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







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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














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







share|cite|improve this question



















  • 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












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







share|cite|improve this question















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






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edited Sep 4 at 9:08









Armando j18eos

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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












  • 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










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LordSharks's comment above actually answers the question by explaining why the question was ill-formed in the first place.






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    LordSharks's comment above actually answers the question by explaining why the question was ill-formed in the first place.






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      LordSharks's comment above actually answers the question by explaining why the question was ill-formed in the first place.






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        LordSharks's comment above actually answers the question by explaining why the question was ill-formed in the first place.






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        LordSharks's comment above actually answers the question by explaining why the question was ill-formed in the first place.







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        answered Dec 6 '17 at 22:06









        David Halitsky

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