Nodal Curves Research Articles

$\mathcal D$-bundles and integrable hierarchies

We study the geometry of \mathcal D - bundles —locally projective \mathcal D -modules—on algebraic curves, and apply them to the study of integrable hierarchies, specifically the multicomponent Kadomtsev-Petviashvili (KP) and spin Calogero-Moser (CM) hierarchies. We show that KP hierarchies have a geometric description as flows on moduli spaces of \mathcal D -bundles; in particular, we prove that the local structure of \mathcal D -bundles is captured by the full Sato Grassmannian. The rational, trigonometric, and elliptic solutions of KP are therefore captured by \mathcal D -bundles on cubic curves E , that is, irreducible (smooth, nodal, or cuspidal) curves of arithmetic genus 1 . We develop a Fourier-Mukai transform describing \mathcal D -modules on cubic curves E in terms of (complexes of) sheaves on a twisted cotangent bundle {E^{\natural}} over E . We then apply this transform to classify \mathcal D -bundles on cubic curves, identifying their moduli spaces with phase spaces of general CM particle systems (realized through the geometry of spectral curves in {E^{\natural}} ). Moreover, it is immediate from the geometric construction that the flows of the KP and CM hierarchies are thereby identified and that the poles of the KP solutions are identified with the positions of the CM particles. This provides a geometric explanation of a much-explored, puzzling phenomenon of the theory of integrable systems: the poles of meromorphic solutions to KP soliton equations move according to CM particle systems.

Classifying quadratic quantum [formula omitted]s by using graded skew Clifford algebras

We prove that quadratic regular algebras of global dimension three on degree-one generators are related to graded skew Clifford algebras. In particular, we prove that almost all such algebras may be constructed as a twist of either a regular graded skew Clifford algebra or of an Ore extension of a regular graded skew Clifford algebra of global dimension two. In so doing, we classify all quadratic regular algebras of global dimension three that have point scheme either a nodal cubic curve or a cuspidal cubic curve in P 2 .

Coherent systems on a nodal curve of genus one

AbstractWe study coherent systems on an irreducible nodal curve of arithmetic genus 1. We determine conditions for their non‐emptiness and study properties like irreducibility, smoothness, seminormality and rationality.

The 16th Hilbert problem on algebraic limit cycles

For real planar polynomial differential systems there appeared a simple version of the 16th Hilbert problem on algebraic limit cycles: Is there an upper bound on the number of algebraic limit cycles of all polynomial vector fields of degree m? In [J. Llibre, R. Ramírez, N. Sadovskaia, On the 16th Hilbert problem for algebraic limit cycles, J. Differential Equations 248 (2010) 1401–1409] Llibre, Ramírez and Sadovskaia solved the problem, providing an exact upper bound, in the case of invariant algebraic curves generic for the vector fields, and they posed the following conjecture: Is 1 + ( m − 1 ) ( m − 2 ) / 2 the maximal number of algebraic limit cycles that a polynomial vector field of degree m can have? In this paper we will prove this conjecture for planar polynomial vector fields having only nodal invariant algebraic curves. This result includes the Llibre et al.ʼs as a special one. For the polynomial vector fields having only non-dicritical invariant algebraic curves we answer the simple version of the 16th Hilbert problem.

Conformal Invariance in Three-Dimensional Rotating Turbulence

We examine turbulent flows in the presence of solid-body rotation and helical forcing in the framework of stochastic Schramm-Löwner evolution (SLE) curves. The data stem from a run with 1536³ grid points, with Reynolds and Rossby numbers of, respectively, 5100 and 0.06. We average the parallel component of the vorticity in the direction parallel to that of rotation and examine the resulting <ω(z)>(z) field for scaling properties of its zero-value contours. We find for the first time for three-dimensional fluid turbulence evidence of nodal curves being conformal invariant, belonging to a SLE class with associated Brownian diffusivity κ = 3.6 ± 0.1. SLE behavior is related to the self-similarity of the direct cascade of energy to small scales and to the partial bidimensionalization of the flow because of rotation. We recover the value of κ with a heuristic argument and show that this is consistent with several nontrivial SLE predictions.

Generating stable modular graphs

We present and prove the correctness of the program boundary, whose sources are available at http://people.sissa.it/~maggiolo/boundary/. Given two natural numbers g and n satisfying 2g+n−2>0, the program generates all genus g stable graphs with nunordered marked points. Each such graph determines the topological type of a nodal stable curve of arithmetic genus g with n unordered marked points. Our motivation comes from the fact that the boundary of the moduli space of stable genus g, n-pointed curves can be stratified by taking loci of curves of a fixed topological type.

