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Another good summary of vanishing cycles and Lefschetz pencils is found in Sections 4 and 5 of P. Deligne's "La conjecture de Weil, I" (available online http://www.numdam.org/numdam-bin/fitem?id=PMIHES_1974__43__273_0). He discusses briefly both the theory over the complex numbers and the version in étale cohomology; and then immediately proceeds to putting these to pretty good use.