Suppose that $T$ is a smooth complex projective algebraic surface, such that the finite cyclic group $\mathbb{Z_n}$ acts on it. Also, suppose that $S$ is another complex projective algebraic surface (not necessarily smooth) such that $S$ is birational to $T/\mathbb{Z}_{n}$. What can one say about the relation between the (sheaf theoretic) Euler characteristics of $T$ and $S$? in particular, can we claim that the Euler characteristic of $T$, $e(T)$ is greater than or equal to $ne(S)$?
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$\begingroup$ what is the "sheaf theoretic Euler characteristic"? $\endgroup$– Vivek ShendeCommented Jun 26, 2011 at 15:58
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$\begingroup$ I mean the Euler characteristic of the structure sheaf as defined in Hartshorne p.230. $\endgroup$– CyrusCommented Jun 26, 2011 at 16:05
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2$\begingroup$ Let $T$ be an abelian surface with $\{\pm1\}$ acting it. The quotient is a singular K3-surface and a resolution is a non-singular K3-surface. Both of them have Euler characteristic $2$ and $T$ has Euler characteristic $0$ so the answer to the last question is no. $\endgroup$– Torsten EkedahlCommented Jun 26, 2011 at 16:15
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1$\begingroup$ In the case where $S$ is the minimal resolution of $T$, this is pretty classical stuff and it is related in some fascinating way to continuous fractions. You need of course to know how your group acts (there are several inequivalent actions of $\mathbb{Z}_n$ on $\mathbb{C}^2$). For a (by no means original) treatment, have a look at Sections $2$ and $3$ of my paper with E. Mistretta "Standard isotrivial fibrations with $p_g=q=1$ - II" (it is on the arXiv), in particular at Proposition 3.5. $\endgroup$– Francesco PolizziCommented Jun 26, 2011 at 16:25
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$\begingroup$ Dear Francesco. Thank you for your answer and also for introducing the paper. It was very helpful for me. $\endgroup$– CyrusCommented Jun 26, 2011 at 22:19
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