Generic Fiber Research Articles

Algebraic Complete Integrability of the $a_4^{(2)}$ Toda Lattice

The aim of this work is focused on the investigation of the algebraic complete integrability of the Toda lattice associated with the twisted affine Lie algebra $a_4^{(2)}$. First, we prove that the generic fiber of the momentum map for this system is an affine part of an abelian surface. Second, we show that the flows of integrable vector fields on this surface are linear. Finally, using the formal Laurent solutions of the system, we provide a detailed geometric description of these abelian surfaces and the divisor at infinity.

A miniaturized microfluidic nanoplasmonic sensor with cavity reflection enhancement for ultrasensitive molecular interaction analysis

Molecular binding kinetics quantification is pivotal in the design of antibodies, small-molecule drugs, and early diagnosis of complex disease. This study presents a portable microfluidic system, featuring a nanoplasmonic sensor with cavity reflection enhancement (NSCRE), designed for the ultrasensitive detection and analysis of molecular interactions. This device comprises liquid flow paths, generic fiber spectrometers, a versatile NSCRE sensor chip, and a novel data analysis system. Furthermore, a neural network data processing system is developed based on a deep learning model to decrease significant data fluctuations caused by air segments. This “all-in-one” platform, integrating multifunctional, easy-to-use, and renewable NSCRE biosensors, boasts detection capabilities for both slow and rapid kinetics. It can automatically monitor interactions between molecules across a broad range of molecular weights (0.259–150 kDa), allowing for real-time, ultra-sensitive, and highly selective analysis. In addition, the platform with CRE effect facilitates a 7.2-fold increase in reflection intensity signals compared to that of a sensor without CRE. The immunoglobulin G detection with a limit of detection of 73 pM based on this device, showing a 50-fold increase in sensitivity compared to previously reported values. The miniaturized automatic detection system paves a new path for the use of molecular interaction analysis in life science research, drug discovery, early diagnosis, and other fields.

On regular but non-smooth integral curves

Let C be a regular geometrically integral curve over an imperfect field K and assume that it admits a non-smooth point p which — seen as a prime of the separable function field K(C)|K — is non-decomposed in the base field extension K‾⊗KK(C)|K‾. In this paper we establish a bound for the number of iterated Frobenius pullbacks needed in order to transform p into a rational point. This provides an algorithm to compute geometric δ-invariants of non-smooth points and a procedure to construct fibrations with moving singularities of prescribed δ-invariants. We show that the bound is sharp in characteristic 2. We further study the geometry of a pencil of plane projective rational quartics in characteristic 2 whose generic fibre attains our bound. On our way, we prove several results on separable and non-decomposed points that might be of independent interest.

On subvarieties of degenerations of Fano varieties

The goal of this work is to study geometric properties of geometrically irreducible subschemes on degenerations of Fano varieties (more generally, of separably rationally connected varieties). It is known that these geometrically irreducible subschemes exist when the ground field has characteristic zero or contains an algebraically closed subfield. We show that the dimension of this geometrically irreducible subscheme has a lower bound by the Fano index of the generic fiber.

A conjecture of Flach and Morin

A conjecture recently stated by Flach and Morin relates the action of the monodromy on the Galois invariant part of the p-adic Beilinson–Hyodo–Kato cohomology of the generic fiber of a scheme defined over a DVR of mixed characteristic to (the cohomology of) its special fiber. We prove the conjecture in the case that the special fiber of the given arithmetic scheme is also a fiber of a geometric family over a curve in positive characteristic.

Riemannian Maps From Kaehler Manifold With Generic Fibers

We study Riemannian maps from almost Hermitian manifolds to Riemannian manifolds for the case when the fibers are genericsubmanifold of the total space. We obtain the integrability conditionsfor the distributions while vertical distribution is always integrable. Wealso study the geometry of the leaves of the distribution which arisefrom such maps, and obtain the necessary and sufficient conditions forthe fibers as well as the total manifold to be generic product manifolds. We, further, obtain the necessary and sufficient condition for aRiemannian maps with generic fibers to be totally geodesic map.

