Singular Scheme Research Articles

Dimension two holomorphic distributions on four-dimensional projective space

We study two-dimensional holomorphic distributions on P4. We classify dimension two distributions, of degree at most 2, with either locally free tangent sheaf or locally free conormal sheaf and whose singular scheme has pure dimension one. We show that the corresponding sheaves are split. Next, we investigate the geometry of such distributions, studying from maximally non–integrable to integrable distributions. In the maximally non–integrable case, we show that the distribution is either of Lorentzian type or a push-forward by a rational map of the Cartan prolongation of a singular contact structure on a weighted projective 3-fold. We study distributions of dimension two in P4 whose conormal sheaves are the Horrocks–Mumford sheaves, describing the numerical invariants of their singular schemes which are smooth and connected. Such distributions are maximally non–integrable, uniquely determined by their singular schemes and invariant by a group H5⋊SL(2,Z5)⊂Sp(4,Q), where H5 is the Heisenberg group of level 5. We prove that the moduli spaces of Horrocks–Mumford distributions are irreducible quasi-projective varieties and we determine their dimensions. Finally, we observe that the space of codimension one distributions, of degree d≥6, on P4 has a family of degenerate flat holomorphic Riemannian metrics. Moreover, the degeneracy divisors of such metrics consist of codimension one distributions invariant by H5⋊SL(2,Z5) and singular along a degenerate abelian surface with (1,5)-polarization and level-5-structure.

Classification of the invariants of foliations by curves of low degree on the three-dimensional projective space

We study foliations by curves on the three-dimensional projective space with no isolated singularities, which is equivalent to assuming that the conormal sheaf is locally free. We provide a classification of the topological and algebraic invariants of the conormal sheaves and singular schemes for such foliations by curves, up to degree 3. In particular, we prove that foliations by curves of degree 1 or 2 are contained in a pencil of planes or are Legendrian, and are given by the complete intersection of two codimension one distributions. Furthermore, we prove that the conormal sheaf of a foliation by curves of degree 3 with reduced singular scheme either splits as a sum of line bundles or is an instanton bundle. For degree larger than 3, we focus on two classes of foliations by curves, namely Legendrian foliations and those whose conormal sheaf is a twisted null-correlation bundle. We give characterizations of such foliations, describe their singular schemes and their moduli spaces.

Foliations by curves on threefolds

AbstractWe study the conormal sheaves and singular schemes of one‐dimensional foliations on smooth projective varieties X of dimension 3 and Picard rank 1. We prove that if the singular scheme has dimension 0, then the conormal sheaf is μ‐stable whenever the tangent bundle is stable, and apply this fact to the characterization of certain irreducible components of the moduli space of rank 2 reflexive sheaves on and on a smooth quadric hypersurface . Finally, we give a classification of local complete intersection foliations, that is, foliations with locally free conormal sheaves, of degree 0 and 1 on Q3.

Non-torsion Brauer groups in positive characteristic

Unlike the classical Brauer group of a field, the Brauer-Grothendieck group of a singular scheme need not be torsion. We show that there exist integral normal projective surfaces over a large field of positive characteristic with non-torsion Brauer group. In contrast, we demonstrate that such examples cannot exist over the algebraic closure of a finite field.

Moduli of distributions via singular schemes

Let X be a smooth projective variety. We show that the map that sends a codimension one distribution on X to its singular scheme is a morphism from the moduli space of distributions into a Hilbert scheme. We describe its fibers and, when \(X = {\mathbb {P}^{n}}\), compute them via syzygies. As an application, we describe the moduli spaces of degree 1 distributions on \({\mathbb {P}^{3}}\). We also give the minimal graded free resolution for the ideal of the singular scheme of a generic distribution on \({\mathbb {P}^{3}}\).

