Singular Curves Research Articles

Polynomial Analogs of Theta Functions and Rational Solutions of the KP Equation

Abstract In this paper we classify the singular curves whose theta divisors in their generalized Jacobians are algebraic, meaning that they are cut out by polynomial analogs of theta functions. We also determine the degree of an algebraic theta divisor in terms of the singularities of the curve. Furthermore, we show a precise relation between such algebraic theta functions and the corresponding tau functions for the KP hierarchy.

Boundedness of geometric invariants near a singularity which is a suspension of a singular curve

Boundedness of geometric invariants near a singularity which is a suspension of a singular curve

The Bourbaki degree of a plane projective curve

Bourbaki sequences and Bourbaki ideals have been studied by several authors since its inception sixty years ago circa. Generic Bourbaki sequences have been thoroughly examined by the senior author with B. Ulrich and W. Vasconcelos, but due to their nature, no numerical invariant was immediately available. Recently, J. Herzog, S. Kumashiro, and D. Stamate introduced the Bourbaki number in the category of graded modules as the shifted degree of a Bourbaki ideal corresponding to submodules generated in degree at least the maximal degree of a minimal generator of the given module. The present work introduces the Bourbaki degree as the algebraic multiplicity of a Bourbaki ideal corresponding to choices of minimal generators of minimal degree. The main intent is a study of plane curve singularities via this new numerical invariant. Accordingly, quite naturally, the focus is on the case where the standing graded module is the first syzygy module of the gradient ideal of a reduced form f ∈ k [ x , y , z ] f\in k[x,y,z] – i.e., the main component of the module of logarithmic derivations of the corresponding curve. The overall goal of this project is to allow for a facet of classification of projective plane curves based on the behavior of this new numerical invariant, with emphasis on results about its lower and upper bounds.

Classification of unimodal parametric plane curve singularities in positive characteristic

In 2011 Hefez and Hernandes completed Zariski's analytic classification of plane branches belonging to a given equisingularity class by creating “very short” parameterizations over the complex numbers. Their results were used by Mehmood and Pfister to classify unimodal plane branches in characteristic 0 by giving lists of normal forms. The aim of this paper is to give a complete classification of unimodal plane branches over an algebraically closed field of positive characteristic. Since the methods of Hefez and Hernandes cannot be used in positive characteristic, we use a different approach and, for some sporadic singularities in small characteristic, computations with Singular. Our methods are characteristic-independent and provide a different proof of the classification in characteristic 0 showing at the same time that this classification holds also in large characteristic. The main theoretical ingredients are the semicontinuity of the semigroup and of the modality, which we prove and which may be of independent interest.

On free and plus-one generated curves arising from free curves by addition–deletion of a line

In a recent paper, after introducing the notion of plus-one generated hyperplane arrangements, Takuro Abe has shown that if we add (respectively, delete) a line to (respectively, from) a free line arrangement, then the resulting line arrangement is either free or plus-one generated. In this paper, we prove that the same properties hold when we replace the line arrangement by a free curve and add (respectively, delete) a line. The proof uses a new version of a key result due originally to H. Schenck, H. Terao and M. Yoshinaga, in which no quasi-homogeneity assumption is needed. Two conjectures about the Tjurina number of a union of two plane curve singularities are also stated. As a geometric application, we show that, under a mild numerical condition, the projective closure of a contractible, irreducible affine plane curve is either free or plus-one generated, using a deep result due to U. Walther.

Vertex operators of the KP hierarchy and singular algebraic curves

Vertex operators of the KP hierarchy and singular algebraic curves

Modal identification from turbulence response based on improved frequency domain decomposition

Turbulence excitation is an unavoidable form of excitation in flutter flight tests, and it is also a necessary and effective excitation method in high-speed flights, dives, other high-risk test, and high-frequency modal information mining. However, because the turbulence excitation signals are unmeasurable in the time domain, although the turbulence response contains rich and valuable flutter test information, the randomness and low quality of the data often cause difficulties in modal analysis. Therefore, an improved frequency domain decomposition algorithm for turbulence response processing is proposed in this paper. First, due to the irrelevancy of the atmospheric excitation, the power spectral density function matrix of the multi-channel turbulence response is subjected to singular value decomposition. There is a mathematical relationship between the maximum singular value curve and the system frequency response function. Second, the modal assurance criteria are used to calculate the maximum singular value of a single-degree-of-freedom system. Finally, an orthogonal polynomial method is applied to fit the maximum singular value curve, and the system identification is performed directly in the frequency domain. The simulated data and a certain type of aircraft flutter flight test data are used to verify the proposed method, and the results confirm the effectiveness and engineering applicability of the method developed in this work.

Singular Curves on K3 Surfaces

Singular Curves on K3 Surfaces

Asymmetric Image Encryption Based on Singular Cubic Curve with Chaotic Map

Asymmetric Image Encryption Based on Singular Cubic Curve with Chaotic Map

Reading the log canonical threshold of a plane curve singularity from its Newton polyhedron

There is a proposition due to Kollár as reported by Kollár (Proceedings of the summer research institute, Santa Cruz, CA, USA, July 9–29, 1995, American Mathematical Society, Providence, 1997) on computing log canonical thresholds of certain hypersurface germs using weighted blowups, which we extend to weighted blowups with non-negative weights. Using this, we show that the log canonical threshold of a convergent complex power series is at most 1/c, where (c,…,c)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(c, \\ldots , c)$$\\end{document} is a point on a facet of its Newton polyhedron. Moreover, in the case n=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n = 2$$\\end{document}, if the power series is weakly normalised with respect to this facet or the point (c, c) belongs to two facets, then we have equality. This generalises a theorem of Varchenko 1982 to non-isolated singularities.

