Milnor fibers and characteristic varieties of 3-arrangements
speaker: Michael Falk (Northern Arizona University, Flagstaff)abstract: Let A be a union of linear hyperplanes in C3 with defining equation Q(x,y,z)=0. The Milnor fiber F of A is the affine variety defined by Q(x,y,z)=1. Let S be the projective closure of F in P3, defined by Q(x,y,z)=wn, n=deg(Q). We define a resolution of S determined by the incidence graph of the irreducible components of A, and show that the rational homology of F is determined by the local cohomology of the boundary divisor F \cap {w=0}. As a consequence we obtain a combinatorial algorithm to compute the first betti number of F. The same argument applies when Q is not reduced, which implies that rational points on the first characteristic variety of A can be detected combinatorially. Applying a 2013 result of Artal-Bartolo, Cogolludo-AgustÃn, and Matei, one concludes that the first characteristic variety of A, the jumping locus for cohomology of rank-one local systems over the complement of A, is determined combinatorially. By general results, these statements apply to arrangements of arbitrary rank.
timetable:
Tue 6 Jun, 9:00 - 9:40, Aula Dini
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