Sergey Yuzvinsky
(University of Oregon)
"Special fibers of pencils of hypersurfaces and topology of hyperplane arrangement complements"
Abstract:
Let $F_1$ and $F_2$ be two homogeneous complex polynomials of same degree $d$ and let $\{a_1F_1+a_2F_2\}, a_i$ run through complex numbers be the pencil generated by them. Suppose that $d>1$ and the generic polynomial in the pencil is irreducible. A classically looking question asks how many completely reducible members (fibers) the pencil can have. We give a suprisingly simple answer to this question and discuss related (even equivalent) questions in combinatorics of projective lines and hyperplane arrangements.