Multiplicity Versus Buchsbaumness of the special fiber cone

Let $(A,\mathfrak m)$ be a Noetherian local ring of dimension $d>0$ with infinite residue field and $I$ an $\mathfrak{m}$-primary ideal. Let $\mathcal I$ be an $I$-good filtration. We study an equality of Hilbert coefficients, first given by Elias and Valla, versus passage of Buchsbaum property from the local ring to the blow-up algebras. Suppose $e_1(\mathcal I)-e_1(Q)=2e_0(\mathcal I)-2\ell(A/I_1)-\ell(I_1/(I_2+Q))$ where $Q\subseteq I$, a minimal reduction of $\mathcal I$, is a standard parameter ideal. Under some mild conditions, we prove that if $A$ is Buchsbaum (generalized Cohen-Macaulay respectively), then the associated graded ring $G(\mathcal I)$ is Buchsbaum (generalized Cohen-Macaulay respectively). Our results settle a question of Corso in general for an $I$-good filtration. Further, let $f_0(I)= e_1(I)-e_0(I)-e_1(Q)+\ell(A/I)+\mu(I)-d+1$ and $e_1(I)-e_1(Q)=2e_0(I)-2\ell(A/I)-\ell(I/(I^2+Q))$. We prove, under mild conditions, that (1) if $A$ is generalized Cohen-Macaulay, then the special fiber ring $F_{\mathfrak{m}}(I)$ is generalized Cohen-Macaulay; In addition, if depth of $A$ is positive, then depth of $F_{\mathfrak {m}}(I)$ is same as depth of $A$ and (2) if $A$ is Buchsbaum and depth A$\geq d-1$, then $F_{\mathfrak{m}}(I)$ is Buchsbaum and the $I$-invariant of $F_{\mathfrak{m}}(I)$ is same as that of $A$.