Let $K$ be an algebraically closed field of null characteristic and $p(z)$ a Hilbert polynomial. We look for the minimal Castelnuovo-Mumford regularity $\minReg{p(z)}$ 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 {\em minimal Hilbert function}. We give a constructive proof of this fact. Moreover, we are able to compute the minimal Castelnuovo-Mumford regularity $\minRho{p(z)}{\varrho}$ of schemes with Hilbert polynomial $p(z)$ and given regularity $\varrho$ 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 \emph{minimal Hilbert functions} and of two new constructive methods which are based on the notion of growth-height-lexicographic Borel set and called \emph{ideal graft} and \emph{extended lifting}.
Minimal Castelnuovo-Mumford regularity for a given Hilbert polynomial / Cioffi, Francesca; P., Lella; M. G., Marinari; M., Roggero. - In: EXPERIMENTAL MATHEMATICS. - ISSN 1058-6458. - 24:4(2015), pp. 424-437. [10.1080/10586458.2015.1020577]
Minimal Castelnuovo-Mumford regularity for a given Hilbert polynomial
CIOFFI, FRANCESCA;
2015
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 $\minReg{p(z)}$ 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 {\em minimal Hilbert function}. We give a constructive proof of this fact. Moreover, we are able to compute the minimal Castelnuovo-Mumford regularity $\minRho{p(z)}{\varrho}$ of schemes with Hilbert polynomial $p(z)$ and given regularity $\varrho$ 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 \emph{minimal Hilbert functions} and of two new constructive methods which are based on the notion of growth-height-lexicographic Borel set and called \emph{ideal graft} and \emph{extended lifting}.File | Dimensione | Formato | |
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