Foliations with isolated singularities on Hirzebruch surfaces

Abstract We study foliations ℱ {\mathcal{F}} on Hirzebruch surfaces S δ {S_{\delta}} and prove that, similarly to those on the projective plane, any ℱ {\mathcal{F}} can be represented by a bi-homogeneous polynomial affine 1-form. In case ℱ {\mathcal{F}} has isolated singularities, we show that, for δ = 1 {\delta=1} , the singular scheme of ℱ {\mathcal{F}} does determine the foliation, with some exceptions that we describe, as is the case of foliations in the projective plane. For δ ≠ 1 {\delta\neq 1} , we prove that the singular scheme of ℱ {\mathcal{F}} does not determine the foliation. However, we prove that, in most cases, two foliations ℱ {\mathcal{F}} and ℱ ′ {\mathcal{F}^{\prime}} given by sections s and s ′ {s^{\prime}} have the same singular scheme if and only if s ′ = Φ ⁢ ( s ) {s^{\prime}=\Phi(s)} , for some global endomorphism Φ of the tangent bundle of S δ {S_{\delta}} .

IMPROVING A POSTERIORI SPATIAL DISCRETIZATION ERROR ESTIMATION FOR SN TRANSPORT BY SINGULAR CHARACTERISTICS TRACKING

Previously, we have developed a novel spatial discretization error estimator, the “residual source” estimator, in which an error transport problem, analogous to the discretized transport equation, is solved to acquire an estimate of the error, with a residual term acting as a fixed source. Like all error estimators, the residual source estimator suffers inaccuracy and imprecision in the proximity of singular characteristics, lines across which the solution is irregular. Estimator performance worsens as the irregularities become more pronounced, especially so if the true solution itself is discontinuous. This work introduces a modification to the residual approximation procedure that seeks to reduce the adverse effects of the singular characteristics on the error estimate. A partial singular characteristic tracking scheme is implemented to reduce the portion of the error in the numerical solution born by irregularities in the true solution. This treated numerical solution informs the residual approximations. The partial singular characteristic tracking scheme greatly enhances the numerical solution for a problem with prominent singular characteristics. The residual approximation and resultant residual source error estimate are likewise improved by the scheme, which only incurs the computational cost of an extra inner iteration.

Codimension one distributions and stable rank 2 reflexive sheaves on threefolds.

% We show that codimension one distributions with at most isolated singularities on certain smooth projective threefolds with Picard rank one have stable tangent sheaves. The ideas in the proof of this fact are then applied to the characterization of certain irreducible components of the moduli space of stable rank 2 reflexive sheaves on $\p3$, and to the construction of stable rank 2 reflexive sheaves with prescribed Chern classes on general threefolds. We also prove that if $\sG$ is a subfoliation of a codimension one distribution $\sF$ with isolated singularities, then $\sing(\sG)$ is a curve. As a consequence, we give a criterion to decide whether $\sG$ is globally given as the intersection of $\sF$ with another codimension one distribution. Turning our attention to codimension one distributions with non isolated singularities, we determine the number of connected components of the pure 1-dimensional component of the singular scheme.

Pelas redes da tradução: um estudo do conceito de transcriação, de Haroldo de Campos, no poema “Quisera no meu canto ser tão áspero”, de Dante Alighieri

O conceito de tradução aplicado por Haroldo de Campos é uma importante vertente da sua obra. Partindo da noção de Bense, ele defende que a identidade de um texto literário reside na articulação das redes de informação estética, enfatizando que cada texto possui uma ordenação singular de signos. Ao tradutor cabe ler e reescrever essa informação, com os olhos firmados na tradução entendida como crítica e criação (CAMPOS, 2006). Transgressiva, a perspectiva de tradução luciferina (CAMPOS, 2005, p. 180) baseia-se em um gesto de insubmissão ao conteúdo do texto de partida, pensando o exercício de tradução do signo em si. A ideia é tornar o texto de chegada em uma espécie de original, transformando o tradutor em “criador”. Aliás, para Campos, essa seria a única saída, principalmente, no caso da poesia. Esse artigo tem, pois, por objetivo, o estudo dessa vertente de tradução no poema “Quisera no meu canto ser tão áspero”, de Dante Alighieri, transcriação conduzida por Campos. A recriação da “lira pétrea” – esquema singular de rimas que culmina no isomorfismo: mulher/pedra – a melopeia posta a serviço da violência da linguagem etc., convidam-nos a refletir sobre o método de tradução de Campos, percebendo aqui o poeta que habita o fazer do tradutor.

