The scheme of liftings and applications

We study the locus of the liftings of a homogeneous ideal $H$ in a polynomial ring over any field. We prove that this locus can be endowed with a structure of scheme $\mathrm L_H$ by applying the constructive methods of Gr\"obner bases, for any given term order. Indeed, this structure does not depend on the term order, since it can be defined as the scheme representing the functor of liftings of $H$. We also provide an explicit isomorphism between the schemes corresponding to two different term orders. Our approach allows to embed $\mathrm L_H$ in a Hilbert scheme as a locally closed subscheme, and, over an infinite field, leads to find interesting topological properties, as for instance that $\mathrm L_H$ is connected and that its locus of radical liftings is open. Moreover, we show that every ideal defining an arithmetically Cohen-Macaulay scheme of codimension two has a radical lifting, giving in particular an answer to an open question posed by L. G. Roberts in 1989.Comment: the presentation of the results has been improved, new section (Section 6 of this version) concerning the torus action on the scheme of liftings, more detailed proofs in Section 7 of this version (Section 6 in the previous version), new example added (Example 8.5 of this version

Upgraded methods for the effective computation of marked schemes on a strongly stable ideal

Let $J\subset S=K[x_0,...,x_n]$ be a monomial strongly stable ideal. The collection \Mf(J) of the homogeneous polynomial ideals $I$, such that the monomials outside $J$ form a $K$-vector basis of $S/I$, is called a {\em $J$-marked family}. It can be endowed with a structure of affine scheme, called a {\em $J$-marked scheme}. For special ideals $J$, $J$-marked schemes provide an open cover of the Hilbert scheme \hilbp, where $p(t)$ is the Hilbert polynomial of $S/J$. Those ideals more suitable to this aim are the $m$-truncation ideals $\underline{J}_{\geq m}$ generated by the monomials of degree $\geq m$ in a saturated strongly stable monomial ideal $\underline{J}$. Exploiting a characterization of the ideals in \Mf(\underline{J}_{\geq m}) in terms of a Buchberger-like criterion, we compute the equations defining the $\underline{J}_{\geq m}$-marked scheme by a new reduction relation, called {\em superminimal reduction}, and obtain an embedding of \Mf(\underline{J}_{\geq m}) in an affine space of low dimension. In this setting, explicit computations are achievable in many non-trivial cases. Moreover, for every $m$, we give a closed embedding \phi_m: \Mf(\underline{J}_{\geq m})\hookrightarrow \Mf(\underline{J}_{\geq m+1}), characterize those $\phi_m$ that are isomorphisms in terms of the monomial basis of $\underline{J}$, especially we characterize the minimum integer $m_0$ such that $\phi_m$ is an isomorphism for every $m\geq m_0$.Comment: 28 pages; this paper contains and extends the second part of the paper posed at arXiv:0909.2184v2[math.AG]; sections are now reorganized and the general presentation of the paper is improved. Final version accepted for publicatio

Segments and Hilbert schemes of points

Using results obtained from the study of homogeneous ideals sharing the same initial ideal with respect to some term order, we prove the singularity of the point corresponding to a segment ideal with respect to the revlex term order in the Hilbert scheme of points in $\mathbb{P}^n$. In this context, we look inside properties of several types of "segment" ideals that we define and compare. This study led us to focus our attention also to connections between the shape of generators of Borel ideals and the related Hilbert polynomial, providing an algorithm for computing all saturated Borel ideals with the given Hilbert polynomial.Comment: 19 pages, 2 figures. Comments and suggestions are welcome

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 $m_p(z)^{\varrho}$ 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

A Review on Montmorillonite-Based Nanoantimicrobials: State of the Art

One of the crucial challenges of our time is to effectively use metal and metal oxide nanoparticles (NPs) as an alternative way to combat drug-resistant infections. Metal and metal oxide NPs such as Ag, Ag2O, Cu, Cu2O, CuO, and ZnO have found their way against antimicrobial resistance. However, they also suffer from several limitations ranging from toxicity issues to resistance mechanisms by complex structures of bacterial communities, so-called biofilms. In this regard, scientists are urgently looking for convenient approaches to develop heterostructure synergistic nanocomposites which could overcome toxicity issues, enhance antimicrobial activity, improve thermal and mechanical stability, and increase shelf life. These nanocomposites provide a controlled release of bioactive substances into the surrounding medium, are cost effective, reproducible, and scalable for real life applications such as food additives, nanoantimicrobial coating in food technology, food preservation, optical limiters, the bio medical field, and wastewater treatment application. Naturally abundant and non-toxic Montmorillonite (MMT) is a novel support to accommodate NPs, due to its negative surface charge and control release of NPs and ions. At the time of this review, around 250 articles have been published focusing on the incorporation of Ag-, Cu-, and ZnO-based NPs into MMT support and thus furthering their introduction into polymer matrix composites dominantly used for antimicrobial application. Therefore, it is highly relevant to report a comprehensive review of Ag-, Cu-, and ZnO-modified MMT. This review provides a comprehensive overview of MMT-based nanoantimicrobials, particularly dealing with preparation methods, materials characterization, and mechanisms of action, antimicrobial activity on different bacterial strains, real life applications, and environmental and toxicity issues

A computational and experimental study inside microfluidic systems: the role of shear stress and flow recirculation in cell docking

Abstract In this paper, microfluidic devices containing microwells that enabled cell docking were investigated. We theoretically assessed the effect of geometry on recirculation areas and wall shear stress patterns within microwells and studied the relationship between the computational predictions and experimental cell docking. We used microchannels with 150 μm diameter microwells that had either 20 or 80 μm thickness. Flow within 80 μm deep microwells was subject to extensive recirculation areas and low shear stresses (<0.5 mPa) near the well base; whilst these were only presented within a 10 μm peripheral ring in 20 μm thick microwells. We also experimentally demonstrated that cell docking was significantly higher (p<0.01) in 80 μm thick microwells as compared to 20 μm thick microwells. Finally, a computational tool which correlated physical and geometrical parameters of microwells with their fluid dynamic environment was developed and was also experimentally confirmed