Nonsingular Algebraic Curve Research Articles

Operator index of a nonsingular algebraic curve

Operator index of a nonsingular algebraic curve

Algebraic surfaces determine analyticity of functions

Let f :X rightarrow mathbb {R} be a function defined on a nonsingular real algebraic set X of dimension at least 3. We prove that f is an analytic (resp. a Nash) function whenever the restriction f|_{S} is an analytic (resp. a Nash) function for every nonsingular algebraic surface S subset X whose each connected component is homeomorphic to the unit 2-sphere. Furthermore, the surfaces S can be replaced by compact nonsingular algebraic curves in X, provided that dimX ge 2 and f is of class mathcal {C}^{infty }.

Аналитические и теоретико-числовые свойства двумерных сигма-функций

Обзор посвящен классическим и современным задачам, связанным с целой функцией $\sigma({\bf u};\lambda)$, которая определяется семейством неособых алгебраических кривых рода 2, где ${\bf u}= (u_1,u_3)$, $\lambda=(\lambda_4, \lambda_6, \lambda_8, \lambda_{10})$. Эта функция является аналогом сигма-функции Вейерштрасса $\sigma({{u}};g_2,g_3)$ семейства эллиптических кривых. Логарифмические производные порядка 2 и выше функции ${\sigma({\mathbf{u}};\lambda)}$ порождают поле гиперэллиптических функций от ${\mathbf{u}} = (u_1,u_3)$ на якобианах кривых с фиксированным значением вектора параметров $\lambda$. Мы рассматриваем три ряда Гурвица $\sigma({\mathbf{u}};\lambda)=\sum_{m,n\ge0}a_{m,n}(\lambda)\frac{u_1^mu_3^n}{m!n!}$, $\sigma({\mathbf{u}};\lambda) = \sum_{k\ge 0}\xi_k(u_1;\lambda)\frac{u_3^k}{k!}$ и $\sigma({\mathbf{u}};\lambda) = \sum_{k\ge 0}\mu_k(u_3;\lambda)\frac{u_1^k}{k!}$. Обзор посвящен теоретико-числовым свойствам функций $a_{m,n}(\lambda)$, $\xi_k(u_1;\lambda)$ и $\mu_k(u_3;\lambda)$. Он включает самые последние результаты, доказательства которых использует тот фундаментальный факт, что функция $\sigma ({\mathbf{u}};\lambda)$ определяется системой четырех уравнений теплопроводности в неголономном репере шестимерного пространства.

On the disposition of cubic and pair of conics in a real projective plane

In the first part of the 16th Hilbert problem the question about the topology of nonsingular projective algebraic curves and surfaces was formulated. The problem on topology of algebraic manifolds with singularities belong to this subject too. The particular case of this problem is the study of curves that are decompozable into the product of curves in a general position. This paper deals with the problem of topological classification of mutual positions of a nonsingular curve of degree three and two nonsingular curves of degree two in the real projective plane. Additiolal conditions for this problem include general position of the curves and its maximality; in particular, the number of common points for each pair of curves-factors reaches its maximum. It is proved that the classification contains no more than six specific types of positions of the species under study. Four position types are built, and the question of realizability of the two remaining ones is open.

Criteria for Algebraicity of Analytic Functions

Abstract We consider functions defined on an open subset of a nonsingular, either real or complex, algebraic set. We give criteria for an analytic function to be a Nash (resp. regular, resp. polynomial) function. Our criteria depend only on the behavior of such a function along irreducible nonsingular algebraic curves passing through a given point. In the proofs we use results on algebraicity of formal power series, which are also established in this paper.

Polynomial non-integrability of magnetic billiards on the sphere and the hyperbolic plane

Magnetic billiards in a convex domain with smooth boundary on a constant-curvature surface in a constant magnetic field is considered in this paper. The question of the existence of an integral of motion which is a polynomial in the components of the velocity is investigated. It is shown that if such an integral exists, then the boundary of the domain defines a non-singular algebraic curve in . It is also shown that for a domain other than a geodesic disk, magnetic billiards does not admit a polynomial integral for all but perhaps finitely many values of the magnitude of the magnetic field. To prove our main theorems a new dynamical system, ‘outer magnetic billiards’, on a constant-curvature surface is introduced, a system ‘dual’ to magnetic billiards. By passing to this dynamical system one can apply methods of algebraic geometry to magnetic billiards. Bibliography: 30 titles.

Bounds for the number of points on curves over finite fields

Let $$\mathcal{X}$$ be a projective irreducible nonsingular algebraic curve defined over a finite field $$\mathbb{F}_q$$ . This paper presents a variation of the Stohr–Voloch theory and sets new bounds to the number of $${F_{{q^r}}}$$ -rational points on X. In certain cases, where comparison is possible, the results are shown to improve other bounds such as Weil’s, Stohr–Voloch’s and Ihara’s.

