4 problems

Kernel-generation conjecture. For pencils of sufficiently large degree $(k\gg 0)$, the kernel of $\rho$ is generated by the three types of elements described in Propositions 3.2--3…

Kernel-relations completeness conjecture. The listed relations for the kernel are conjectured to be complete in a suitable $k \rightarrow \infty$ sense, where $k$ is the degree par…

Topological breeding conjecture. For every $d\geq 4$, the holomorphic Lefschetz pencil

Let $(X,f)$ be a non-trivial genus-$g$ Lefschetz pencil, and let $b_1(X)$ denote the first Betti number of its total space. For fixed $g$, consider the supremum of $b_1(X)$ over al…