3

Let $X,X',X''$ be algebraic varieties and let $X''\to X'$ and $X'\to X$ be two ramified covers. Is the ramification divisor of the composition $X''\to X'\to X$ reducible?

Bonanza
  • 1,203

1 Answers1

4

Let $X=X'=X''=\Bbb{A}^1$ be the affine line, and let the two covers be $z \mapsto z^2$. Their composite is the (ramified) cover $z \mapsto z^4$, and the ramification divisor of this morphism is supported at $0$ only. This shows in particular that the ramification divisor can be irreducible.

Nils Matthes
  • 4,444
  • Ah, that's true. Is there some hypothesis I'm missing that will ensure that the ramification divisor is reducible though? – Bonanza May 19 '13 at 19:06
  • e.g., if the ramification divisor is reduced? – Bonanza May 19 '13 at 19:20
  • Dear @hilbert, I know of no condition which will ensure reducibility of the ramification divisor. However, since reducedness is a condition on the structure sheaf of the divisor, and irreducibility a purely topological notion, I don't believe the two are related by any means. – Nils Matthes May 19 '13 at 20:50
  • Ok, thanks for the answer! – Bonanza May 19 '13 at 21:26