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The Abel differential equation $y'=p(x)y^3 + q(x) y^2$ with polynomialcoefficients $p,q$ is said to have a center on $[a,b]$ if all its solutions,with the initial value $y(a)$ small enough, satisfy the condition $y(a)=y(b)$.The problem of giving conditions on $(p,q,a,b)$ implying a center for the Abelequation is analogous to the classical Poincar\'e Center-Focus problem forplane vector fields. Center conditions are provided by an infinite system of"Center Equations". An important new information on these equations has beenobtained via a detailed analysis of two related structures: Composition Algebraand Moment Equations (first order approximation of the Center ones). Recentlyone of the basic open questions in this direction - the "Polynomial momentsproblem" - has been completely settled in \cite{mp1,pak}.  In this paper we present a progress in the following two main directions:First, we translate the results of \cite{mp1,pak} into the language ofAlgebraic Geometry of the Center Equations. On this base we obtain newinformation on the center conditions, significantly extending, in particular,the results of \cite{broy}. Second, we study the "second Melnikov coefficients"(second order approximation of the Center equations) showing that in many casesvanishing of the moments and of these coefficients is sufficient in order tocompletely characterize centers.

Algebraic Geometry of the Center-Focus problem for Abel Differential Equation