Distance Estimation Scheme Exploiting IR-UWB Radar with Clutter Suppressing Algorithm in Indoor Environments

This paper proposes a distance estimation scheme with an impulse radio ultrawide-band (IR-UWB) radar for a long-range target in an indoor environment. When the pulse is transmitted from the IR-UWB radar and is reflected from the target, fading effects make it difficult to estimate the time-of-arrival of a direct path, which can be distorted by the clutter in indoor environments. The proposed scheme can improve the performance of distance estimation by mitigating the effect of the clutter with a low-rank approximation scheme and a singular value decomposition scheme. The performance of the proposed scheme is evaluated through experiments in a general office building, and the results of the experiment show that the proposed scheme can track the dynamic distance of a target from the radar up to 11 m in an open space and 14 m in a narrow corridor, respectively, with an average error of less than 0.2 m.

Extraction of Multimode S-Parameters by Using a Hybrid FDTD/SVD Technique

In this communication, a method for extracting multimode S-parameters in guided-wave structures is presented based on the finite-difference time-domain electromagnetic method in conjunction with the frequency domain (FD) singular value decomposition scheme. The present method can calculate the multimode S-parameters directly from the field distribution in the FD. It is found that the proposed modal analysis and multimode S-parameter extraction agree excellently with the analytic value and other available simulation techniques. Also, the method for multimode S-parameter extraction is validated by various microstrip discontinuities.

El tatuaje en el mundo contemporáneo: un recurso significante para la subjetividad individual

En este documento se realiza un ejercicio de aproximación al esquema singular que conduce a algunas personas hacia la práctica del tatuaje, para ello se elige un grupo de supuestos del psicoanálisis para ubicar los procesos que permiten que el tatuaje participe de la construcción de la subjetividad individual. La propuesta aquí es que el tatuaje puede llegar a ser un eslabón dentro de una cadena más amplia de significantes, con lo cual se convierte en un mecanismo que soporta una búsqueda individual en la que el tatuaje puede posibilitar el desplazamiento de una metáfora.

Accumulator Capacitor Converter for a Switched Reluctance Generator

In this paper, a novel converter is presented to improve the excitation process in switched reluctance generators (SRGs). Accumulator capacitor converter (ACC) is based on a singular scheme of flyback converters. ACC performs two main roles. First, in an initial step, it charges an accumulator capacitor with a determined value of voltage; then, during the magnetizing period, it discharges the stored energy into machine coils. Therefore, ACC provides the SRG with a determined magnetizing energy and uses the forced magnetizing phenomenon to magnetize its coils. The processes yield some outstanding superiority such as excellent operation at high speeds, simple controlling scheme, lower stress on the excitation source, etc., over conventional converters. To analyze the performance of ACC, the required analytical expressions for charging the capacitor and discharging it into the coils are developed and the constraints for switching periods are demonstrated in detail. Afterward, through MATLAB/Simulink and finite-element analysis (FEA) simulations, its operation is evaluated and compared to conventional asymmetric bridge and bifilar converters (BFCs); moreover, its test bed is assembled and tested experimentally, and its results, along with the simulation ones, validates ACC performance.

Foliations on $$\mathbb {CP}^2$$ CP 2 of degree d with a singular point with Milnor number $$d^2+d+1$$ d 2 + d + 1

We study holomorphic foliations on $$\mathbb {CP}^2$$ of degree d with a singular point with Milnor number $$d^2+d+1$$ and non-zero linear part. Given a foliation, using the singular scheme of the foliation and the lexicographical monomial order, we give necessary and sufficient conditions to have this kind of singularities. We show that if the singularity is nilpotent, the Grobner basis with respect to this order gives us the normal form around the singular point. Finally, we prove that these foliations have no invariant lines and we exhibit a family of foliations with a nilpotent singularity with Milnor number $$d^2+d+1$$ , for d odd.