Automorphisms of moduli spaces of principal bundles over a smooth curve

For any almost-simple group [Formula: see text] over an algebraically closed field [Formula: see text] of characteristic zero, we describe the automorphism group of the moduli space of semistable [Formula: see text]-bundles over a connected smooth projective curve [Formula: see text] of genus at least [Formula: see text]. The result is achieved by studying the singular fibers of the Hitchin fibration. As a byproduct, we provide a description of the irreducible components of two natural closed subsets in the Hitchin basis: the divisor of singular cameral curves and the divisor of singular Hitchin fibers.

Gaussian maps for singular curves on Enriques surfaces

AbstractA marked Prym curve is a triple $$(C,\alpha ,T_d)$$ ( C , α , T d ) where C is a smooth algebraic curve, $$\alpha $$ α is a $$2-$$ 2 - torsion line bundle on C, and $$T_d$$ T d is a divisor of degree d. We give obstructions—in terms of Gaussian maps—for a marked Prym curve $$(C,\alpha ,T_d)$$ ( C , α , T d ) to admit a singular model lying on an Enriques surface with only one ordinary singular point of multiplicity d, such that $$T_d$$ T d is the pull-back of the singular point by the normalization map. More precisely, let (S, H) be a polarized Enriques surface and let (C, f) be a smooth curve together with a morphism $$f:C \rightarrow S$$ f : C → S birational onto its image and such that $$f(C) \in |H|$$ f ( C ) ∈ | H | , f(C) has exactly one ordinary singular point of multiplicity d. Let $$\alpha =f^*\omega _S$$ α = f ∗ ω S and $$T_d$$ T d be the divisor over the singular point of f(C). We show that if H is sufficiently positive then certain natural Gaussian maps on C, associated with $$\omega _C$$ ω C , $$\alpha $$ α , and $$T_d$$ T d are not surjective. On the contrary, we show that for the general triple in the moduli space of marked Prym curves $$(C,\alpha ,T_d)$$ ( C , α , T d ) , the same Gaussian maps are surjective.

Galois covers of singular curves in positive characteristics

Galois covers of singular curves in positive characteristics

On the ∂—-Equation with L2 Estimates on Singular Complex Spaces

Abstract In this paper, we present the unsolvability of $\overline \partial $-equation with weighted $L^{2}$ estimates involved curvature terms on any singular normal complex space in general. Moreover, in the non-normal case, we also give a complete description on $L^{2}$-solvability of the $\overline \partial $-equation with weighted $L^{2}$ estimates for plane curve singularities and their variants in the higher dimension.

Pedal and contrapedal curves in equi-affine plane

In this paper, we define equi-affine pedal and contrapedal curves in equi-affine plane. Then, we also consider the relationships among equi-affine evolutes, involutes, parallels, pedal and contrapedal curves. In addition, we investigate the classifications of singularities of equi-affine pedal and contrapedal curves.

Solving modular cubic equations with Coppersmith's method

Several cryptosystems based on Elliptic Curve Cryptography such as KMOV and Demytko process the message as a point M=(x0,y0) of an elliptic curve with an equation of the form y2≡x3+ax+b(modn) over a finite field when n is a prime number, or over a finite ring when n=pq is an RSA modulus. Other systems use singular cubic curves such as y2≡x3+ax2(modn) and y2+axy≡x3(modn). In this paper, we present a method to find the small solutions of the former modular cubic equations. Our method is based on Coppersmith's technique and enables one to find the solutions (x0,y0) when |x0|3|y0|2 is smaller than the modulus.

A note on complex plane curve singularities up to diffeomorphism and their rigidity

We prove that if two germs of plane curves (C, 0) and (C′,0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(C',0)$$\\end{document} with at least one singular branch are equivalent by a (real) smooth diffeomorphism, then C is complex isomorphic to C′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C'$$\\end{document} or to C′¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{C'}$$\\end{document}. A similar result was shown by Ephraim for irreducible hypersurfaces before, but his proof is not constructive. Indeed, we show that the complex isomorphism is given by the Taylor series of the diffeomorphism. We also prove an analogous result for the case of non-irreducible hypersurfaces containing an irreducible component that is non-factorable. Moreover, we provide a general overview of the different classifications of plane curve singularities.

Counting singular curves with prescribed tangency

We obtain an explicit formula for the characteristic number of degree d curves in P2 with prescribed singularities (of type Ak) that are tangent to a given line. The formula is in terms of the characteristic number of curves with exactly those singularities. We are not aware of any explicit formula to enumerate plane curves of degree d with any number of Ak singularities (beyond codimension 8); however, combined with the results of S. Basu and R. Mukherjee ([1], [2], and [3]), this gives us a complete formula for the characteristic number of curves with δ-nodes and one singularity of type Ak, tangent to a given line, provided δ+k≤8. We use a topological method to compute the degenerate contribution to the Euler class. We have made several low degree checks to verify special cases of our result. When the singularities are only nodes, we have verified that our numbers are logically consistent with those computed by L. Caporaso and J. Harris ([7]). We also verify that our answers for the characteristic number of cubics with a cusp tangent to a given line and the characteristic number of quartics with two nodes and a cusp, tangent to a given line is logically consistent with the characteristic number of rational cubics and quartics tangent to a given line that was computed by L. Ernström and G. Kennedy ([9]).

On perturbations of singular complex analytic curves

On perturbations of singular complex analytic curves

KP solitons and the Riemann theta functions

We show that the τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}-functions of the regular KP solitons from the totally nonnegative Grassmannians can be expressed by the Riemann theta functions on singular curves. We explicitly write the parameters in the Riemann theta function in terms of those of the KP soliton. We give a short remark on the Prym theta function on a double covering of singular curves. We also discuss the KP soliton on quasi-periodic background, which is obtained by applying the vertex operators to the Riemann theta function.