On the complexity group of stable curves

Abstract In this paper, we study combinatorial properties of stable curves. To the dual graph of any nodal curve, there is naturally associated a group, which is the group of components of the Néron model of the generalized Jacobian of the curve. We study the order of this group, called the complexity. In particular, we provide a partial characterization of the stable curves having maximal complexity, and we provide an upper bound, depending only on the genus g of the curve, on the maximal complexity of stable curves; this bound is asymptotically sharp for g ≫ 0. Eventually, we state some conjectures on the behavior of stable curves with maximal complexity, and prove partial results in this direction.

A short proof of the Göttsche conjecture

We prove that for a sufficiently ample line bundle $L$ on a surface $S$, the number of $\delta$-nodal curves in a general $\delta$-dimensional linear system is given by a universal polynomial of degree $\delta$ in the four numbers $L^2,\,L.K_S,\,K_S^2$ and $c_2(S)$. The technique is a study of Hilbert schemes of points on curves on a surface, using the BPS calculus of [PT3] and the computation of tautological integrals on Hilbert schemes by Ellingsrud, G\"ottsche and Lehn. We are also able to weaken the ampleness required, from G\"ottsche's $(5\delta-1)$-very ample to $\delta$-very ample.

Modular properties of nodal curves on K3 surfaces

In this article we are going to address the following issues: (1) the first is a rigidity property for pairs (S, C) consisting of a general projective K3 surface S, and a curve C obtained as the normalization of a nodal, hyperplane section \({\hat C\hookrightarrow S}\). We prove that a non-trivial deformation of a pair (S, C) induces a non-trivial deformation of C; (2) the second question concerns the Wahl map of curves C obtained as above. We prove that the Wahl map of the normalization of a nodal curve contained in a general projective K3 surface is non-surjective. In both cases, we impose upper bounds on the number of nodes of \({\hat C}\).

Global bifurcation and stable two-phase separation for a phase field model in a disk

Let D:$=\{(x,y);\ x^2+y^2 $E(u)=\int_{D}(\frac{|\nabla u|^2}{2}+\lambda W(u)) dx\ \ $subject to$ \ \ m=\frac{1}{|D|}\int_Dudx$and study the global structure of critical points. We show that for an arbitrary potential $W\in C^4$ every level set of every nonconstant local minimizer is a $C^1$-curve and it divides $D$ into exactly two simply connected subdomains. Next we consider the case $W(u)=(u^2-1)^2/4$ (Cahn-Hilliard equation). When $\lambda$ varies and $m$ is fixed, we show that this problem has an unbounded continuum of critical points. When $m$ varies and $\lambda$ is fixed, we show that this problem has a bounded continuum meeting at two different points on the trivial branch. Moreover, we show that in each case a bifurcating critical point is stable (a local minimizer) near the bifurcation point in a certain parameter range. The main technique is the nodal curve which relates the shape with the Morse index. We do not use a small parameter or the $\Gamma$-convergence technique.

Reliability-Based Grid Expansion Planning of Power Systems Considering Wind Turbine Generators

AbstractThis paper describes a new methodology for grid expansion planning considering the probabilistic reliability of a composite power system including wind turbine generators (WTG). The proposed model includes capacity limitations and transmission line unavailability. The importance of renewable energy sources has been growing at a rapid rate due to environmental concerns. In particular, wind energy is one of the most successfully utilized such source in producing electrical energy. The random nature of wind speed can be incorporated in the evaluation by representing the WTG as a multi-state model. This type of model comes from a nodal equivalent load duration curve based on a nodal effective load model developed in recent years. In this paper, a heuristic method is used to search for the best reliability level among various scenarios with candidate lines.

Minitwistor spaces, Severi varieties, and Einstein–Weyl structure

In this article, we show that the space of nodal rational curves, which is so called a Severi variety (of rational curves), on any non-singular projective surface is always equipped with a natural Einstein–Weyl structure, if the space is 3-dimensional. This is a generalization of the Einstein–Weyl structure on the space of smooth rational curves on a complex surface, given by Hitchin. As geometric objects naturally associated to Einstein–Weyl structure, we investigate null surfaces and geodesics on the Severi varieties. Also, we see that if the projective surface has an appropriate real structure, then the real locus of the Severi variety becomes a positive definite Einstein–Weyl manifold. Moreover, we construct various explicit examples of rational surfaces having 3-dimensional Severi varieties of rational curves.

