On the Number of Ordinary Lines Determined by Sets in Complex Space

Kelly\u27s theorem states that a set of n points affinely spanning C^3 must determine at least one ordinary complex line (a line passing through exactly two of the points). Our main theorem shows that such sets determine at least 3n/2 ordinary lines, unless the configuration has n-1 points in a plane and one point outside the plane (in which case there are at least n-1 ordinary lines). In addition, when at most n/2 points are contained in any plane, we prove a theorem giving stronger bounds that take advantage of the existence of lines with four and more points (in the spirit of Melchior\u27s and Hirzebruch\u27s inequalities). Furthermore, when the points span four or more dimensions, with at most n/2 points contained in any three dimensional affine subspace, we show that there must be a quadratic number of ordinary lines

Ordinary planes, coplanar quadruples, and space quartics

An ordinary plane of a finite set of points in real 3-space with no three collinear is a plane intersecting the set in exactly three points. We prove a structure theorem for sets of points spanning few ordinary planes. Our proof relies on Green and Tao's work on ordinary lines in the plane, combined with classical results on space quartic curves and non-generic projections of curves. This gives an alternative approach to Ball's recent results on ordinary planes, as well as extending them. We also give bounds on the number of coplanar quadruples determined by a finite set of points on a rational space quartic curve in complex 3-space, answering a question of Raz, Sharir, and De Zeeuw [Israel J. Math. 227 (2018) 663–690]

Three Dimensional N=2 Gauge Theories and Degenerations of Calabi-Yau Four-Folds

Three dimensional N=2 gauge theories with arbitrary gauge group and fundamental flavors are engineered from degenerations of Calabi-Yau four-folds. We show how Coulomb and Higgs branches emerge in the geometric picture. The analysis of instanton generated superpotentials unravels interesting aspects of the five-brane effective action in M theory.Comment: subsections on Sp(N) and Spin(2N+1) theories removed, 32 pages, harvmac, 6 postscript figure