On The Number Of Ordinary Conics

We prove a lower bound on the number of ordinary conics determined by a finite point set in R-2. An ordinary conic for S subset of R-2 is a conic that is determined by five points of S and contains no other points of S. Wiseman and Wilson proved the Sylvester Gallai-type statement that if a finite point set is not contained in a conic, then it determines at least one ordinary conic. We give a simpler proof of this statement and then combine it with a theorem of Green and Tao to prove our main result: If S is not contained in a conic and has at most c vertical bar S vertical bar points on a line, then S determines Omega(c)(vertical bar 5 vertical bar(4)) ordinary conics. We also give constructions, based on the group law on elliptic curves, that show that the exponent in our bound is best possible.