# Semi-stable model and Neron model for family of elliptic curves

I am looking for an "easy-to-understand" reference for Neron Models. Specifically if I have a semi-stable family of elliptic curves over $Spec {O}_K$ , with generic fibre $E_K$ and special fibre $E_k$ , then $E_k$ is an $N$-gon of $\mathbb{P}^1$'s. In this context, what is the Neron model of $E_K$? I guess what I am asking is for a geometric description of the special fibre of the Neron model for $E_K$.

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Neron model is always the smooth locus of minimal regular proper model; these matters are discussed (with references) in the later part of Q. Liu's book "Algebraic geometry and arithmetic curves". The sst model you have above may be non-regular, for example. If you begin with a regular sst proper flat model and make a ramified base change then regularity is generally lost but is regained by blow-ups; that's why even in mult. reduction case the structure of the component group of geometric special fiber is affected by ramified base change even though relative identity component is not. – BCnrd May 17 '10 at 15:31
By the way, the "N-gon" description of the special fiber is not quite right; that is only for the geometric special fiber. – BCnrd May 18 '10 at 5:40

Thus in your case the connected component is a rational curve with two points removed: as a group it is $\mathbb{G}_m$, the multiplicative group. The component group here is cyclic of order $N$.