Incidences Between Points and Lines on Two- and Three-Dimensional Varieties.

Let P be a set of m points and L a set of n lines in $${\mathbb {R}}^4,$$ such that the points of P lie on an algebraic three-dimensional variety of degree $$D$$ that does not contain hyperplane or quadric components (a quadric is an algebraic variety of degree two), and no 2-flat contains more than s lines of L. We show that the number of incidences between P and L is for some absolute constant of proportionality. This significantly improves the bound of the authors (Sharir, Solomon, Incidences between points and lines in $${\mathbb {R}}^4.$$ Discrete Comput Geom 57(3), 702-756, 2017), for arbitrary sets of points and lines in $${\mathbb {R}}^4,$$ when $$D$$ is not too large. Moreover, when $$D$$ and s are constant, we get a linear bound. The same bound holds when the three-dimensional surface is embedded in any higher-dimensional space. The bound extends (with a slight deterioration, when $$D$$ is large) to the complex field too. For a complex three-dimensional variety, of degree $$D,$$ embedded in $${\mathbb {C}}^4$$ (or in any higher-dimensional $${\mathbb {C}}^d$$ ), under the same assumptions as above, we have For the proof of these bounds, we revisit certain parts of [36], combined with the following new incidence bound, for which we present a direct and fairly simple proof. Going back to the real case, let P be a set of m points and L a set of n lines in $${\mathbb {R}}^d,$$ for $$d\ge 3,$$ which lie in a common two-dimensional algebraic surface of degree $$D$$ that does not contain any 2-flat, so that no 2-flat contains more than s lines of L (here we require that the lines of L also be contained in the surface). Then the number of incidences between P and L is When $$d=3,$$ this improves the bound of Guth and Katz (On the Erdős distinct distances problem in the plane. Ann Math 181(1), 155-190, 2015) for this special case, when $$D\ll n^{1/2}.$$ Moreover, the bound does not involve the term O( nD). This term arises in most standard approaches, and its removal is a significant aspect of our result. Again, the bound is linear when $$D= O(1).$$ This bound too extends (with a slight deterioration, when $$D$$ is large) to the complex field. For a complex two-dimensional variety, of degree $$D,$$ when the ambient space is $${\mathbb {C}}^3$$ (or any higher-dimensional $${\mathbb {C}}^d$$ ), under the same assumptions as above, we have These new incidence bounds are among the very few bounds, known so far, that hold over the complex field. The bound for two-dimensional (resp., three-dimensional) varieties coincides with the bound in the real case when $$D= O(m^{1/3})$$ (resp., $$D= O(m^{1/6})$$ ). [ABSTRACT FROM AUTHOR]