The Morse complex of a line arrangement

Let M(A) be the complement in C2 of a complexified line arrangement. We provide compact formulas for a Morse complex which computes the (co)homology of M(A) with coefficients in an abelian local system. This refines and simplifies, in the two-dimensional case, a general construction appeared in [M. Salvetti, S. Settepanella, Combinatorial Morse theory and minimality of hyperplane arrangements, Geom. Topol. 11 (2007) 1733–1766], giving also a direct geometrical interpretation.