On the Brauer groups of fibrations

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Completed prismatic F-crystals and crystalline Zp-local systems

We introduce the notion of completed $F$-crystals on the absolute prismatic site of a smooth $p$-adic formal scheme. We define a functor from the category of completed prismatic $F$-crystals to that of crystalline étale $\mathbf {Z}_p$-local systems on the generic fiber of the formal scheme and show that it gives an equivalence of categories. This generalizes the work of Bhatt and Scholze, which treats the case of a mixed characteristic complete discrete valuation ring with perfect residue field.

An explicit deformation of a plane branch with constant δ-invariant

We construct an explicit deformation of a plane branch with constant δ-invariant, where δ denotes the double point number of the singularity. The generic fibers of the deformation are the union of two irreducible curves. Our construction works over algebraically closed fields of arbitrary characteristic.

Mixed Hodge Structures on Alexander Modules

Motivated by the limit mixed Hodge structure on the Milnor fiber of a hypersurface singularity germ, we construct a natural mixed Hodge structure on the torsion part of the Alexander modules of a smooth connected complex algebraic variety. More precisely, let U U be a smooth connected complex algebraic variety and let f : U → C ∗ f\colon U\to \mathbb {C}^* be an algebraic map inducing an epimorphism in fundamental groups. The pullback of the universal cover of C ∗ \mathbb {C}^* by f f gives rise to an infinite cyclic cover U f U^f of U U . The action of the deck group Z \mathbb {Z} on U f U^f induces a Q [ t ± 1 ] \mathbb {Q}[t^{\pm 1}] -module structure on H ∗ ( U f ; Q ) H_*(U^f;\mathbb {Q}) . We show that the torsion parts A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) of the Alexander modules H ∗ ( U f ; Q ) H_*(U^f;\mathbb {Q}) carry canonical Q \mathbb {Q} -mixed Hodge structures. We also prove that the covering map U f → U U^f \to U induces a mixed Hodge structure morphism on the torsion parts of the Alexander modules. As applications, we investigate the semisimplicity of A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) , as well as possible weights of the constructed mixed Hodge structures. Finally, in the case when f : U → C ∗ f\colon U\to \mathbb {C}^* is proper, we prove the semisimplicity and purity of A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) , and we compare our mixed Hodge structure on A ∗ ( U f ; Q ) A_*(U^f;\mathbb {Q}) with the limit mixed Hodge structure on the generic fiber of f f .

A note on some $p$-adic analytic Hecke actions

We show that the action of Hecke operators away from p on the space of ( p -adic) overconvergent modular forms is ( p -adically) locally analytic in a certain sense. As a corollary, the action of the Hecke algebra can be extended naturally to an action of rigid functions on its generic fiber. This directly determines the Hodge–Tate–Sen weights of Galois representation associated to an overconvergent eigenform and confirms a conjecture of Gouvêa.

On integral class field theory for varieties over p-adic fields

Let K be a finite extension of the p-adic numbers Qp with ring of integers OK and residue field κ. Let X a regular scheme, proper, flat, and geometrically irreducible over OK of dimension d, and XK its generic fiber. We show, under some assumptions on X, that there is a reciprocity isomorphism of locally compact groups Har2d−1(XK,Z(d))≃π1ab(XK)W from the cohomology theory defined in [10] to an integral model π1ab(XK)W of the abelianized fundamental group π1ab(XK). After removing the contribution from the base field, the map becomes an isomorphism of finitely generated abelian groups. The key ingredient is the duality result in [10].

