# Addition law on moduli space of curves

Dislaimer: I know very little about this, so if parts of my question don't make sense, please feel free to edit in a way that does, or ask me to clarify.

Let $\mathcal{M}_{1,2}$ be the moduli space of genus 1 curves (plane curves with base field $\mathbb{C}$) with 2 marked points. I think there is a map (thought of as addition in a sense like addition on an elliptic curve) that is \begin{align*} \mathcal{M}_{1,2} \times \mathcal{M}_{1,2} &\to \mathcal{M}_{1,1} \\ (C,p_1,p_2), (C,p_1,p_3) &\mapsto (C,p_2+p_3 - p_1) \end{align*} where $p_1$ is sort of an identity element. My question is: is this correct? If so, how should I think about it, and where can I read more about it?

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How would you add $(C,p_1,p_2)$ to $(C',p'_1,p'_2)$ where $C$ and $C'$ are nonisomorphic curves? –  Brad Nov 17 '12 at 0:27
I don't know but I think I only care about the case when $C= C'$, so assume we're just looking at this subset. In particular, the only curves I am looking at are smooth. –  Derek Allums Nov 17 '12 at 0:37
Well, there is a map $\mathcal{M}_{1,2} \times \mathcal{M}_{1,2}\to \overline{\mathcal{M}}_{1,2}$ where you map into a compactification of the moduli space and the map just glues the two curves at a marked point... –  Max Nov 17 '12 at 17:07
The compactification is for smooth curves. If your $\mathcal{M}$ already includes singular curves then you probably don't need that, but I'm not so familiar with that case. –  Max Nov 17 '12 at 17:10