Semistable Lefschetz pencils

Abstract We study the geometry and cohomology of Lefschetz pencils for semistable schemes over a discrete valuation ring. We relate the global cohomological properties of the Lefschetz pencil and the monodromy-weight conjecture; in particular, we show that if one assumes the monodromy-weight conjecture in smaller dimensions, then one can obtain a rather complete understanding of the relative cohomology of the pencil. This reduces the monodromy-weight conjecture to an arithmetic variant of a conjecture of Kashiwara for the projective line.