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 ).
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?
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: