The article under review is aimed at studying self-intersection properties of sections of surface bundles over surfaces, and of Lefschetz fibrations.
Let \(\Sigma_g\) be the closed orientable surface of genus \(g\), and let \(f : X \to \Sigma_h\) be either a \(\Sigma_g\)-bundle over \(\Sigma_h\), or a (positive) Lefschetz fibration with regular fiber \(\Sigma_g\), and suppose that \(f\) admits a section, which can be identified with a surface \(S \subset X\) intersecting any fiber of \(f\) transversely in a single point.
In Proposition 4 it is proved, for \(g, h \geq 1\), that the self-intersection of \(S\) satisfies the following inequalities (which are called adjunction bounds): \([S]^2 \leq 2h-2\) if \(f\) is a Lefschetz fibration (that is, with a non-empty set of critical points), and \(2-2h \leq [S]^2 \leq 2h-2\) if \(f\) is a surface bundle.
As the name suggests, these bounds are derived from the adjunction inequality for Seiberg-Witten basic classes, by taking a symplectic structure on \(X\) (which exists by the Gompf-Thurston construction). Indeed, for \(b^+(X) > 1\), this inequality immediately implies \([S]^2 \leq -\chi(S)\), which is the bound for Lefschetz fibrations. If \(f\) is a genuine bundle, the other inequality can be easily obtained by applying the same argument to \(X\) with reversed orientation (for Lefschetz fibrations this trick does not work, because the fibration would become achiral, making it not symplectic).
The main Theorem 1 states that for any \(h\geq 1\) and \(g\geq 2\) and for any \(k\) such that \(|k| \leq 2h-2\), there is a \(\Sigma_g\)-bundle over \(\Sigma_h\) with a section of self-intersection \(k\). Moreover, for \(g \geq 8h-8\), there is a \(\Sigma_g\)-bundle over \(\Sigma_h\) admitting sections with all possible self-intersections allowed by the adjunction bounds (Theorem 18).
Then, there are no other universal bounds for self-intersections that can be expressed in terms of fiber and base genera (actually, the only parameter is the base genus), and the adjunction bounds are sharp.
In Theorem 3 the authors prove that, for \(g \geq 2\) and \(h\geq 1\), there is no bound on the number of critical points of relatively minimal genus-\(g\) Lefschetz fibrations over \(\Sigma_h\) admitting a maximal section. The proof is based on known relations in the mapping class group of a genus-\(g\) surface with one boundary component, that are used to construct factorizations of the \((2-2h)^{\text{th}}\) power of the boundary parallel Dehn twist in terms of commutators and non-separating Dehn twists, so that the number of the Dehn twists is arbitrarily large. Such factorizations represent the monodromy sequences of the desired Lefschetz fibrations.
In the same spirit Theorem 2 (Theorem 15) is proved, which represents the first exact non-zero computation of the commutator length of certain elements of the mapping class group. Specifically, the authors prove that, for a boundary parallel Dehn twist \(t_\delta\) of a surface with non-empty boundary and genus \(g \geq 2\), the commutator length of \(t_\delta^n\), \(n \neq 0\), is the integer part of \((|n| + 3)/2\), and so the stable commutator length of \(t_\delta\) is \(1/2\).