Ordinary lines in space

Frank de Zeeuw, Baruch College/CUNY

Date and time: 2pm, Friday, October 5, 2018

Place: CUNY Graduate Center, Rm 4419

The Sylvester-Gallai theorem states that if a finite set of points in the real plane is not contained in a line, then it spans at least one ordinary line, i.e., a line containing exactly two of the points. In fact, by a result of Green and Tao, any finite point set in the real plane spans a linear number of ordinary lines, and that is best possible because there are point sets on cubic curves that determine only a linear number of ordinary lines. One can consider the same question in three-dimensional space. By projection, the results in the plane hold word-for-word in space, but the known constructions with a linear number of ordinary lines are contained in a plane. I will show that if one assumes that the point set does not have too many points on a plane, then it spans a quadratic number of ordinary lines. More precisely, for any $a\lt1$ there is a $c\gt0$ such that if we have $n$ points in real space with at most $an$ points on a plane, then there are at least $cn^2$ ordinary lines. The proof uses projection and Beck’s theorem.