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

The Abel differential equation $y'=p(x)y^3 + q(x) y^2$ with polynomial coefficients $p,q$ 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 Abel equation is analogous to the classical Poincar\'e Center-Focus problem for plane vector fields. Center conditions are provided by an infinite system of "Center Equations". An important new information on these equations has been obtained via a detailed analysis of two related structures: Composition Algebra and Moment Equations (first order approximation of the Center ones). Recently one of the basic open questions in this direction - the "Polynomial moments problem" - 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 of Algebraic Geometry of the Center Equations. On this base we obtain new information 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 cases vanishing of the moments and of these coefficients is sufficient in order to completely characterize centers.