Two questions on three quadrics in $P^5$ whose intersection is a genus $5$ K3 surface.
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It is well known that the intersection of three quadrics in $P^5$ yields a genus $5$ K3 surface. (See this link: https://en.wikipedia.org/wiki/K3_surface ).
Question I:
Does anyone have an example (or a link to an example) in which one of the quadrics is a hyperboloid of one sheet and one is a hyperbolic paraboloid?
Question II:
Assuming that such a case can exist, is it possible that the intersection can contain one line which is a ruling on both surfaces (the hyperboloid and the paraboloid)?
Or am I not visualizing the situation correctly in the first place?
Thanks as always for whatever time you can afford to spend considering this matter.
Note: in the comments below, I asked LordShark the following question:
So in this context, "quadric" is the kind of "quadric" which appears in this discussion of Plucker coordinates:
https://en.wikipedia.org/wiki/Pl%C3%BCcker_coordinates
algebraic-geometry algebraic-groups quadrics k3-surfaces
edited Sep 4 at 9:08
Armando j18eos
2,48311226
asked Dec 6 '17 at 21:50
David Halitsky
130115
add a comment |Â
up vote
1
down vote
favorite
It is well known that the intersection of three quadrics in $P^5$ yields a genus $5$ K3 surface. (See this link: https://en.wikipedia.org/wiki/K3_surface ).
Question I:
Does anyone have an example (or a link to an example) in which one of the quadrics is a hyperboloid of one sheet and one is a hyperbolic paraboloid?
Question II:
Assuming that such a case can exist, is it possible that the intersection can contain one line which is a ruling on both surfaces (the hyperboloid and the paraboloid)?
Or am I not visualizing the situation correctly in the first place?
Thanks as always for whatever time you can afford to spend considering this matter.
Note: in the comments below, I asked LordShark the following question:
So in this context, "quadric" is the kind of "quadric" which appears in this discussion of Plucker coordinates:
https://en.wikipedia.org/wiki/Pl%C3%BCcker_coordinates
algebraic-geometry algebraic-groups quadrics k3-surfaces
edited Sep 4 at 9:08
Armando j18eos
2,48311226
asked Dec 6 '17 at 21:50
David Halitsky
130115
1
These are quadrics in $P^5$; hyperboloids etc., exist inside 3-dimensional space.
– Lord Shark the Unknown
Dec 6 '17 at 22:02
@LordSharktheUnknown - Thank you for correcting my misconception so quickly! I will mark the question as answered, of course.
– David Halitsky
Dec 6 '17 at 22:04
@LordSharktheUnknown - so in this context, "quadric" is the kind of "quadric" which appears in this discussion of Plucker coordinates: en.wikipedia.org/wiki/Pl%C3%BCcker_coordinates
– David Halitsky
Dec 7 '17 at 0:10
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
It is well known that the intersection of three quadrics in $P^5$ yields a genus $5$ K3 surface. (See this link: https://en.wikipedia.org/wiki/K3_surface ).
Question I:
Does anyone have an example (or a link to an example) in which one of the quadrics is a hyperboloid of one sheet and one is a hyperbolic paraboloid?
Question II:
Assuming that such a case can exist, is it possible that the intersection can contain one line which is a ruling on both surfaces (the hyperboloid and the paraboloid)?
Or am I not visualizing the situation correctly in the first place?
Thanks as always for whatever time you can afford to spend considering this matter.
Note: in the comments below, I asked LordShark the following question:
So in this context, "quadric" is the kind of "quadric" which appears in this discussion of Plucker coordinates:
https://en.wikipedia.org/wiki/Pl%C3%BCcker_coordinates
algebraic-geometry algebraic-groups quadrics k3-surfaces
edited Sep 4 at 9:08
Armando j18eos
2,48311226
asked Dec 6 '17 at 21:50
David Halitsky
130115
It is well known that the intersection of three quadrics in $P^5$ yields a genus $5$ K3 surface. (See this link: https://en.wikipedia.org/wiki/K3_surface ).
Question I:
Does anyone have an example (or a link to an example) in which one of the quadrics is a hyperboloid of one sheet and one is a hyperbolic paraboloid?
Question II:
Assuming that such a case can exist, is it possible that the intersection can contain one line which is a ruling on both surfaces (the hyperboloid and the paraboloid)?
Or am I not visualizing the situation correctly in the first place?
Thanks as always for whatever time you can afford to spend considering this matter.