Fibres in the nasal cavity: A pilot study of the recovery, background, and transfer in smothering scenarios

In cases where the suspected cause of death is smothering, fibre traces recovered from the nasal cavity are hypothesised to refute or support this proposition. In order to carry out such evaluations, an efficient recovery method must first be established. This pilot study tested five different recovery methods on 3D printed models of nasal cavities. Among which, the use of the transparent AccuTrans® polyvinyl Siloxane casts demonstrated the best recovery efficiency with a median of 90% of deposited fibres recovered. The efficacy of this method was then verified on cadavers. Apart from a reliable recovery method, an understanding of the background population of fibres in nasal cavities, as well as the mechanisms of the transfer from the purported smothering textile to the nasal cavity is essential to evaluate the findings in these cases of suspected smothering. Samplings of the nasal cavities of 20 cadavers were thus carried out to gather data on the background population of fibres. Results showed that nasal cavities are not void of fibres, but the quantities are expected to be low, with a mean of 3.8 fibres per cavity recovered. Information on generic fibre class, colour, and length of these background fibres were also obtained with the use of low and high-power microscopy. The frequencies found in this population of fibres closely align with data from other population studies where black cotton was the most common. Finally, transfer experiments using the 3D printed models fitted with a respiratory pump to simulate breathing were carried out, along with testing on live volunteers in-vivo. The results demonstrated a verifiable transfer of fibres into the nasal cavity in smothering scenarios. Textiles of various shedding capacities were used in these tests and the findings suggest an influence of this variable on the quantities of fibres transferred.

Development of a Handheld Optical Fusion Splicer with a Wing Sleeve Optical Connector

FTTH (Fibre to the Home) uses a fusion splice field-assembled optical connector. The fusion splice field-assembled optical connector is connected and assembled using a generic fibre fusion splicer. However, general purpose fusion splicers make the device difficult to operate in the installation field because the fusion splicer is too large and heavy to handle. As a result, the fibre optic splice often breaks during the optical connector assembly process. This makes it difficult to apply fusion-spliced optical connectors in the FTTH field. To solve this problem, this paper proposes a fusion splicer for FTTH that can perform fusion splicing using a wing-type sleeve optical connector. The proposed fusion splicer, with a connection module with a lifting/lowering function, is implemented to connect and protect the wing-type sleeve field-assembled optical connector. In addition, by eliminating the tube heater used as a connection protection method in the existing fusion splicer, the power module is reduced. The developed fusion splicer was evaluated for assembly reliability through splice loss measurements and a comparison of assembling time with the existing fusion splicer.

The asymptotic weak gravity conjecture in M-theory

The tower Weak Gravity Conjecture predicts infinitely many super-extremal states along every ray in the charge lattice of a consistent quantum gravity theory. We show this far-reaching claim in five-dimensional compactifications of M-theory on Calabi-Yau 3-folds for gauge groups with a weak coupling limit. We first characterize the possible weak coupling limits, building on an earlier classification of infinite distance limits in the Kähler moduli space of M-theory compactifications. We find that weakly coupled gauge groups are associated to curves on the compactification space contained in generic fibers or in fibers degenerating at finite distance in their moduli space. These always admit an interpretation as a Kaluza-Klein or winding U(1) in a dual frame or as part of a dual perturbative heterotic gauge group, in agreement with the Emergent String Conjecture. Using the connection between Donaldson-Thomas invariants and Noether-Lefschetz theory, we then show that every ray in the associated charge lattice either supports a tower of BPS states or of non-BPS states, and prove that these satisfy the super-extremality condition, at least in the weak coupling regime.

Compactified Jacobians as Mumford models

We show that relative compactified Jacobians of one-parameter smoothings of a nodal curve of genus g g are Mumford models of the generic fiber. Each such model is given by an admissible polytopal decomposition of the skeleton of the Jacobian. We describe the decompositions corresponding to compactified Jacobians explicitly in terms of the auxiliary stability data and find, in particular, that in degree g g there is a unique compactified Jacobian encoding slope stability, and it is induced by the tropical break divisor decomposition.

