Koszul homology and extremal properties of Gin and Lex

For every homogeneous ideal $I$ in a polynomial ring $R$ and for every $p\leq\dim R$ we consider the Koszul homology $H_i(p,R/I)$ with respect to a sequence of $p$ of generic linear forms. The Koszul-Betti number $\beta_{ijp}(R/I)$ is, by definition, the dimension of the degree $j$ part of $H_i(p,R/I)$. In characteristic $0$, we show that the Koszul-Betti numbers of any ideal $I$ are bounded above by those of the gin-revlex $\mathrm{Gin}(I)$ of $I$ and also by those of the Lex-segment $\mathrm{Lex}(I)$ of $I$. We show that $\beta_{ijp}(R/I)=\beta_{ijp}(R/\mathrm{Gin}(I))$ iff $I$ is componentwise linear and that and $\beta_{ijp}(R/I)=\beta_{ijp}(R/\mathrm{Lex}(I))$ iff $I$is Gotzmann. We also investigate the set $\mathrm{Gins}(I)$ of all the gin of $I$ and show that the Koszul-Betti numbers of any ideal in $\mathrm{Gins}(I)$are bounded below by those of the gin-revlex of $I$. On the other hand, we present examples showing that in general there is no $J$ is $\mathrm{Gins}(I)$such that the Koszul-Betti numbers of any ideal in $\mathrm{Gins}(I)$ are bounded above by those of $J$.