Generic Fiber Research Articles

On the projectivity of proper normal curves over valuation domains

A theorem of Lichtenbaum states, that every proper, regular curve [Formula: see text] over a discrete valuation domain [Formula: see text] is projective. This theorem is generalized to the case of an arbitrary valuation domain [Formula: see text] using the following notion of regularity for non-noetherian rings introduced by Bertin: the local ring [Formula: see text] of a point [Formula: see text] is called regular, if every finitely generated ideal [Formula: see text] has finite projective dimension. The generalization is a particular case of a projectivity criterion for proper, normal [Formula: see text]-curves: such a curve [Formula: see text] is projective if for every irreducible component [Formula: see text] of its closed fiber [Formula: see text] there exists a closed point [Formula: see text] of the generic fiber of [Formula: see text] such that the Zariski closure [Formula: see text] meets [Formula: see text] and meets [Formula: see text] in regular points only.

The fibers of the ramified Prym map

We study the ramified Prym map [Formula: see text] which assigns to a ramified double cover of a smooth irreducible curve of genus [Formula: see text] ramified in [Formula: see text] points the Prym variety of the covering. We focus on the six cases where the dimension of the source is strictly greater than the dimension of the target giving a geometric description of the generic fiber. We also give an explicit example of a totally geodesic curve which is an irreducible component of a fiber of the Prym map [Formula: see text].

Semistable abelian varieties and maximal torsion 1-crystalline submodules

Let p be a prime, let K be a discretely valued extension of ℚ p , and let A K be an abelian K-variety with semistable reduction. Extending work by Kim and Marshall from the case where p>2 and K/ℚ p is unramified, we prove an l=p complement of a Galois cohomological formula of Grothendieck for the l-primary part of the Néron component group of A K . Our proof involves constructing, for each m∈ℤ ≥0 , a finite flat 𝒪 K -group scheme with generic fiber equal to the maximal 1-crystalline submodule of A K [p m ]. As a corollary, we have a new proof of the Coleman–Iovita monodromy criterion for good reduction of abelian K-varieties.

Comparison between rigid and crystalline syntomic cohomology for strictly semistable log schemes with boundary

We introduce rigid syntomic cohomology for strictly semistable log schemes over a complete discrete valuation ring of mixed characteristic (0,p) In case a good compactification exists, we compare this cohomology theory to Nekovář–Nizioł’s crystalline syntomic cohomology of the generic fibre. The main ingredients are a modification of Große-Klönne’s rigid Hyodo–Kato theory and a generalization of it for strictly semistable log schemes with boundary.

A remark on the generalized Franchetta conjecture for K3 surfaces

A family of K3 surfaces \({\mathscr {X}}\rightarrow B\) has the Franchetta property if the Chow group of 0-cycles on the generic fiber is cyclic. The generalized Franchetta conjecture proposed by O’Grady asserts that the universal family \({\mathscr {X}}_g\rightarrow {\mathscr {F}}_g\) of polarized K3 of degree \(2g-2\) has the Franchetta property. While this is known only for small g thanks to [7], we prove that for all g there is a hypersurface in \( {\mathscr {F}}_g\) such that the corresponding family has the Franchetta property.

Algebraic surfaces with Eisenbud–Harris general fibration of genus 4 containing fibers of rank 3

Algebraic surfaces with Eisenbud–Harris general fibration of genus 4 containing fibers of rank 3

On semistable degenerations of Fano varieties

Consider a family of Fano varieties \(\pi :X \rightarrow B\ni o\) over a curve germ with a smooth total space X. Assume that the generic fiber is smooth and the special fiber \(F=\pi ^{-1}(o)\) has simple normal crossings. Then F is called a semistable degeneration of Fano varieties. We show that the dual complex of F is a simplex of dimension \(\hbox {\,\char 054\,}\mathrm {dim}\, F\). Simplices of any admissible dimension can be realized for any dimension of the fiber. Using this result and the Minimal Model Program in dimension 3 we reproduce the classification of the semistable degenerations of del Pezzo surfaces obtained by Fujita. We also show that the maximal degeneration is unique and has trivial monodromy in dimension \(\hbox {\,\char 054\,}\,3\).

Characteristic cycle and wild ramification for nearby cycles of étale sheaves

Abstract In this article, we give a bound for the wild ramification of the monodromy action on the nearby cycles complex of a locally constant étale sheaf on the generic fiber of a smooth scheme over an equal characteristic trait in terms of Abbes and Saito’s logarithmic ramification filtration. This provides a positive answer to the main conjecture in [24] for smooth morphisms in equal characteristic. We also study the ramification along vertical divisors of étale sheaves on relative curves and abelian schemes over a trait.

SPECIAL VALUES OF THE ZETA FUNCTION OF AN ARITHMETIC SURFACE

AbstractWe prove that the special-value conjecture for the zeta function of a proper, regular, flat arithmetic surface formulated in [6] at $s=1$ is equivalent to the Birch and Swinnerton-Dyer conjecture for the Jacobian of the generic fibre. There are two key results in the proof. The first is the triviality of the correction factor of [6, Conjecture 5.12], which we show for arbitrary regular proper arithmetic schemes. In the proof we need to develop some results for the eh-topology on schemes over finite fields which might be of independent interest. The second result is a different proof of a formula due to Geisser, relating the cardinalities of the Brauer and the Tate–Shafarevich group, which applies to arbitrary rather than only totally imaginary base fields.

