Two inequalities for a sequence of blow-ups (Q745649)

This short communication gives some inequalites relating the multiplicity of a germ of an effective 1-cycle on a germ of a nonsingular surface, the full multiplicity of the 1-cycle with respect to the order and the discrepancies of a prime divisor over a point with respect to the ``lower storeys'' of a resolution of the surface. The proof uses the oriented graph associated to the resolution and the inequalities are rewritten in this language, they are application of inequalities of [\textit{A. Pukhlikov}, Birationally rigid varieties. Mathematical Surveys and Monographs 190. Providence, RI: American Mathematical Society (AMS) (2013; Zbl 1297.14001)].