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 Gin(I) of I and also by those of the Lex-segment Lex(I) of I. We show that \beta_{ijp}(R/I) = \beta_{ijp}(R/Gin(I)) iff I is componentwise linear and that and \beta_{ijp}(R/I) = \beta_{ijp}(R/Lex(I)) iff I is Gotzmann. We also investigate the set Gins(I) of all the gin of I and show that the Koszul-Betti numbers of any ideal in 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 Gins(I) such that the Koszul-Betti numbers of any ideal in Gins(I) are bounded above by those of J.