Newton transformations and motivic invariants at infinity of plane curves

In this article we give an expression of the motivic Milnor fiber at infinity and the motivic nearby cycles at infinity of a polynomial f in two variables with coefficients in an algebraic closed field of characteristic zero. This expression is given in terms of some motives associated to the faces of the Newton polygons appearing in the Newton algorithm at infinity of f without any condition of convenience or non degeneracy. In the complex setting, we compute the Euler characteristic of the generic fiber of f in terms of the area of the surfaces associated to faces of the Newton polygons. Furthermore, if f has isolated singularities, we compute similarly the classical invariants at infinity \(\lambda _{c}(f)\) which measures the non equisingularity at infinity of the fibers of f in \({\mathbb {P}}^2\), and we prove the equality between the topological and the motivic bifurcation sets and give an algorithm to compute them.

Limits of dual curves via foliations

We develop a method to compute limits of dual plane curves in Zeuthen families of any kind. More precisely, we compute the limit 0-cycle of the ramification scheme of a general linear system on the generic fiber, only assumed geometrically reduced, of a Zeuthen family of any kind.

Entropic Competition between Supercoiled and Torsionally Relaxed Chromatin Fibers Drives Loop Extrusion through Pseudo-Topologically Bound Cohesin.

Simple SummaryChromatin dynamics and chromatin structure are a two-way relationship governed by polymer physics and active biological processes. Thanks to the research in the field of computational biology and modeling, computer simulations became indispensable in studying these complex relationships. It is now generally accepted that looped structures occurring in the intermediate range of ordering of chromatin are formed by a loop extrusion mechanism involving specialized proteins (structural maintenance complexes or SMCs). Although the motor activity of SMCs has been speculated for a long time, the motor activity of cohesin was discovered only recently (Davidson 2019). While evidence of the cohesin’s motor activity is missing, other mechanisms that could efficiently drive the loop extrusion without motor activity of SMCs have been discovered by computer simulations. These mechanisms account for transcriptionally driven loop extrusion or entropically driven loop extrusion by osmotic pressure. In our previous model, we have shown that the cohesin in handcuffed conformation can be pushed mechanically by emerging plectoneme formed during transcription, exerting pressure on the joint section of handcuffs. In the current work, we use coarse-grained molecular simulation to further explore the extrusion driven by supercoiling while employing much lower levels of supercoiling. Moreover, recent works favor non-topological binding of cohesin on fibers, which would solve a range of topological problems while bypassing other molecular machinery sitting on DNA. We show by means of computer simulations that supercoiling can drive loop extrusion without taking advantage of mechanic push on the joint section of cohesin handcuffs. As such, the work addresses current problems in molecular biology and employs advanced methods and original solutions in the study.We propose a model for cohesin-mediated loop extrusion, where the loop extrusion is driven entropically by the energy difference between supercoiled and torsionally relaxed chromatin fibers. Different levels of negative supercoiling are controlled by varying imposed friction between the cohesin ring and the chromatin fiber. The speed of generation of negative supercoiling by RNA polymerase associated with TOP1 is kept constant and corresponds to 10 rotations per second. The model was tested by coarse-grained molecular simulations for a wide range of frictions between 2 to 200 folds of that of generic fiber and the surrounding medium. The higher friction allowed for the accumulation of higher levels of supercoiling, while the resulting extrusion rate also increased. The obtained extrusion rates for the given range of investigated frictions were between 1 and 10 kbps, but also a saturation of the rate at high frictions was observed. The calculated contact maps indicate a qualitative improvement obtained at lower levels of supercoiling. The fits of mathematical equations qualitatively reproduce the loop sizes and levels of supercoiling obtained from simulations and support the proposed mechanism of entropically driven extrusion. The cohesin ring is bound on the fibers pseudo-topologically, and the model suggests that the topological binding is not necessary.

On the standard conjecture for projective compactifications of Néron models of -dimensional Abelian varieties

We prove that the Grothendieck standard conjecture of Lefschetz type holds for a smooth complex projective -dimensional variety fibred by Abelian varieties (possibly, with degeneracies) over a smooth projective curve if the endomorphism ring of the generic geometric fibre is not an order of an imaginary quadratic field. This condition holds automatically in the cases when the reduction of the generic scheme fibre at some place of the curve is semistable in the sense of Grothendieck and has odd toric rank or the generic geometric fibre is not a simple Abelian variety.