Scrollar invariants, syzygies and representations of the symmetric group

Abstract We give an explicit minimal graded free resolution, in terms of representations of the symmetric group S d {S_{d}} , of a Galois-theoretic configuration of d points in 𝐏 d - 2 {\mathbf{P}^{d-2}} that was studied by Bhargava in the context of ring parametrizations. When applied to the geometric generic fiber of a simply branched degree d cover of 𝐏 1 {\mathbf{P}^{1}} by a relatively canonically embedded curve C, our construction gives a new interpretation for the splitting types of the syzygy bundles appearing in its relative minimal resolution. Concretely, our work implies that all these splitting types consist of scrollar invariants of resolvent covers. This vastly generalizes a prior observation due to Casnati, namely that the first syzygy bundle of a degree 4 cover splits according to the scrollar invariants of its cubic resolvent. Our work also shows that the splitting types of the syzygy bundles, together with the multi-set of scrollar invariants, belong to a much larger class of multi-sets of invariants that can be attached to C → 𝐏 1 {C\to\mathbf{P}^{1}} : one for each irreducible representation of S d {S_{d}} , i.e., one for each partition of d.

Motivic Euler Products and Motivic Height Zeta Functions

A motivic height zeta function associated to a family of varieties parametrised by a curve is the generating series of the classes, in the Grothendieck ring of varieties, of moduli spaces of sections of this family with varying degrees. This text is devoted to the study of the motivic height zeta function associated to a family of varieties with generic fiber having the structure of an equivariant compactification of a vector group. Our main theorem describes the convergence of this motivic height zeta function with respect to a topology on the Grothendieck ring of varieties coming from the theory of weights in cohomology. We deduce from it the asymptotic behaviour, as the degree goes to infinity, of a positive proportion of the coefficients of the Hodge-Deligne polynomial of the above moduli spaces: in particular, we get an estimate for their dimension and the number of components of maximal dimension. The main tools for this are a notion of motivic Euler product for series with coefficients in the Grothendieck ring of varieties, an extension of Hrushovski and Kazhdan’s motivic Poisson summation formula, and a motivic measure on the Grothendieck ring of varieties with exponentials constructed using Denef and Loeser’s motivic vanishing cycles.

Ultra-fine self-powered interactive fiber electronics for smart clothing

Fiber electronics discretely integrated into daily life is taking center stage for its enormous potentials for healthcare, physiological monitoring, and human-machine interfaces. However, the state-of-the-art fiber electronics are remarkably distinguished from generic fibers in diameter, stiffness, smoothness, and productivity. Here, we developed a universal scalable manufacturing process and generic fiber-like fine and soft self-powered interactive electronics with a diameter of only dozens of μm and a continuous length of hundreds of kilometers. These fiber electronics are prepared with a large-scale manufacturing speed (>20 m/min), and adapt to modern textile technology including weaving, knitting, sewing and embroidering. Moreover, Kawabata evaluation system for fabrics was systematically employed to analyze the fiber electronics enabled e-textile’s sensorial comfort, suggesting that adding the fiber electronics to fabric benefits its tactile sensations (e.g., softness, fluffy and fullness). Based on tribotronics, the interactive fiber electronics-based smart clothing demonstrates versatile applications including biomechanical energy harvesting, vital signal detecting, human motion recognizing and tactile sensing.

Deformations of Log Terminal and Semi Log Canonical Singularities

Abstract In this paper, we prove that klt singularities are invariant under deformations if the generic fiber is $\mathbb {Q}$ -Gorenstein. We also obtain a similar result for slc singularities. These are generalizations of results of Esnault-Viehweg [Math. Ann.271 (1985), 439–449] and S. Ishii [Math. Ann.275 (1986), 139–148; Singularities (Iowa City, IA, 1986) Contemporary Mathematics, vol. 90 (American Mathematical Society, Providence, RI, 1989), 135–145].