Note: in the comments below, I asked LordShark the following question:
So in this context, "quadric" is the kind of "quadric" which appears in this discussion of Plucker coordinates:
https://en.wikipedia.org/wiki/Pl%C3%BCcker_coordinates
algebraic-geometry algebraic-groups quadrics k3-surfaces
algebraic-geometry algebraic-groups quadrics k3-surfaces
edited Sep 4 at 9:08
Armando j18eos
2,48311226
asked Dec 6 '17 at 21:50
David Halitsky
130115
edited Sep 4 at 9:08
Armando j18eos
2,48311226
asked Dec 6 '17 at 21:50
David Halitsky
130115
edited Sep 4 at 9:08
Armando j18eos
2,48311226
edited Sep 4 at 9:08
Armando j18eos
2,48311226
edited Sep 4 at 9:08
Armando j18eos
2,48311226
2,48311226
asked Dec 6 '17 at 21:50
David Halitsky
130115
asked Dec 6 '17 at 21:50
David Halitsky
130115
asked Dec 6 '17 at 21:50
David Halitsky
130115
130115
1
These are quadrics in $P^5$; hyperboloids etc., exist inside 3-dimensional space.
– Lord Shark the Unknown
Dec 6 '17 at 22:02
@LordSharktheUnknown - Thank you for correcting my misconception so quickly! I will mark the question as answered, of course.
– David Halitsky
Dec 6 '17 at 22:04
@LordSharktheUnknown - so in this context, "quadric" is the kind of "quadric" which appears in this discussion of Plucker coordinates: en.wikipedia.org/wiki/Pl%C3%BCcker_coordinates
– David Halitsky
Dec 7 '17 at 0:10
add a comment |Â
1
These are quadrics in $P^5$; hyperboloids etc., exist inside 3-dimensional space.
– Lord Shark the Unknown
Dec 6 '17 at 22:02
@LordSharktheUnknown - Thank you for correcting my misconception so quickly! I will mark the question as answered, of course.
– David Halitsky
Dec 6 '17 at 22:04
@LordSharktheUnknown - so in this context, "quadric" is the kind of "quadric" which appears in this discussion of Plucker coordinates: en.wikipedia.org/wiki/Pl%C3%BCcker_coordinates
– David Halitsky
Dec 7 '17 at 0:10
1
1
These are quadrics in $P^5$; hyperboloids etc., exist inside 3-dimensional space.
– Lord Shark the Unknown
Dec 6 '17 at 22:02
These are quadrics in $P^5$; hyperboloids etc., exist inside 3-dimensional space.
– Lord Shark the Unknown
Dec 6 '17 at 22:02
@LordSharktheUnknown - Thank you for correcting my misconception so quickly! I will mark the question as answered, of course.
– David Halitsky
Dec 6 '17 at 22:04
@LordSharktheUnknown - Thank you for correcting my misconception so quickly! I will mark the question as answered, of course.
– David Halitsky
Dec 6 '17 at 22:04
@LordSharktheUnknown - so in this context, "quadric" is the kind of "quadric" which appears in this discussion of Plucker coordinates: en.wikipedia.org/wiki/Pl%C3%BCcker_coordinates
– David Halitsky
Dec 7 '17 at 0:10
@LordSharktheUnknown - so in this context, "quadric" is the kind of "quadric" which appears in this discussion of Plucker coordinates: en.wikipedia.org/wiki/Pl%C3%BCcker_coordinates
– David Halitsky
Dec 7 '17 at 0:10
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
0
down vote
accepted
LordSharks's comment above actually answers the question by explaining why the question was ill-formed in the first place.
answered Dec 6 '17 at 22:06
David Halitsky
130115
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
LordSharks's comment above actually answers the question by explaining why the question was ill-formed in the first place.
answered Dec 6 '17 at 22:06
David Halitsky
130115
add a comment |Â
up vote
0
down vote
accepted
LordSharks's comment above actually answers the question by explaining why the question was ill-formed in the first place.
answered Dec 6 '17 at 22:06
David Halitsky
130115
add a comment |Â
up vote
0
down vote
accepted
up vote
0
down vote
accepted
LordSharks's comment above actually answers the question by explaining why the question was ill-formed in the first place.
answered Dec 6 '17 at 22:06
David Halitsky
130115
LordSharks's comment above actually answers the question by explaining why the question was ill-formed in the first place.
answered Dec 6 '17 at 22:06
David Halitsky
130115
answered Dec 6 '17 at 22:06
David Halitsky
130115
answered Dec 6 '17 at 22:06
David Halitsky
130115
answered Dec 6 '17 at 22:06
David Halitsky
130115
130115
add a comment |Â
add a comment |Â
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1
These are quadrics in $P^5$; hyperboloids etc., exist inside 3-dimensional space.
– Lord Shark the Unknown
Dec 6 '17 at 22:02
@LordSharktheUnknown - Thank you for correcting my misconception so quickly! I will mark the question as answered, of course.
– David Halitsky
Dec 6 '17 at 22:04
@LordSharktheUnknown - so in this context, "quadric" is the kind of "quadric" which appears in this discussion of Plucker coordinates: en.wikipedia.org/wiki/Pl%C3%BCcker_coordinates
– David Halitsky
Dec 7 '17 at 0:10