Minimal Castelnuovo–Mumford Regularity for a Given Hilbert Polynomial

Abstract

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 closed subschemes of projective spaces over K with Hilbert polynomial p(z). Experimental evidences led us to consider the idea that m_p(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^ϱ_p(z) of schemes with Hilbert polynomial p(z) and given regularity ϱ of the Hilbert function, and also the minimal Castelnuovo–Mumford regularity m_u of schemes with Hilbert function u. These results find applications in the study of Hilbert schemes. They are obtained by 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.

2000 AMS Subject Classification::

• 14Q99
• 68W30
• 11Y55

Keywords:

• Borel ideal
• closed projective subscheme
• Castelnuovo–Mumford regularity
• Hilbert polynomial
• minimal function
• regularity of a Hilbert function

FUNDING

The first and fourth authors are members of GNSAGA and were partially supported by PRIN 2010-11 Geometria delle varietà algebriche, cofinanced by MIUR (Italy). The second author was partially supported by PRIN 2010-11 Geometria delle varietà algebriche, cofinanced by MIUR (Italy), by FIRB 2012 Moduli spaces and Applications and by GNSAGA.