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

Abstract

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 $$\begin{aligned} I(P,L) = O\bigl (m^{1/2}n^{1/2}D + m^{2/3}n^{1/3}s^{1/3} + nD + m\bigr ) \end{aligned}$$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 $$\begin{aligned} I(P,L) = O\bigl (m^{1/2}n^{1/2}D + m^{2/3}n^{1/3}s^{1/3} + D^6 + nD + m \bigr ). \end{aligned}$$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 $$\begin{aligned} I(P,L) = O\bigl (m^{1/2}n^{1/2}D^{1/2} + m^{2/3}D^{2/3}s^{1/3} + m + n\bigr ). \end{aligned}$$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 $$\begin{aligned} I(P,L) = O\bigl (m^{1/2}n^{1/2}D^{1/2} + m^{2/3}D^{2/3}s^{1/3} + D^3 + m + n\bigr ). \end{aligned}$$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., \(D= O(m^{1/6})\)).