Higher symmetries of powers of the Laplacian and rings of differential operators

We study the interplay between the minimal representations of the orthogonal Lie algebra $\mathfrak{g}=\mathfrak{so}(n+2,\mathbb{C})$ and the algebra of symmetries$\mathscr{S}(\Box ^{r})$ of powers of the Laplacian $\Box$ on $\mathbb{C}^{n}$. The connection is made through the construction of a highest-weight representation of $\mathfrak{g}$ via the ring of differential operators ${\mathcal{D}}(X)$ on the singular scheme $X=(\mathtt{F}^{r}=0)\subset \mathbb{C}^{n}$, for $\mathtt{F}=\sum _{j=1}^{n}X_{i}^{2}\in \mathbb{C}[X_{1},\ldots ,X_{n}]$. In particular, we prove that $U(\mathfrak{g})/K_{r}\cong \mathscr{S}(\Box ^{r})\cong {\mathcal{D}}(X)$ for a certain primitive ideal $K_{r}$. Interestingly, if (and only if) $n$ is even with $r\geqslant n/2$, then both $\mathscr{S}(\Box ^{r})$ and its natural module ${\mathcal{A}}=\mathbb{C}[\unicode[STIX]{x2202}/\unicode[STIX]{x2202}X_{n},\ldots ,\unicode[STIX]{x2202}/\unicode[STIX]{x2202}X_{n}]/(\Box ^{r})$ have a finite-dimensional factor. The same holds for the ${\mathcal{D}}(X)$-module ${\mathcal{O}}(X)$. We also study higher-dimensional analogues $M_{r}=\{x\in A:\Box ^{r}(x)=0\}$ of the module of harmonic elements in $A=\mathbb{C}[X_{1},\ldots ,X_{n}]$ and of the space of ‘harmonic densities’. In both cases we obtain a minimal $\mathfrak{g}$-representation that is closely related to the $\mathfrak{g}$-modules ${\mathcal{O}}(X)$ and ${\mathcal{A}}$. Essentially all these results have real analogues, with the Laplacian replaced by the d’Alembertian $\Box _{p}$ on the pseudo-Euclidean space $\mathbb{R}^{p,q}$ and with $\mathfrak{g}$ replaced by the real Lie algebra $\mathfrak{so}(p+1,q+1)$.

Foliations singular along a curve

A general one-dimensional foliation in the complex projective space has finitely many singularities. For an appropriately good family of subschemes in ℙ n , we study the loci in the space of foliations of degree d defined by the requirement that the singularities contain a member of the family. We give a formula for the dimensions of such loci. We show that their degrees are expressed by a polynomial in d. We compute it explicitly in a few examples. Next we provide a formula for the number of isolated singular points of a foliation containing a prescribed positive-dimensional subscheme in its singular scheme under mild assumptions. We include an appendix by Steven L. Kleiman on a theorem of Bertini suitable for sections of vector bundles with rank equal to the dimension of the base.

Image-based spacecraft pointing model using single-bank dual-band registration

The growing interest in the development of small satellites and the demand for high-resolution imaging has made the pointing and drift rate requirements of a satellite more stringent. To achieve high pointing accuracy, star sensors can be used, but their size and weight are too large for small satellites. The need for keeping the overall cost of the spacecraft down and still achieve adequate pointing accuracies has provoked the development of relatively inexpensive and high-performance attitude systems that can provide competitive pointing accuracies during imaging operations. In order to realize such a system, this research describes a novel approach for finding the attitude of a satellite at any arbitrary rotation by using inter-band offsets from a single multi-spectral imager (MSI). For Earth observational imagery, UK Disaster Monitoring Constellation Earth-pointing MSI is used. This research focuses on the potential use of a narrow angle between the bands of a pushbroom sensor for determining the attitude of a spacecraft. The technique investigated does not require ground control points or knowledge of any ground features, but rather estimates the orientation of the platform through analysis of perspective and timing-based distortions between images. These distortions are assumed translational and affine in nature, with two-dimensional shifts being extracted from imagery using a Singular Value Decomposition-based registration scheme. In order to better understand the effect of attitude on imagery, a model was developed for predicting inter-band shifts given an attitude. This was then used to estimate the shifts between imagery at nominal attitude and given a series of simulated manoeuvres. Several simulations have shown that the row and column offsets represented a straight line. Hence, we expressed the row and column shifts in terms of straight line parameters. These geometric attributes are then represented in terms of Euler axis/angle to find the mapping for the elements of the general rotation matrix. Once the mapping is computed for the elements of the rotation matrix, we used the standard equations to determine the angle of rotation and Euler axis. The accuracy of attitude estimates depends on the magnitude of angular separation between the cameras, orientation of spacecraft, sensor resolution, image texture, and image registration method. The technique proposed in this study may however be applied to any satellite with pushbroom sensors that have a discernable along track separation and sufficient overlap.

