Nodal Curves Research Articles

Improved Statistics for F-theory Standard Models

AbstractMuch of the analysis of F-theory-based Standard Models boils down to computing cohomologies of line bundles on matter curves. By varying parameters one can degenerate such matter curves to singular ones, typically with many nodes, where the computation is combinatorial and straightforward. The question remains to relate the (a priori possibly smaller) value on the original curve to the singular one. In this work, we introduce some elementary techniques (pruning trees and removing interior edges) for simplifying the resulting nodal curves to a small collection of terminal ones that can be handled directly. When applied to the QSMs, these techniques yield optimal results in the sense that obtaining more precise answers would require currently unavailable information about the QSM geometries. This provides us with an opportunity to enhance the statistical bounds established in earlier research regarding the absence of vector-like exotics on the quark-doublet curve.

Rational surfaces with a non‐arithmetic automorphism group

AbstractIn [Algebraic surfaces and hyperbolic geometry, Cambridge University Press, Cambridge, 2012], Totaro gave examples of a K3 surface such that its automorphism group is not commensurable with an arithmetic group, answering a question of Mazur in Section 7 of [Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 14–50]. We give examples of rational surfaces with the same property. Our examples are Looijenga pairs, that is, there is a connected singular nodal curve such that .

Remarks on the nodal quiver

We study an admissible subcategory of the nodal quiver which conjecturally does not admit any Bridgeland stability conditions. Specifically, we prove that its Serre functor coincides with the spherical twist associated with a 3-spherical object. As a consequence, we obtain a classification of the spherical objects, deduce the non-existence of Serre-invariant stability conditions, and construct a natural spherical functor from its structure as a categorical resolution of the nodal cubic curve.

On the Jacobian scheme of a plane curve

We study the Jacobian scheme of a plane algebraic curve at an ordinary singularity, characterizing it through a geometric property. We give an algorithm that gives the analytic type of a double point using the algebraic version of the Mather-Yau Theorem. We give a characterization of plane nodal curves with no infinitely near points through their Tjurina number.

Obstructions to deforming maps from curves to surfaces

This paper studies the obstructions to deforming a map from a complex variety to another variety which is an immersion of codimension one. We extend the classical notion of semiregularity of subvarieties to maps between varieties, and prove the unobstructedness of the deformation of such a map, even when the image is non-reduced. As an application, we give a simple but effective criterion for the vanishing of the obstructions to equisingular deformations of nodal curves on surfaces.

A note on polarized varieties with high nef value

Abstract We study the classification problem for polarized varieties with high nef value. We give a complete list of isomorphism classes for normal polarized varieties with high nef value. This generalizes classical work on the smooth case by Fujita, Beltrametti and Sommese. As a consequence we obtain that polarized varieties with slc singularities and high nef value are birationally equivalent to projective bundles over nodal curves.

Stability conditions for line bundles on nodal curves

Abstract We introduce the abstract notion of a smoothable fine compactified Jacobian of a nodal curve, and of a family of nodal curves whose general element is smooth. Then we introduce the combinatorial notion of a stability assignment for line bundles and their degenerations. We prove that smoothable fine compactified Jacobians are in bijection with these stability assignments. We then turn our attention to fine compactified universal Jacobians – that is, fine compactified Jacobians for the moduli space $\overline {\mathcal {M}}_g$ of stable curves (without marked points). We prove that every fine compactified universal Jacobian is isomorphic to the one first constructed by Caporaso, Pandharipande and Simpson in the nineties. In particular, without marked points, there exists no fine compactified universal Jacobian unless $\gcd (d+1-g, 2g-2)=1$ .

Quasimaps to moduli spaces of sheaves on a surface

Abstract In this article, we study quasimaps to moduli spaces of sheaves on a $K3$ surface S. We construct a surjective cosection of the obstruction theory of moduli spaces of $\epsilon $ -stable quasimaps. We then establish reduced wall-crossing formulas which relate the reduced Gromov–Witten theory of moduli spaces of sheaves on S and the reduced Donaldson–Thomas theory of $S\times C$ , where C is a nodal curve. As applications, we prove the Hilbert-schemes part of the Igusa cusp form conjecture; higher-rank/rank-one Donaldson–Thomas correspondence with relative insertions on $S\times C$ , if $g(C)\leq 1$ ; Donaldson–Thomas/Pandharipande–Thomas correspondence with relative insertions on $S\times \mathbb {P}^1$ .

