We prove effective upper bounds on the number of effective sections of a Hermitian line bundle over an arithmetic surface. It is an effective version of the arithmetic Hilbert–Samuel formula in the nef case. As a consequence, we obtain effective lower bounds on the Faltings height and on the self-intersection of the canonical bundle in terms of the number of singular points on fibers of the arithmetic surface.
Xinyi Yuan. Tong Zhang. "Effective bound of linear series on arithmetic surfaces." Duke Math. J. 162 (10) 1723 - 1770, 15 July 2013. https://doi.org/10.1215/00127094-2322779