– James
Sep 7 at 11:49
add a comment |Â
Do you means "surfaces" (subvarieties of dimension two) or "hypersurfaces" (subvarieties defined by a single equation). Two hypersurfaces in $Bbb P^n$ meet in a set of dimension at least $n-2$.
– Lord Shark the Unknown
Sep 7 at 9:29
I mean surfaces, i.e. two-dimensional subvarieties.
– James
Sep 7 at 11:49
Do you means "surfaces" (subvarieties of dimension two) or "hypersurfaces" (subvarieties defined by a single equation). Two hypersurfaces in $Bbb P^n$ meet in a set of dimension at least $n-2$.
– Lord Shark the Unknown
Sep 7 at 9:29
Do you means "surfaces" (subvarieties of dimension two) or "hypersurfaces" (subvarieties defined by a single equation). Two hypersurfaces in $Bbb P^n$ meet in a set of dimension at least $n-2$.
– Lord Shark the Unknown
Sep 7 at 9:29
I mean surfaces, i.e. two-dimensional subvarieties.
– James
Sep 7 at 11:49
I mean surfaces, i.e. two-dimensional subvarieties.
– James
Sep 7 at 11:49
add a comment |Â
1 Answer
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3
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I am assuming the base field is algebraically closed.
You say rather casually in the question "we may assume that $n=3$", but the answer depends crucially on whether $n=3$ or not.
If $n=3$ then any two surfaces intersect in a set of dimension at least 1 (they cannot be disjoint, and there is no such thing as "parallel" in projective space). A reference is Hartshorne Theorem I.7.2.
If $n>3$ then certainly it is possible for two surfaces in $mathbf P^n$ to intersect in a finite set of points --- for example, try coordinate planes in $mathbf P^4$. Indeed, by the Theorem already mentioned, any two surfaces in $mathbf P^4$ must have nonempty intersection, and in general the intersection will be a finite set.
If $n>4$ then any dimension up to 2 is possible for the intersection of two surfaces, but there are statements analogous to the above for intersections of higher-dimensional algebraic subsets.
answered Sep 7 at 9:10
Asal Beag Dubh
37814
OK, I didn't know its depending so much on the dimension and was just assuming that we could assume $n=3$. Actually, I was thinking way to much of a plane in $mathbb P^n$ as a three-dimensional object in $mathbb C^n+1$. The reference you mentioned is actually very helpful! By intersection of coordinate planes you mean for example $S_1=x_0=x_1=0, S_2=x_2=x_3=0subsetmathbb P^4$ which intersect in the point $(0:0:0:0:1)$, correct?
– James
Sep 7 at 11:46
@James: yes, that's right.
– Asal Beag Dubh
Sep 7 at 12:02
Thank you very much! :)
– James
Sep 7 at 12:12
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
I am assuming the base field is algebraically closed.
You say rather casually in the question "we may assume that $n=3$", but the answer depends crucially on whether $n=3$ or not.
If $n=3$ then any two surfaces intersect in a set of dimension at least 1 (they cannot be disjoint, and there is no such thing as "parallel" in projective space). A reference is Hartshorne Theorem I.7.2.
If $n>3$ then certainly it is possible for two surfaces in $mathbf P^n$ to intersect in a finite set of points --- for example, try coordinate planes in $mathbf P^4$. Indeed, by the Theorem already mentioned, any two surfaces in $mathbf P^4$ must have nonempty intersection, and in general the intersection will be a finite set.
If $n>4$ then any dimension up to 2 is possible for the intersection of two surfaces, but there are statements analogous to the above for intersections of higher-dimensional algebraic subsets.
answered Sep 7 at 9:10
Asal Beag Dubh
37814
OK, I didn't know its depending so much on the dimension and was just assuming that we could assume $n=3$. Actually, I was thinking way to much of a plane in $mathbb P^n$ as a three-dimensional object in $mathbb C^n+1$. The reference you mentioned is actually very helpful! By intersection of coordinate planes you mean for example $S_1=x_0=x_1=0, S_2=x_2=x_3=0subsetmathbb P^4$ which intersect in the point $(0:0:0:0:1)$, correct?
– James
Sep 7 at 11:46
@James: yes, that's right.
– Asal Beag Dubh
Sep 7 at 12:02
Thank you very much! :)
– James
Sep 7 at 12:12
add a comment |Â
up vote
3
down vote
accepted
I am assuming the base field is algebraically closed.
You say rather casually in the question "we may assume that $n=3$", but the answer depends crucially on whether $n=3$ or not.
If $n=3$ then any two surfaces intersect in a set of dimension at least 1 (they cannot be disjoint, and there is no such thing as "parallel" in projective space). A reference is Hartshorne Theorem I.7.2.
If $n>3$ then certainly it is possible for two surfaces in $mathbf P^n$ to intersect in a finite set of points --- for example, try coordinate planes in $mathbf P^4$. Indeed, by the Theorem already mentioned, any two surfaces in $mathbf P^4$ must have nonempty intersection, and in general the intersection will be a finite set.
If $n>4$ then any dimension up to 2 is possible for the intersection of two surfaces, but there are statements analogous to the above for intersections of higher-dimensional algebraic subsets.
answered Sep 7 at 9:10
Asal Beag Dubh
37814
OK, I didn't know its depending so much on the dimension and was just assuming that we could assume $n=3$. Actually, I was thinking way to much of a plane in $mathbb P^n$ as a three-dimensional object in $mathbb C^n+1$. The reference you mentioned is actually very helpful! By intersection of coordinate planes you mean for example $S_1=x_0=x_1=0, S_2=x_2=x_3=0subsetmathbb P^4$ which intersect in the point $(0:0:0:0:1)$, correct?
