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In this paper we study the intersection theory on surfaces with abelianquotient singularities and we derive properties of quotients of weightedprojective planes. We also use this theory to study weighted blow-ups in orderto construct embedded $\mathbf{Q}$-resolutions of plane curve singularities andabstract $\mathbf{Q}$-resolutions of surfaces.

Intersection Theory on Abelian-Quotient $V$-Surfaces and $\mathbf{Q}$-Resolutions