  We initiate a study of positive multisections of Lefschetz fibrations via
positive factorizations in framed mapping class groups of surfaces. Using our
methods, one can effectively capture various interesting symplectic surfaces in
symplectic 4-manifolds as multisections, such as Seiberg-Witten basic classes
and exceptional classes, or branched loci of compact Stein surfaces as branched
coverings of the 4-ball. Various problems regarding the topology of symplectic
4-manifolds, such as the smooth classification of symplectic Calabi-Yau
4-manifolds, can be translated to combinatorial problems in this manner. After
producing special monodromy factorizations of Lefschetz pencils on symplectic
Calabi-Yau K3 and Enriques surfaces, and introducing monodromy substitutions
tailored for generating multisections, we obtain several novel applications,
allowing us to construct: new counter-examples to Stipsicz's conjecture on
fiber sum indecomposable Lefschetz fibrations, non-isomorphic Lefschetz pencils
of the same genera on the same new symplectic 4-manifolds, the very first
examples of exotic Lefschetz pencils, and new exotic embeddings of surfaces.
