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We prove new cases of Vojta's conjectures for surfaces in the context offunction fields, with truncation equal to one and providing an effectiveexplicit description of the exceptional set. We also prove a general andexplicit result towards Campana's conjecture over complex function fields ofcurves. Our methods rely on a local study of $\omega$-integral varieties.

On the conjectures of Vojta and Campana over function fields with explicit exceptional sets