Let P be a set of m points and L a set of n lines in R4, 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 I(P, L) = O m1/2n1/2D + m2/3n1/3s1/3 + n D + m for some absolute constant of
roportionality. This significantly improves the bound of the authors (Sharir, Solomon, Incidences between points and lines in R4. Discrete Comput Geom 57(3), 702–756, 2017), for arbitrary sets of points and lines in R4,
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 C4 (or in any higher-dimensional Cd ) under the same assumptions as

EB00000003456K

Available