For foliated algebraic surfaces, the paper computes the Zariski decomposition of adjoint divisors K_F+D and gives explicit very-ampleness bounds for their canonical models.
Brunella, Some remarks on indices of holomorphic vector fields, Publicacions matematiques, 41 (2) (1997), 527-544
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Canonical Models of Adjoint Foliated Structures on Surfaces
For foliated algebraic surfaces, the paper computes the Zariski decomposition of adjoint divisors K_F+D and gives explicit very-ampleness bounds for their canonical models.