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curves which Curves whose stable reduction dose reductions do not contain rational curves

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

show/hide this revision's text 1

curves which stable reduction dose not contain rational curves

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 is the conditions of $X$ that the $\cal X_{s}$ dose not contain any rational curves?