Rights-Based Constitutional Review in France

A well-developed system of constitutional review is a ‘must-have’ feature of liberal democracies aimed at realizing constitutional justice and human rights protection. In Europe, constitutional review mechanisms have now developed to be part of a multilevel system of rights protection involving the domestic judiciary, including the constitutional court, the Court of Justice of the European Union (CJEU) and the European Court of Human Rights (ECtHR). France is generally categorised in the European continental model of constitutional review of legislation operated by a specialised court. Yet, the French system has very much stood apart in this regard, on a tradition of Legicentrisme and a fear of government by judges. There was, as for other things, an ‘Exception Francaise’ featuring a system with, originally, no proper constitutional court and, up until recently, a weak form of constitutional review. The protection of rights offered a combination of ex ante abstract constitutional review of legislation operated by the Constitutional Council (CC) against the provisions of the Constitution, and the subsequently developed conventional review performed by ordinary judges against rights contained in international and European conventions, including the ECHR. This has changed with the introduction of ex post concrete constitutional review in 2008. The Priority Question on Constitutionality (PQC) is a new constitutional remedy which allows individuals to challenge laws already in force that infringe their constitutional rights and freedoms before ordinary courts in the course of proceedings. Subject to a filtering mechanism involving the two highest courts (the Council of State, the supreme administrative court, and the Court of Cassation, the supreme judicial court), the issue on constitutionality is decided by the CC under Article 61-1 of the 1958 Constitution: ‘If, during proceedings in progress before a court of law, it is claimed that a legislative provision infringes the rights and freedoms guaranteed by the Constitution, the matter may be referred by the Conseil d'Etat or by the Cour de Cassation to the Constitutional Council which shall rule within a determined period’. Entered into force in March 2010, its three years of operations reveal a ‘French legal success story’. With nearly 280 PQC decisions rendered by the CC, the new review has solidly and rapidly established itself in domestic constitutional law. It has also been a qualitative success. The PQC has affected all areas of law, with the most significant activity in tax and criminal law. Significant rulings have enhanced the protection of rights and provoked changes in the legislation on, for examples, police custody, involuntary confinement and sexual harassment. The introduction of ex post review has profoundly modified the tenets of constitutional review. It represents a landmark in further development of constitutionalisation and in the achievement of constitutionalism in France. It has helped to ingrain the Constitution politically, civically and, of course, judicially. The French system of rights-based constitutional review is heading towards completeness and convergence with the dominant European model of constitutional review addressing the requirements of modern democracy. Yet, the PQC had to ‘fit in’ an already settled and complex domestic courts system and, as outlined above, a singular scheme of rights adjudication involving review of constitutionality and review of conventionality. Various issues arose at drafting stage and in the first years of operation posing interesting challenges in terms of constitutional justice and governance. The study is divided into two main parts about the evolution of the model of constitutional review, on one hand, and the practice and endurance of the new form of constitutional review, on the other hand. The first part aims at explaining the original form of constitutional review and its development with emphasis on the PQC. The second part examines specific issues raised by the new mechanism from an institutional perspective, in particular the changing role of the CC and its relationship with other courts, domestic and European, as well as with the legislator. The aim of the study is to provide a better understanding of the French system of right-based constitutional review in light of most recent developments The methodology adopted is not comparative per se, although specific comparisons will be made with relevant European systems. Also, the approach will focus on institutional aspects rather than on the substance of rights.

On the singular scheme of split foliations

We prove that the tangent sheaf of a codimension one locally free distribution splits as a sum of line bundles if and only if its singular scheme is arithmetically Cohen-Macaulay. In addition, we show that a foliation by curves is given by an intersection of generically transversal holomorphic distributions of codimension one if and only if its singular scheme is arithmetically Buchsbaum. Finally, we establish that these foliations are determined by their singular schemes, and deduce that the Hilbert scheme of certain arithmetically Buchsbaum schemes of codimension $2$ is birational to a Grassmannian.

On Milnor classes via invariants of singular subschemes

We derive a formula for the Milnor class of certain scheme-theoretic global complete intersections (with arbitrary singularities) in a smooth variety in terms of the Segre class of its singular scheme. In codimension one the formula recovers a formula of Aluffi for the Milnor class of a hypersurface.