Re:Any smooth cubic surface contains 27 lines
by infogulch on Wednesday February 17, 2016 @12:40AM
(#11115597)
So what can we say about 27 lines? Is it enough to just exclude the counterexample you gave? Say, is it true that:
All sets of 27 distinct lines describe a smooth cubic surface as long as no subset of 3 of them both intersect at the same point and are mutually orthogonal.
What additional restrictions would it take to know that the lines describe a distinct smooth cubic surface? (if that's possible at all)
eg. consider three lines through a single point and mutually orthogonal (the three axes say). There is no smooth surface that contains all three of these lines.
That said, you may be able to find conditions under which 27 lines define a smooth cubic surface, however I suspect you probably need as much information as you would need to construct the cubic surface in the first place.
A treatment of how you can parameterise cubic surfaces (which is what we are really talking about here) is at http://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=2208&...