Tilting on non-commutative rational projective curves

In this article we introduce a new class of non-commutative projective curves and show that in certain cases the derived category of coherent sheaves on them has a tilting complex. In particular, we prove that the right bounded derived category of coherent sheaves on a reduced rational projective curve with only nodes and cusps as singularities, can be fully faithfully embedded into the right bounded derived category of the finite dimensional representations of a certain finite dimensional algebra of global dimension two. As an application of our approach we show that the dimension of the bounded derived category of coherent sheaves on a rational projective curve with only nodal or cuspidal singularities is at most two. In the case of the Kodaira cycles of projective lines, the corresponding tilted algebras belong to a well-known class of gentle algebras. We work out in details the tilting equivalence in the case of the Weierstrass nodal curve zy2 = x3 + x2z.

Torsion-free sheaves on nodal curves and triples

Let X be a reduced irreducible curve with at most nodes as singularities with normalization π: \( \tilde X \) → X. We study the description of torsion free sheaves on X in terms of vector bundles with an additional structure on \( \tilde X \) which was introduced by Seshadri.

The Chow Ring of the Stack of Rational Curves with at Most 3 Nodes

In this article, we explain the rational Chow ring of the stack consisting of nodal curves of genus 0 with at most 3 nodes: it is a ℚ-algebra with 10 generators and 11 relations.

Tautological Classes of the Stack of Rational Nodal Curves

This is the second in a series of three papers in which we investigate the rational Chow ring of the stack 𝔐0 consisting of nodal curves of genus 0. Here we define the basic classes: the classes of strata and the Mumford classes.

The Stack of Rational Curves

In this series of three article, we start to investigate the rational Chow ring of the stack 𝔐0 consisting of nodal curves of genus 0. In particular, we determine completely the rational Chow ring of the substack consisting of curves with at most 3 nodes. In the first article, we construct the stack 𝔐0 and its stratification by nodes, and show that the map from the universal curve to the stack is not representable in the category of schemes.

Koszul cohomology and singular curves

We investigate Koszul cohomology on irreducible nodal curves following the lines of [2]. In particular, we prove both Green and Green-Lazarsfeld conjectures for any k-gonal nodal curve which is general in the sense of [4].

Twistor-strings, Grassmannians and leading singularities

We derive a systematic procedure for obtaining an explicit, L-loop leading singularities of planar N=4 super Yang-Mills scattering amplitudes in twistor space directly from their momentum space channel diagrams. The expressions are given as integrals over the moduli of connected, nodal curves in twistor space whose degree and genus matches expectations from twistor-string theory. We propose that a twistor-string theory for pure N=4 super Yang-Mills, if it exists, is determined by the condition that these leading singularity formulae arise as residues when an unphysical contour for the path integral is used, by analogy with the momentum space leading singularity conjecture. We go on to show that the genus g twistor-string moduli space for g-loop N^{k-2}MHV amplitudes may be mapped into the Grassmannian G(k,n). Restricting to a leading singularity, the image of this map is a 2(n-2)-dimensional subcycle of G(k,n) of exactly the type found from the Grassmannian residue formula of Arkani-Hamed, Cachazo, Cheung and Kaplan. Based on this correspondence and the Grassmannian conjecture, we deduce restrictions on the possible leading singularities of multi-loop N^pMHV amplitudes. In particular, we argue that no new leading singularities can arise beyond 3p loops.

Eigenmodes and eigenfrequencies of vibrating elliptic membranes: a Klein oscillation theorem and numerical calculations

We give a complete proof of the existence of an infinite set of eigenmodes for a vibrating elliptic membrane in one to one correspondence with the well-known eigenmodes for a circular membrane. More exactly, we show that for each pair $(m,n) \in \{0,1,2, \cdots\}^2$ there exists a unique even eigenmode with $m$ ellipses and $n$ hyperbola branches as nodal curves and, similarly, for each $(m,n) \in \{0,1,2, \cdots\}\times \{1,2, \cdots\}$ there exists a unique odd eigenmode with $m$ ellipses and $n$ hyperbola branches as nodal curves. Our result is based on directly using the separation of variables method for the Helmholtz equation in elliptic coordinates and in proving that certain pairs of curves in the plane of parameters $a$ and $q$ cross each other at a single point. As side effects of our proof, a new and precise method for numerically calculating the eigenfrequencies of these modes is presented and also approximate formulae which explain rather well the qualitative asymptotic behavior of the eigenfrequencies for large eccentricities.