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Let $X$ be a smooth projective curve over $K:=K(A)$. $A$ is a strict henselian ring, $A/m=k=\bar k$. Suppose $\cal X$ is a stable model of $X$, ${\cal X}_{s}$ is the special fiber.

My question is:

what are the conditions on $X$ so that $\cal X_{s}$ does not contain any rational curves?

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Kiseki, could you specify a bit more what you are looking for? For instance, are you looking for conditions expressed in terms of the monodromy action of the Galois group of $K$ on the first $l$-adic cohomology group of $X$? –  Jason Starr Jul 11 '12 at 14:34
1  
By the adjunction formula, a rational curve $E$ on $\mathcal X$ will have self-intersection $-E^2 = -2 - K\cdot E$, where $K$ is a canonical divisor of $\mathcal X /O_K$. So a necessary and sufficient condition to have no rational curves is that there aren't any vertical divisors $E$ such that $E^2 = -2 - (K,E)$. –  Harry Jul 15 '12 at 22:14

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