the method from algebraic geometry and algebraic topology have a effect on incidence combinatorics this years.espatialy on finite field case.there is some examples of the achievement follow this idea.
Dvir-Finite Kakeya conjecture
Guth-Katz-Erdos Distance problem
there is a example with classical algebraic geometry,cubic curve in fact.
Ben Green:
$ P\subset R^2$,$ P$ is a set consist with n points.
A k-rich line is a line in $ R^2$ which contain k points of $ P$
$ N_k=$#k-rich lines,$ k\geq 2$.we call 2-rich line as original line.
there is a classical theorem:
Sylvester-Gallai theorem:if the points in $ P$ is not collinear,then $ N_2\geq 2$.
this theorem is not true in other fields.
there is a lots of counterexample.
the original proof of sylvester-Galli theorem:
find the pair of point and line minimize the distance from the point to the line.if the line is not original we can get a contradiction!
this proof is very pretty,but too clever to extend to a system method to deal with similar problem in incidence geometry…