– James
Sep 7 at 11:46
@James: yes, that's right.
– Asal Beag Dubh
Sep 7 at 12:02
Thank you very much! :)
– James
Sep 7 at 12:12
add a comment |Â
up vote
3
down vote
accepted
up vote
3
down vote
accepted
I am assuming the base field is algebraically closed.
You say rather casually in the question "we may assume that $n=3$", but the answer depends crucially on whether $n=3$ or not.
If $n=3$ then any two surfaces intersect in a set of dimension at least 1 (they cannot be disjoint, and there is no such thing as "parallel" in projective space). A reference is Hartshorne Theorem I.7.2.
If $n>3$ then certainly it is possible for two surfaces in $mathbf P^n$ to intersect in a finite set of points --- for example, try coordinate planes in $mathbf P^4$. Indeed, by the Theorem already mentioned, any two surfaces in $mathbf P^4$ must have nonempty intersection, and in general the intersection will be a finite set.
If $n>4$ then any dimension up to 2 is possible for the intersection of two surfaces, but there are statements analogous to the above for intersections of higher-dimensional algebraic subsets.
answered Sep 7 at 9:10
Asal Beag Dubh
37814
I am assuming the base field is algebraically closed.
You say rather casually in the question "we may assume that $n=3$", but the answer depends crucially on whether $n=3$ or not.
If $n=3$ then any two surfaces intersect in a set of dimension at least 1 (they cannot be disjoint, and there is no such thing as "parallel" in projective space). A reference is Hartshorne Theorem I.7.2.
If $n>3$ then certainly it is possible for two surfaces in $mathbf P^n$ to intersect in a finite set of points --- for example, try coordinate planes in $mathbf P^4$. Indeed, by the Theorem already mentioned, any two surfaces in $mathbf P^4$ must have nonempty intersection, and in general the intersection will be a finite set.
If $n>4$ then any dimension up to 2 is possible for the intersection of two surfaces, but there are statements analogous to the above for intersections of higher-dimensional algebraic subsets.
answered Sep 7 at 9:10
Asal Beag Dubh
37814
edited Sep 7 at 9:21
answered Sep 7 at 9:10
Asal Beag Dubh
37814
answered Sep 7 at 9:10
Asal Beag Dubh
37814
answered Sep 7 at 9:10
Asal Beag Dubh
37814
37814
OK, I didn't know its depending so much on the dimension and was just assuming that we could assume $n=3$. Actually, I was thinking way to much of a plane in $mathbb P^n$ as a three-dimensional object in $mathbb C^n+1$. The reference you mentioned is actually very helpful! By intersection of coordinate planes you mean for example $S_1=x_0=x_1=0, S_2=x_2=x_3=0subsetmathbb P^4$ which intersect in the point $(0:0:0:0:1)$, correct?
– James
Sep 7 at 11:46
@James: yes, that's right.
– Asal Beag Dubh
Sep 7 at 12:02
Thank you very much! :)
– James
Sep 7 at 12:12
add a comment |Â
OK, I didn't know its depending so much on the dimension and was just assuming that we could assume $n=3$. Actually, I was thinking way to much of a plane in $mathbb P^n$ as a three-dimensional object in $mathbb C^n+1$. The reference you mentioned is actually very helpful! By intersection of coordinate planes you mean for example $S_1=x_0=x_1=0, S_2=x_2=x_3=0subsetmathbb P^4$ which intersect in the point $(0:0:0:0:1)$, correct?
– James
Sep 7 at 11:46
@James: yes, that's right.
– Asal Beag Dubh
Sep 7 at 12:02
Thank you very much! :)
– James
Sep 7 at 12:12
OK, I didn't know its depending so much on the dimension and was just assuming that we could assume $n=3$. Actually, I was thinking way to much of a plane in $mathbb P^n$ as a three-dimensional object in $mathbb C^n+1$. The reference you mentioned is actually very helpful! By intersection of coordinate planes you mean for example $S_1=x_0=x_1=0, S_2=x_2=x_3=0subsetmathbb P^4$ which intersect in the point $(0:0:0:0:1)$, correct?
– James
Sep 7 at 11:46
OK, I didn't know its depending so much on the dimension and was just assuming that we could assume $n=3$. Actually, I was thinking way to much of a plane in $mathbb P^n$ as a three-dimensional object in $mathbb C^n+1$. The reference you mentioned is actually very helpful! By intersection of coordinate planes you mean for example $S_1=x_0=x_1=0, S_2=x_2=x_3=0subsetmathbb P^4$ which intersect in the point $(0:0:0:0:1)$, correct?
– James
Sep 7 at 11:46
@James: yes, that's right.
– Asal Beag Dubh
Sep 7 at 12:02
@James: yes, that's right.
– Asal Beag Dubh
Sep 7 at 12:02
Thank you very much! :)
– James
Sep 7 at 12:12
Thank you very much! :)
– James
Sep 7 at 12:12
add a comment |Â
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Do you means "surfaces" (subvarieties of dimension two) or "hypersurfaces" (subvarieties defined by a single equation). Two hypersurfaces in $Bbb P^n$ meet in a set of dimension at least $n-2$.
– Lord Shark the Unknown
Sep 7 at 9:29
I mean surfaces, i.e. two-dimensional subvarieties.
– James
Sep 7 at 11:49