Incidences Between Points and Lines in $${\\mathbb {R}}^4$$R4

We show that the number of incidences between m distinct points and n distinct lines in $${\\mathbb {R}}^4$$R4 is $$O(2^{c\\sqrt{\\log m}} (m^{2/5}n^{4/5}+m) + m^{1/2}n^{1/2}q^{1/4} + m^{2/3}n^{1/3}s^{1/3} + n)$$O(2clogm(m2/5n4/5+m)+m1/2n1/2q1/4+m2/3n1/3s1/3+n), for a suitable absolute constant c, provided that no 2-plane contains more than s input lines, and no hyperplane or quadric contains more than q lines. The bound holds without the factor $$2^{c\\sqrt{\\log m}}$$2clogm when $$m \\le n^{6/7}$$m≤n6/7 or $$m \\ge n^{5/3}$$mźn5/3. Except for the factor $$2^{c\\sqrt{\\log m}}$$2clogm, the bound is tight in the worst case.