MINIMAL CASTELNUOVO-MUMFORD REGULARITY FOR A GIVEN HILBERT POLYNOMIAL

Let K be an algebraically closed field of null characteristic and p(z) a Hilbert polynomial. We look for the minimal Castelnuovo-Mumford regularity m_p(z) of closedsubschemes of projective spaces over K with Hilbert polynomial p(z). Experimental evidences led us to consider the idea that mp(z) could be achieved by schemes having a suitable minimal Hilbert function. We give a constructive proof of this fact. Moreover, we are able to compute the minimal Castelnuovo-Mumford regularity m_p(z) of schemeswith Hilbert polynomial p(z) and given regularity % of the Hilbert function, and also the minimal Castelnuovo-Mumford regularity mu of schemes with Hilbert function u. These results find applications in the study of Hilbert schemes. They are obtainedby means of minimal Hilbert functions and of two new constructive methods which are based on the notion of growth-height-lexicographic Borel set and called ideal graft and extended lifting.