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I'd like to get my hands on some simple examples of families of stable curves. Ideally these would come in the form of a projective curve $C$ over a 1 dimensional base $B$, say $B = \mathbb{A}^1$. The generic fiber would be smooth and the special fiber would be nodal.

For genus 0 there is the nice example of the closure in $\mathbb{P}^2_{\mathbb{A}^1}$ of Spec $k[x,y,t]/(xy - t)$ where $t$ is the coordinate on $\mathbb{A}^1$.

For genus 1 there is the closure in $\mathbb{P}^2_{\mathbb{A}^1}$ of Spec $k[x,y,t]/(y^2 = x(x-t)(x+1))$.

I know for genus $2$ you can't expect to have an example in $\mathbb{P}^2_{\mathbb{A}^1}$ but what about something in $\mathbb{P}^3_{\mathbb{A}^1}$?

Whenever $g = (d-1)(d-2)/2$ I would think you can get an example in $\mathbb{P}^2_{\mathbb{A}^1}$, is there anyway to control the number of nodes you get in the special fiber?

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    The fibre over $t=0$ of your curve given by $xy=t$ is the curve $xy=0$. Is this really a stable curve in the usual sense? It seems to consist of two rational lines. I thought stable curves had the property that each rational curve intersects at least three other components. Maybe I'm wrong.2012-07-21
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    You are correct but this is fixed just by adding marked points. I'm not too concerned with the marked points so I didn't mention it but your point is absolutely right.2012-07-21

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