Let S be a fibred surface. We prove that the existence of morphisms from non countably many fibres to curves implies, up to base change, the existence of a rational map from S to another surface fibred over the same base reflecting the properties of the original morphisms. Under some conditions of unicity base change is not needed and one recovers exactly the initial maps.
@article{urn:eudml:doc:41239,
title = {Extension of maps defined on many fibres.},
journal = {Collectanea Mathematica},
volume = {49},
year = {1998},
pages = {227-238},
zbl = {0937.14003},
mrnumber = {MR1677164},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41239}
}
Barja, Miguel A.; Naranjo, Juan Carlos. Extension of maps defined on many fibres.. Collectanea Mathematica, Tome 49 (1998) pp. 227-238. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41239/