On combinatorial invariance of the cohomology of the Milnor fiber of arrangements and the Catalan equation over function fields

Abstract

We discuss combinatorial invariance of the betti numbers of the Milnor fiber for arrangements of lines with points of multiplicity two and three and describe a link between this problem and enumeration of solutions of the Catalan equation over function field in the case when its coefficients are products of linear forms and the equation defines an elliptic curve.