On the Jacobians of plane cubics

Let S be a scheme and f a ternary cubic form whose ten coefficients are sections of OS without common zero. The equation f=0 defines a family of plane cubic curves X⊂PS2→S parametrized by S. We prove that the family of generalized Jacobians of those cubic curves is a group scheme J/S which is the locus of smoothness of a scheme f*=0, where f* is a Weierstrass cubic formf*=f*(x,y,z)=y2z+a1xyz+a2yz2-x3-a2x2z-a4xz2-a6z3, in which the coefficient ai is a homogeneous polynomial with integral coefficients, of degree i in the ten coefficients of f, which we give explicitly. A key ingredint of the proof is a characterization, over sufficiently nice bases, of group algebraic spaces which can be described by such a Weierstrass equation.