What are the moduli space of elliptic curves and its compactification? We present an example of plane cubic curves in order to explain the problems. A basic idea of compactification is stability of Mumford. What is stability and why stability?
Stability canonically compactifies the moduli space of abelian varieties over Z[ςN, 1/N] in arbitrary dimension for some N≥3. In the smallest possible case, dimension one and N=3, the problem is reduced to the study of plane cubic curves, more specifically Hesse cubic curves. Thus Hesse cubic curves revive from stability of Mumford.

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