Every non-singular complex projective cubic surface has $27$ lines. Many such surfaces contain points where three lines intersect (called Eckardt points). There are even surfaces with many Eckardt points, like the Fermat cubic, which has $18$. Is there any such non-singular complex projective cubic surface where four, five, or six lines intersect at a point?


[Post on Math StackExchange](https://math.stackexchange.com/questions/4585682/is-there-a-non-singular-cubic-surface-that-has-a-point-where-four-lines-intersec)



Is there a non-singular cubic surface that has a point where four lines intersect?
No, this is not possible. If p is a smooth point on any surface S, and is
contained in a line l on S, then l is contained in the tangent plane at p,
call it T\_p. Now if S is a cubic then it intersects T\_p in a cubic curve
(with some singularity at p, even though S is smooth at p); and a cubic curve
can contain at most three lines.