A compact G2-calibrated manifold with first Betti number b1 = 1

We construct a compact, formal 7-manifold with a closed G2-structure and with first Betti number b1=1, which does not admit any torsion-free G2-structure, that is, it does not admit any G2-structure such that the holonomy group of the associated metric is a subgroup of G2. We also construct associative calibrated (hence volume-minimizing) 3-tori with respect to this closed G2-structure and, for each of those 3-tori, we show a 3-dimensional family of non-trivial associative deformations. We also construct a fibration of our 7-manifold over S2×S1 with generic fiber a (non-calibrated) coassociative 4-torus and some singular fibers.

Optical sensor network interrogation system based on nonuniform microwave photonic filters.

Based on the nonuniformly spaced microwave photonic delay-line filter technology, a new design of a generic optical fiber sensor network interrogation platform is proposed and demonstrated. Sensing information from different types of optical sensors embedded in filter taps is converted into the variations of delay time and amplitude of each filter tap individually. Information to be measured can be decoded from the complex temporal impulse response of the microwave photonic filter. As proof-of-concept, our proposed approach is verified by simulations and experimental demonstrations successfully. Four optical sensors of different types are simultaneously interrogated via inverse Fourier transform of the filter frequency response. The experiment results show good linearity between the variation of temporal impulse response and the variations of the twist, the lateral pressure, the transversal loading and the temperature. The sensitivity of the sensors in the proposed platform is -2.130×10-5 a.u/degree, 6.1039 ps/kPa, -1.9146×10-5 a.u/gram, and 5.1497 ps/°C, respectively. Compared to the conventional optical sensors interrogation system, the presented approach provides a centralized solution that works for different types of optical sensors and can be easily expanded to cover larger optical sensor networks.

Log smooth curves over discrete valuation rings

We give necessary and sufficient conditions for log smoothness of a proper regular arithmetic surface with smooth geometrically connected generic fibre over a discrete valuation ring with perfect residue field. As an application, we recover known criteria for log smooth reduction of minimal normal crossings models of curves.

CM liftings of surfaces over finite fields and their applications to the Tate conjecture

AbstractWe give applications of integral canonical models of orthogonal Shimura varieties and the Kuga-Satake morphism to the arithmetic of$K3$surfaces over finite fields. We prove that every$K3$surface of finite height over a finite field admits a characteristic$0$lifting whose generic fibre is a$K3$surface with complex multiplication. Combined with the results of Mukai and Buskin, we prove the Tate conjecture for the square of a$K3$surface over a finite field. To obtain these results, we construct an analogue of Kisin’s algebraic group for a$K3$surface of finite height and construct characteristic$0$liftings of the$K3$surface preserving the action of tori in the algebraic group. We obtain these results for$K3$surfaces over finite fields of any characteristics, including those of characteristic$2$or$3$.

Explicit Families of K3 Surfaces Having Real Multiplication

For families of K3 surfaces, we establish a sufficient criterion for real or complex multiplication. Our criterion is arithmetic in nature. It may show, at first, that the generic fibre of the family has a nontrivial endomorphism field. Moreover, the endomorphism field does not shrink under specialization. As an application, we present two explicit families of K3 surfaces having real multiplication by Q( 2) and Q( 5), respectively.

The Kodaira dimension and singularities of moduli of stable sheaves on some elliptic surfaces

Let [Formula: see text] be an elliptic surface over [Formula: see text] with [Formula: see text], and [Formula: see text] be the moduli scheme of rank-two stable sheaves [Formula: see text] on [Formula: see text] with [Formula: see text] in [Formula: see text]. We look into defining equations of [Formula: see text] at its singularity [Formula: see text], partly because if [Formula: see text] admits only canonical singularities, then the Kodaira dimension [Formula: see text] can be calculated. We show the following:(A)[Formula: see text] is at worst canonical singularity of [Formula: see text] if the restriction of [Formula: see text] to the generic fiber of [Formula: see text] has no rank-one subsheaf, and if the number of multiple fibers of [Formula: see text] is a few.(B)We obtain that [Formula: see text] and the Iitaka program of [Formula: see text] can be described in purely moduli-theoretic way for [Formula: see text], when [Formula: see text], [Formula: see text] has just two multiple fibers, and one of its multiplicities equals [Formula: see text].(C)On the other hand, when [Formula: see text] has a rank-one subsheaf, it may be insufficient to look at only the degree-two part of defining equations to judge whether [Formula: see text] is at worst canonical singularity or not.

Hitchin and Calabi–Yau Integrable Systems via Variations of Hodge Structures

Abstract Since its discovery by Hitchin in 1987, G-Hitchin systems for a reductive complex Lie group G have extensively been studied. For example, the generic fibers are nowadays well-understood. In this paper, we show that the smooth parts of G-Hitchin systems for a simple adjoint complex Lie group G are isomorphic to non-compact Calabi–Yau integrable systems extending results by Diaconescu–Donagi–Pantev. Moreover, we explain how Langlands duality for Hitchin systems is related to Poincaré–Verdier duality of the corresponding families of quasi-projective Calabi–Yau threefolds. Even though the statement is holomorphic-symplectic, our proof is Hodge-theoretic. It is based on polarizable variations of Hodge structures that admit so-called abstract Seiberg–Witten differentials. These ensure that the associated Jacobian fibration is an algebraic integrable system.