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
] 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.