Given a non-trivial fibration from a complex projective smooth surface S to a smooth curve B of genus b. Let c(f) be the Clifford index of the general fibre F of f. In Barja et al. (Journal für die reine und angewandte Mathematik, 2016) it is proved that the relative irregularity of f, is less or equal than or equal to g-c_f. In this short note we assume that the general fiber of f is a plane curve of degree d, F a smooth plane curve of degree d and we study the infinitesimal deformation of F preserving the planarity of the curve. Then the rank of the cup-product map is at least d-2. We also show that this bound is sharp.
On the Xiao conjecture for plane curves
FAVALE, FILIPPO FRANCESCO;Pirola, G. P.
2018-01-01
Abstract
Given a non-trivial fibration from a complex projective smooth surface S to a smooth curve B of genus b. Let c(f) be the Clifford index of the general fibre F of f. In Barja et al. (Journal für die reine und angewandte Mathematik, 2016) it is proved that the relative irregularity of f, is less or equal than or equal to g-c_f. In this short note we assume that the general fiber of f is a plane curve of degree d, F a smooth plane curve of degree d and we study the infinitesimal deformation of F preserving the planarity of the curve. Then the rank of the cup-product map is at least d-2. We also show that this bound is sharp.File in questo prodotto:
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