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Let $R$ be a discrete valuation ring with fraction field $K$ of characteristic zero and residue field $k$ of characteristic $p>0$. Assume $k$ is algebraically closed. I want to produce examples of flat families of projective curves $\pi:X \to \mbox{Spec}(R)$ such that the generic fiber is smooth of genus at least $2$ and the special fiber $X_k$ is a semi-stable tree-like curve i.e., $X_k$ is a reduced, connected curve with at most nodal singularities such that any rational component of $X_k$ intersects the complement in at least $2$ points and the dual graph of $X_k$ does not have any loops.

I do not have any restriction on the choice of $p$ or $R$. I also can produce such examples in the case I assume $X_k$ is smooth. But, I want to get some non-trivial examples i.e., when the special fiber is not smooth and reducible. Can somebody give such examples and is there some straightforward way to produce such examples?

Any reference/idea for such examples will be very helpful.

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  • $\begingroup$ @abx: That is ok but I prefer reducible curves. Using the theory given in Qing Liu, I have some idea of producing examples when the special fiber is irreducible. The reason for my obsession with reducible fibers is because I am finding it harder to determine when will certain reducible curves deform to a smooth generic fiber. I edited the question accordingly. $\endgroup$
    – user45397
    Nov 2, 2016 at 11:23
  • $\begingroup$ Take the plane curve $\pi F+ G=0$ in $\mathbb{P}^2_R$, where $\pi $ is a uniformizing parameter of $R$, $G=0$ is your favorite stable plane curve, and $F=0$ is a smooth plane curve. $\endgroup$
    – abx
    Nov 2, 2016 at 11:29
  • $\begingroup$ @abx: This has been my approach. But, I am not able to find a good candidate for $G$ in my case since the curve defining $G$ must be also tree-like. Infact (if I am not wrong) for any choice of $G$, I will never get a tree-like curve, mainly due to Bezout's theorem. Am I wrong? $\endgroup$
    – user45397
    Nov 2, 2016 at 11:34
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    $\begingroup$ See Corollary B.3 in arxiv.org/pdf/math/0701075.pdf $\endgroup$
    – nfdc23
    Nov 2, 2016 at 11:37
  • $\begingroup$ @nfdc23 I think this answers my question. Really nice result. Thanks. Did not know of this article. $\endgroup$
    – user45397
    Nov 2, 2016 at 11:40

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