Algebraic Curves Research Articles

Interpolation by decomposable univariate polynomials

The usual univariate interpolation problem of finding a monic polynomial f of degree n that interpolates n given values is well understood. This paper studies a variant where f is required to be composite, say, a composition of two polynomials of degrees d and e, respectively, with de=n, and with d+e−1 given values. Some special cases are easy to solve, and for the general case, we construct a homotopy between it and a special case. We compute a geometric solution of the algebraic curve presenting this homotopy, and this also provides an answer to the interpolation task. The computing time is polynomial in the geometric data, like the degree, of this curve. A consequence is that for almost all inputs, a decomposable interpolation polynomial exists.

Tame parahoric nonabelian Hodge correspondence in positive characteristic over algebraic curves

Tame parahoric nonabelian Hodge correspondence in positive characteristic over algebraic curves

Vertex operators of the KP hierarchy and singular algebraic curves

Vertex operators of the KP hierarchy and singular algebraic curves

Incidence systems on Cartesian powers of algebraic curves

We show that a non-locally modular reduct of the full Zariski structure of an algebraic curve interprets an infinite field.

A survey on recent advances in the computation of asymptotes

AbstractIn this paper, we summarize two algorithms for computing all the generalized asymptotes of a plane algebraic curve implicitly or parametrically defined. The approach is based on the notion of perfect curve introduced from the concepts and results presented in Blasco and Pérez-Díaz (Comput Aided Geom Des 31(2):81–96, 2014), Blasco and Pérez-Díaz (Comput Aided Geom Des 31(7-8):345–357, 2014), and Blasco and Pérez-Díaz (J Comput Appl Math 278:231–247, 2015). From these results, we derive a new method that allow to easily compute horizontal and vertical asymptotes.

Pureté de l’approximation forte sur le corps des fonctions d’une courbe algébrique complexe

Over the function field of a complex algebraic curve, strong approximation off a non-empty finite set of places holds for the complement of a codimension 2 closed subset in a homogeneous space under a semisimple algebraic group, and for the complement of a codimension 2 closed subset in an affine smooth complete intersection of low degree.

Bases for Riemann–Roch spaces of linearized function fields with applications to generalized algebraic geometry codes

Several applications of function fields over finite fields, or equivalently, algebraic curves over finite fields, require computing bases for Riemann–Roch spaces. In this paper, we determine explicit bases for Riemann–Roch spaces of linearized function fields, and we give a lower bound for the minimum distance of generalized algebraic geometry codes. As a consequence, we construct generalized algebraic geometry codes with good parameters.

Algebraic curves associated with centrosymmetric matrices of orders up to 6

In this paper the boundary generating curves and numerical ranges of centrosymmetric matrices of orders up to 6 are characterized in terms of the matrices entries. These results extend previous ones concerning Kac-Sylvester matrices. The classification of all the possible boundary generating curves for centrosymmetric matrices of higher dimensions remains open.

Felix Klein’s early contributions to anschauliche Geometrie

Between 1873 and 1876, Felix Klein published a series of papers that he later placed under the rubric anschauliche Geometrie in the second volume of his collected works (1922). The present study attempts not only to follow the course of this work, but also to place it in a larger historical context. Methodologically, Klein’s approach had roots in Poncelet’s principle of continuity, though the more immediate influences on him came from his teachers, Plücker and Clebsch. In the 1860s, Clebsch reworked some of the central ideas in Riemann’s theory of Abelian functions to obtain complicated results for systems of algebraic curves, most published earlier by Hesse and Steiner. These findings played a major role in enumerative geometry, whereas Plücker’s work had a strongly qualitative character that imbued Klein’s early studies. A leitmotif in these works can be seen in the interplay between real curves and surfaces as reflected by their transformational properties. During the early 1870s, Klein and Zeuthen began to explore the possibility of deriving all possible forms for real cubic surfaces as well as quartic curves. They did so using continuity methods reminiscent of Poncelet’s earlier approach. Both authors also relied on visual arguments, which Klein would later advance under the banner of intuitive geometry (anschauliche Geometrie).

Studying the Upper Bounds of the Numbers of Zeros of Abelian Integrals by the Law of Polynomials

For the quadratic reversible systems of genus one, all of their periodic orbits are higher-order algebraic curves. When they are perturbed by polynomials of degree [Formula: see text], the numbers of zeros of their Abelian integrals will change and we study the upper bounds of these numbers by using the methods of Riccati equation and Picard–Fuchs equation. We consider both the highest and lowest degrees of polynomials, and more importantly, we consider the law of polynomials and the range of values for their variables. Consequently, some laws of the polynomials are discovered and many upper bounds are obtained, and these upper bounds are sharper than the results obtained by other techniques.

