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: < ). 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: <

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