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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

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    $\begingroup$ These are quadrics in $P^5$; hyperboloids etc., exist inside 3-dimensional space. $\endgroup$ – Lord Shark the Unknown Dec 6 '17 at 22:02
  • $\begingroup$ @LordSharktheUnknown - Thank you for correcting my misconception so quickly! I will mark the question as answered, of course. $\endgroup$ – David Halitsky Dec 6 '17 at 22:04
  • $\begingroup$ @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 $\endgroup$ – 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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