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 $m_p(z)^{\varrho}$ 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 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.Comment: 21 pages. Comments are welcome. More concise version with a slight
  change in the title. A further revised version has been accepted for
  publication in Experimental Mathematic

Cioffi, Francesca

Lella, Paolo

Marinari, M. Grazia

Roggero, Margherita

English

arXiv

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}

CIOFFI, FRANCESCA

P. Lella

M. G. Marinari

M. Roggero

Archivio della ricerca - Università degli studi di Napoli Federico II

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 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 $m_p(z)$ of schemes with Hilbert polynomial $p(z)$ and given regularity 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

Minimal Castelnuovo-Mumford regularity for a given Hilbert polynomial

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