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Cyrus
Cyrus
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Suppose that $T$ is a smooth complex projective algebraic surface, such that the finite cyclic group
  $Z_n$ acts$\mathbb{Z_n}$ acts on it. Also, suppose that $S$ is another complex projective algebraic surface (not necessarily smooth) such that $S$ is birational to $T/{Z}_{n}$$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 $ne(S)$?

Suppose that $T$ is a smooth complex projective algebraic surface, such that the finite cyclic group
  $Z_n$ acts on it. Also, suppose that $S$ is another complex projective algebraic surface (not necessarily smooth) such that $S$ is birational to $T/{Z}_{n}$. 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 $ne(S)$?

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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Cyrus
Cyrus
  • 395
  • 1
  • 11

Euler characteristic of cyclic quotients

Suppose that $T$ is a smooth complex projective algebraic surface, such that the finite cyclic group
$Z_n$ acts on it. Also, suppose that $S$ is another complex projective algebraic surface (not necessarily smooth) such that $S$ is birational to $T/{Z}_{n}$. 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 $ne(S)$?