On Algebraic Surfaces with Pencils of Curves of Genus 2

Fibrations of Genus Two on Complex Surfaces

“…The new non-trivial solutions occur for k = 5, 8, 10 and they correspond to (k 1 , k 2 , k 3 ) = (2, 2, 1), (3, 1, 1), (4, 3, 1), (5, 4, 1), (5,3,2).…”

“…[2,4,5,11,14] where the reader will find results concerned with the "geography" of manifolds carrying these fibrations as well as results and methods that are aimed at understanding the structure of fibrations itself. Our previous experience seems to indicate that this literature is not widely known to colleagues working in (differential) geometry and topology (and even on certain complex dynamical systems).…”

On surfaces with a canonical pencil

“…Assume that there is a point p with g 0 ( p) = f 0 ( p) = 0. Then (see (8)) p ∈ supp τ . The node of C above p, say P 0 , belongs to s = {y 2 = f 0 y 0 + f 1 y 1 = 0} and therefore has relative coordinates (y 0 : y 1 : y 2 ) = (1 : 0 : 0).…”

“…Indeed it is well known (see [8] or [18,Proposition 4.1] that the only non free pencil of genus 2 curves on a surface of general type is the canonical system of a hypersurface of degree 10 in P(1 : 1 : 2 : 5).…”

“…[8,15]): it grows asymptotically as 2χ. When studying the relative canonical algebra R of genus 2 fibrations over surfaces with fixed invariants (K 2 , χ), big values of H correspond to many a priori possibilities for its second graded piece R 2 , which is the key bundle to be computed.…”