PLANE CUBIC CURVES

FROM HESSE TO MUMFORD

Abstract

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.