On actions of Frobenius morphisms for moduli stacks of principal bundles over algebraic curves

We study the various arithmetic and geometric Frobenius morphisms on the moduli stack of principal bundles over a smooth projective algebraic curve and determine explicitly their actions on the l-adic cohomology of the moduli stack in terms of Chern classes.

Non-vanishing of multiple zeta values for higher genus curves over finite fields

In this paper, we show that ∞-adic multiple zeta values associated to the function field of an algebraic curve of higher genus over a finite field are not zero, under certain assumption on the gap sequence associated to the rational point ∞ on the given curve. Using arguments and results of Sheats and Thakur for the case of the projective line, we calculate the absolute values of power sums in the series defining multiple zeta values, and show that the calculation implies the non-vanishing result.

Generic symmetry defect set of an algebraic curve

Let X ⊂ C 2 n X \subset \mathbb {C}^{2n} be an n n -dimensional algebraic variety. We define the algebraic version of the generic symmetry defect set (Wigner caustic) of X X . Moreover, we compute its singularities as well as degree, genus and Euler characteristic for X d X_d being a generic (smooth and transversal to the line at infinity) curve of degree d d in C 2 \mathbb {C}^2 .

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.

A family of algebraic curves and Appell series over finite fields

A family of algebraic curves and Appell series over finite fields

Spectral Curves for Third-Order ODOs

Spectral curves are algebraic curves associated to commutative subalgebras of rings of ordinary differential operators (ODOs). Their origin is linked to the Korteweg–de Vries equation and to seminal works on commuting ODOs by I. Schur and Burchnall and Chaundy. They allow the solvability of the spectral problem Ly=λy, for an algebraic parameter λ and an algebro-geometric ODO L, whose centralizer is known to be the affine ring of an abstract spectral curve Γ. In this work, we use differential resultants to effectively compute the defining ideal of the spectral curve Γ, defined by the centralizer of a third-order differential operator L, with coefficients in an arbitrary differential field of zero characteristic. For this purpose, defining ideals of planar spectral curves associated to commuting pairs are described as radicals of differential elimination ideals. In general, Γ is a non-planar space curve and we provide the first explicit example. As a consequence, the computation of a first-order right factor of L−λ becomes explicit over a new coefficient field containing Γ. Our results establish a new framework appropriate to develop a Picard–Vessiot theory for spectral problems.

A Numerical Method for Solving Elliptic Equations on Real Closed Algebraic Curves and Surfaces

A Numerical Method for Solving Elliptic Equations on Real Closed Algebraic Curves and Surfaces

The rank of the Cartier operator on Picard curves

For an algebraic curve [Formula: see text] defined over an algebraically closed field of characteristic [Formula: see text], [Formula: see text]-number [Formula: see text] is the dimension of the space of exact holomorphic differentials on [Formula: see text]. We computed the [Formula: see text]-number for a family of certain Picard curves using the action of the Cartier operator on [Formula: see text].

Hurwitz moduli varieties parameterizing Galois covers of an algebraic curve

Given a smooth, projective curve \(Y\), a finite group \(G\) and a positive integer $n$ we study smooth, proper families \(X\to Y\times S\to S\) of Galois covers of \(Y\) with Galois group isomorphic to $G$ branched in \(n\) points, parameterized by algebraic varieties \(S\). When \(G\) is with trivial center we prove that the Hurwitz space \(H^G_n(Y)\) is a fine moduli variety for this moduli problem and construct explicitly the universal family. For arbitrary \(G\) we prove that \(H^G_n(Y)\) is a coarse moduli variety. For families of pointed Galois covers of \((Y,y_0)\) we prove that the Hurwitz space \(H^G_n(Y,y_0)\) is a fine moduli variety, and construct explicitly the universal family, for arbitrary group \(G\). We use classical tools of algebraic topology and of complex algebraic geometry.

Characterization of the Riccati and Abel Polynomial Differential Systems Having Invariant Algebraic Curves

The Riccati polynomial differential systems are differential systems of the form [Formula: see text], [Formula: see text], where [Formula: see text] and [Formula: see text] for [Formula: see text] are polynomial functions. We characterize all the Riccati polynomial differential systems having an invariant algebraic curve. We show that the coefficients of the first four highest degree terms of the polynomial in the variable [Formula: see text] defining the invariant algebraic curve determine completely the Riccati differential system. A similar result is obtained for any Abel polynomial differential system.