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Let $X$ be a smooth projective algebraic surface (over $\mathbb{C}$ ). For all $L\in \mathrm{Pic}(X)$, we have $$\chi(L)=\chi(\mathcal{O}_X)+\frac{1}{2}(L^2-L\cdot \omega_X).$$ This is the famous Riemann-Roch theorem in the flavour I like the most. It usually comes together with the following two formulas: $$\chi(\mathcal{O}_S)=\frac{1}{12}(K_X^2+\chi_{top}(X)),$$ the Noether's formula and $$2p_a(C)-2=C^2-C\cdot 2p_a(C)-2=C^2+C\cdot K_X,$$ the genus formula for $C$ an irreducible (possibly singular) curve.

Is there a similar (or maybe the same) version for a) Smooth quasi-projective surfaces. b) Projective surface with quotient singularities, or A-D-E singularities.

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# Generalisations of Riemann-Roch for surfaces

Let $X$ be a smooth projective algebraic surface (over $\mathbb{C}$ ). For all $L\in \mathrm{Pic}(X)$, we have $$\chi(L)=\chi(\mathcal{O}_X)+\frac{1}{2}(L^2-L\cdot \omega_X).$$ This is the famous Riemann-Roch theorem in the flavour I like the most. It usually comes together with the following two formulas: $$\chi(\mathcal{O}_S)=\frac{1}{12}(K_X^2+\chi_{top}(X)),$$ the Noether's formula and $$2p_a(C)-2=C^2-C\cdot K_X,$$ the genus formula for $C$ an irreducible (possibly singular) curve.

Is there a similar (or maybe the same) version for a) Smooth quasi-projective surfaces. b) Projective surface with quotient singularities, or A-D-E singularities.