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?
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1$\begingroup$ From the title it sounds like you’re asking if all nonsingular cubic surfaces have such a point, but from the post it sounds like you’re asking if there exists even one such surface. $\endgroup$– BmaCommented Nov 27, 2022 at 1:46
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1$\begingroup$ I fixed it. Thanks for your helpful comment. $\endgroup$– mathlanderCommented Nov 27, 2022 at 1:51
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3$\begingroup$ Welcome new contributor. No, there is no such cubic surface. Consider the intersection of the cubic surface with the osculating hyperplane at the point. $\endgroup$– Jason StarrCommented Nov 27, 2022 at 2:17
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$\begingroup$ Can you please elaborate on this in an answer? $\endgroup$– mathlanderCommented Nov 27, 2022 at 2:19
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1 Answer
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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.
answered Nov 27, 2022 at 2:26
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