Beating resonance patterns and extreme power flux skewing in anisotropic elastic plates.

Elastic waves in anisotropic media can exhibit a power flux that is not collinear with the wave vector. This has notable consequences for waves guided in a plate. Through laser-ultrasonic experiments, we evidence remarkable phenomena due to slow waves in a single-crystal silicon wafer. Waves exhibiting power flux orthogonal to their wave vector are identified. A pulsed line source that excites these waves reveals a wave packet radiated parallel to the line. Furthermore, there exist precisely eight plane waves with zero power flux. These so-called zero-group-velocity modes are oriented along the crystal's principal axes. Time acts as a filter in the wave-vector domain that selects these modes. Thus, a point source leads to beating resonance patterns with moving nodal curves on the surface of the infinite plate. We observe this pattern as it emerges naturally after a pulsed excitation.

Brauer groups and Picard groups of the moduli of parabolic vector bundles on a nodal curve

Brauer groups and Picard groups of the moduli of parabolic vector bundles on a nodal curve

Moduli of nodal sextic curves via periods of K3 surfaces

In this paper we study the moduli spaces of nodal sextic curves. We realize each irreducible component of the GIT space of sextic curves with given number of nodes as an open subspace of type IV arithmetic quotients. We then focus on the compactifications of the moduli spaces, one side is the geometric (GIT) compactifications, the other side is the Hodge theoretic compactifications such as Looijenga compactifications and Baily-Borel compactifications. The main result is the isomorphism between GIT and Looijenga compactifications. Some examples are closely related to del Pezzo surfaces. We also extend our results to moduli of nodal sextic curves with specified symmetry.

Hirota varieties and rational nodal curves

The Hirota variety parameterizes solutions to the KP equation arising from a degenerate Riemann theta function. In this work, we study in detail the Hirota variety arising from a rational nodal curve. Of particular interest is the irreducible subvariety defined as the image of a parameterization map, we call this the main component. Proving that this is an irreducible component of the Hirota variety corresponds to solving a weak Schottky problem for rational nodal curves. We solve this problem up to genus nine using computational tools.

Partial degeneration of finite gap solutions to the Korteweg–de Vries equation: soliton gas and scattering on elliptic backgrounds

We obtain Fredholm type formulas for partial degenerations of theta functions on (irreducible) nodal curves of arbitrary genus, with emphasis on nodal curves of genus one. An application is the study of ‘many-soliton’ solutions on an elliptic (cnoidal) background standing wave for the Korteweg–de Vries (KdV) equation starting from a formula that is reminiscent of the classical Kay–Moses formula for N-solitons. In particular, we represent such a solution as a sum of the following two terms: a ‘shifted’ elliptic (cnoidal) background wave and a Kay–Moses type determinant containing Jacobi theta functions for the solitonic content, which can be viewed as a collection of solitary disturbances on the cnoidal background. The expressions for the travelling (group) speed of these solitary disturbances, as well as for the interaction kernel describing the scattering of pairs of such solitary disturbances, are obtained explicitly in terms of Jacobi theta functions. We also show that genus N + 1 finite gap solutions with random initial phases converge in probability to the deterministic cnoidal wave solution as N bands degenerate to a nodal curve of genus one. Finally, we derive the nonlinear dispersion relations and the equation of states for the KdV soliton gas on the residual elliptic background.

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.

Computing Heights via Limits of Hodge Structures

We consider the problem of explicitly computing Beilinson–Bloch heights of homologically trivial cycles on varieties defined over number fields. Recent results have established a congruence, up to the rational span of logarithms of primes, between the height of certain limit mixed Hodge structures and certain Beilinson–Bloch heights obtained from odd-dimensional hypersurfaces with a node. This congruence suggests a new method to compute Beilinson–Bloch heights. Here we explain how to compute the relevant limit mixed Hodge structures in practice, then apply our computational method to a nodal quartic curve and a nodal cubic threefold. In both cases we explain the nature of the primes occurring in the congruence.

