# Two questions on three quadrics in $P^5$ whose intersection is a genus $5$ K3 surface.

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.

• These are quadrics in $P^5$; hyperboloids etc., exist inside 3-dimensional space. – Lord Shark the Unknown Dec 6 '17 at 22:02