The structure of the Sally module of integrally closed ideals

The first two Hilbert coefficients of a primary ideal play an important role in commutative algebra and in algebraic geometry. In this paper we give a complete algebraic structure of the Sally module of integrally closed ideals $I$ in a Cohen-Macaulay local ring $A$ satisfying the equality $\mathrm{e}_1(I)=\mathrm{e}_0(I)-\ell_A(A/I)+\ell_A(I^2/QI)+1,$ where $Q$ is a minimal reduction of $I$, and $\mathrm{e}_0(I)$ and $\mathrm{e}_1(I)$ denote the first two Hilbert coefficients of $I,$ respectively the multiplicity and the Chern number of $I$. This almost extremal value of $\mathrm{e}_1(I)$ with respect classical inequalities holds a complete description of the homological and the numerical invariants of the associated graded ring. Examples are given.