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## Genera of irreducibles in the special fiber vs. genus of the generic fiber

Is the following statement true (and if not, what is?):

If $R$ is a complete DVR, and $K$ is its quotient field, and $C$ is a relative curve over $R$, then the sum of the arithmetic genera of the irreducible components of the special fiber ($C \times_R R/m$) equals the arithmetic genus of $C \times_R K$?

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This is not true: a degeneration of a smooth plane curve into a union of lines is already enough to see this! What is true is that the arithmetic genus of the whole special fiber is the same as the arithmetic genus of the generic fiber. You can also give explicit conditions on the contributions of specific types of singularities in the special fiber to the global arithmetic genus, but you need to take into account "loops in the dual graph" as well (as the example of plane curves above shows), and of course you need to be careful with non-reduced components. – damiano Jul 20 2010 at 18:09