A degree doubling formula for braid monodromies and Lefschetz pencils (Q930811)

The first author [Invent. Math. 139, No.~3, 551--602 (2000; Zbl 1080.53084)] showed that every compact symplectic four-manifold \((X,\omega)\) can be realized as an approximately holomorphic branched covering of \(\mathbb{CP}^2\) whose branch curve is a symplectic curve in \(\mathbb{C P}^2\)with cusps and nodes as singularities. Such a covering can be constructed by some sections of the line bundle \(L^\otimes{^k}\), where \(c_1(L)=\frac{1}{2\pi}[\omega]\) is integral. For large \({k}\) the authors derived the braid monodromy invariants in a previous paper [ibid. 142, No.~3, 631--673 (2000; Zbl 0961.57019)], which describe the symplectic 4-manifold \((X,\omega)\). In this paper the authors derive an explicit formula relating the braid monodromy invariants obtained for given degree \({k}\) to those obtained for the degree \({2k}\). Also they obtain a similar formula for the monodromy of symplectic Lefschetz pencils. This formula gives an answer to a question considered by Donaldson and for which a partial result has been obtained by Smith. To prove the results the authors explain the necessary ingredients such as the braid monodromy invariants, degree doubling process, stably quasiholomorphic covering, folding process, regeneration of the mutual intersections, assembling rule, the degree doubling formula for braid monodromies, mapping class groups and the degree doubling formula for Lefschetz pencils.