The fundamental group of the complement of the Hesse configuration

This note is of supplementary nature to our previous paper [J. Kaneko, K. Matsumoto and K. Ohara, The structure of a local system associated with a hypergeometric system of rank 9, Int. J. Math. 31 (2020) 2050021]. Let [Formula: see text] be the union of the nodal cubic curve and its three inflectional tangents in the complex projective plane [Formula: see text]. Such [Formula: see text] has appeared as the singular locus of certain hypergeometric system introduced in [J. Kaneko, K. Matsumoto and K. Ohara, A system of hypergeometric differential system in two variables of rank 9,Int. J. Math. 28 (2017) 1750015], and we have given generators and defining relations of the fundamental group of [Formula: see text] [J. Kaneko, K. Matsumoto and K. Ohara, The structure of a local system associated with a hypergeometric system of rank 9, Int. J. Math. 31 (2020) 2050021]. [Formula: see text] has a 9-fold Galois covering space [Formula: see text] given by the complement of the Hesse configuration of 12 lines in [Formula: see text]. Hence one can apply the method of Reidemeister–Schreier to derive a finite presentation of [Formula: see text], which we carry out in this note.

Finite element analysis of posterior acetabular column plate and posterior acetabular wall prostheses in treating posterior acetabular fractures

BackgroundThe purpose of this study was to investigate the mechanical stability of the posterior acetabular column plate and different posterior acetabular wall prostheses used in treating posterior acetabular fractures with or without comminution.MethodsThe unilateral normal ilium was reconstructed, and a model of posterior acetabular wall fracture was established on this basis. The fracture fragment accounted for approximately 40% of the posterior acetabular wall. The posterior acetabular column plate and different posterior acetabular wall prostheses were also designed. Using static and dynamic analysis methods, we observed and compared the changes in the stress and displacement values of different models at different hip joint flexion angles under external forces.ResultsAt different hip flexion angles, the stress of each model mainly fluctuated between 37.98 MPa and 1129.00 MPa, and the displacement mainly fluctuated between 0.076 and 6.955 mm. In the dynamic analysis, the nodal stress‒time curves of the models were nonlinear, and the stress changed sharply during the action time. Most of the nodal displacement‒time curves of the models were relatively smooth, with no dramatic changes in displacement during the action time; additionally, most of the curves were relatively consistent in shape.ConclusionsFor simple posterior acetabular wall fractures, we recommend using a posterior acetabular column plate. In the case of comminuted posterior acetabular fractures, we recommend the use of a nonflanked posterior acetabular prosthesis or a biflanked posterior acetabular prosthesis. Regarding the method of acetabular prosthesis design, we propose the concept of “Break up to Make up” as a guide.

Degenerations of bundle moduli

Over a family X of complete curves of genus g, which gives the degeneration of a smooth curve into one with nodal singularities, we build a moduli space which is the moduli space of holomorphic SL(n,C) bundles over the generic smooth curve Xt in the family, and is a moduli space of bundles equipped with extra structure at the nodes for the nodal curves in the family. This moduli space is a quotient by (C⁎)s of a moduli space on the desingularization. Taking a “maximal” degeneration of the curve into a nodal curve built from the glueing of three-pointed spheres, we obtain a degeneration of the moduli space of bundles into a (C⁎)(3g−3)(n−1)-quotient of a (2g−2)-th power of a space associated to the three-pointed sphere. Via the Narasimhan-Seshadri theorem, the moduli space of SL(n,C) bundles on the smooth curve is a space of representations of the fundamental group into SU(n) (the “symplectic picture”). We obtain the degenerations also in this symplectic context, in a way that is compatible with the holomorphic degeneration, so that our limit space is also a (S1)(3g−3)(n−1) symplectic quotient of a (2g−2)-th power of a space associated to the three-pointed sphere.

Curves and stick figures not contained in a hypersurface of a given degree

A stick figure $X\subset \mathbb{P}^r$ is a nodal curve whose irreducible components are lines. For fixed integers $r\ge 3$, $s\ge 2$ and $d$ we study the maximal arithmetic genus of a connected stick figure (or any reduced and connected curve) $X\subset \mathbb{P}^r$ such that $\deg (X)=d$ and $h^0(\mathcal{I}_X(s-1))=0$. We consider Halphen's problem of obtaining all arithmetic genera below the maximal one.

Nonemptiness of severi varieties on enriques surfaces

Abstract Let $(S,L)$ be a general polarised Enriques surface, with L not numerically 2-divisible. We prove the existence of regular components of all Severi varieties of irreducible nodal curves in the linear system $|L|$ , that is, for any number of nodes $\delta =0, \ldots , p_a(L)-1$ . This solves a classical open problem and gives a positive answer to a recent conjecture of Pandharipande–Schmitt, under the additional condition of non-2-divisibility.