Linear series on metrized complexes of algebraic curves

A metrized complex of algebraic curves over an algebraically closed field \(\kappa \) is, roughly speaking, a finite metric graph \(\Gamma \) together with a collection of marked complete nonsingular algebraic curves \(C_v\) over \(\kappa \), one for each vertex \(v\) of \(\Gamma \); the marked points on \(C_v\) are in bijection with the edges of \(\Gamma \) incident to \(v\). We define linear equivalence of divisors and establish a Riemann–Roch theorem for metrized complexes of curves which combines the classical Riemann–Roch theorem over \(\kappa \) with its graph-theoretic and tropical analogues from Amini and Caporaso (Adv Math 240:1–23, 2013); Baker and Norine (Adv Math 215(2):766–788, 2007); Gathmann and Kerber (Math Z 259(1):217–230, 2008) and Mikhalkin and Zharkov (Tropical curves, their Jacobians and Theta functions. Contemporary Mathematics 203–231, 2007), providing a common generalization of all of these results. For a complete nonsingular curve \(X\) defined over a non-Archimedean field \(\mathbb {K}\), together with a strongly semistable model \(\mathfrak {X}\) for \(X\) over the valuation ring \(R\) of \(\mathbb {K}\), we define a corresponding metrized complex \(\mathfrak {C}\mathfrak {X}\) of curves over the residue field \(\kappa \) of \(\mathbb {K}\) and a canonical specialization map \(\tau ^{\mathfrak {C}\mathfrak {X}}_*\) from divisors on \(X\) to divisors on \(\mathfrak {C}\mathfrak {X}\) which preserves degrees and linear equivalence. We then establish generalizations of the specialization lemma from Baker (Algebra Number Theory 2(6):613–653, 2008) and its weighted graph analogue from Amini and Caporaso (Adv Math 240:1–23, 2013), showing that the rank of a divisor cannot go down under specialization from \(X\) to \(\mathfrak {C}\mathfrak {X}\). As an application, we establish a concrete link between specialization of divisors from curves to metrized complexes and the theory of limit linear series due to Eisenbud and Harris (Invent Math 85:337–371, 1986). Using this link, we formulate a generalization of the notion of limit linear series to curves which are not necessarily of compact type and prove, among other things, that any degeneration of a \(\mathfrak {g}^r_d\) in a regular family of semistable curves is a limit \(\mathfrak {g}^r_d\) on the special fiber.

SIGMA FUNCTIONS FOR TELESCOPIC CURVES

In this paper we consider a symplectic basis of the first cohomology group and the sigma functions for algebraic curves expressed by a canonical form using a finite sequence $(a_{1}, \ldots, a_{t})$ of positive integers whose greatest common divisor is equal to one (Miura [13]). The idea is to express a non-singular algebraic curve by affine equations of $t$ variables whose orders at infinity are $(a_{1}, \ldots, a_{t})$. We construct a symplectic basis of the first cohomology group and the sigma functions for telescopic curves, i.e., the curves such that the number of defining equations is exactly $t-1$ in the Miura canonical form. The largest class of curves for which such construction has been obtained thus far is $(n, s)$-curves ([4] [15]), which are telescopic because they are expressed in the Miura canonical form with $t=2$, $a_{1}=n$, and $a_{2}=s$, and the number of defining equations is one.

Characterization of the Fermat curve as the most symmetric nonsingular algebraic plane curve

A projective nonsingular plane algebraic curve of degree \(d\ge 4\) is called maximally symmetric if it attains the maximum order of the automorphism groups for complex nonsingular plane algebraic curves of degree \(d\). For \(d\le 7\), all such curves are known. Up to projectivities, they are the Fermat curve for \(d=5,7\); see Kaneta et al. (RIMS Kokyuroku 1109:182–191, 1999) and Kaneta et al. (Geom. Dedic. 85:317–334, 2001), the Klein quartic for \(d=4\), see Hartshorne (Algebraic Geometry. Springer, New York, 1977), and the Wiman sextic for \(d=6\); see Doi et al. (Osaka J. Math. 37:667–687, 2000). In this paper we work on projective plane curves defined over an algebraically closed field of characteristic zero, and we extend this result to every \(d\ge 8\) showing that the Fermat curve is the unique maximally symmetric nonsingular curve of degree \(d\) with \(d\ge 8\), up to projectivity. For \(d=11,13,17,19\), this characterization of the Fermat curve has already been obtained; see Kaneta et al. (Geom. Dedic. 85:317–334, 2001).

Adaptive Isotopic Approximation of Nonsingular Curves: the Parameterizability and Nonlocal Isotopy Approach

We consider domain subdivision algorithms for computing isotopic approximations of a nonsingular algebraic curve. The curve is given by a polynomial equation f(X,Y)=0. Two algorithms in this area are from Snyder (1992) SIGGRAPH Comput. Graphics, 26(2), 121 and Plantinga and Vegter (2004) In Proc. Eurographics Symposium on Geometry Processing, pp. 245–254. We introduce a new algorithm that combines the advantages of these two algorithms: like Snyder, we use the parameterizability criterion for subdivision, and like Plantinga and Vegter, we exploit nonlocal isotopy. We further extend our algorithm in two important and practical directions: first, we allow subdivision cells to be rectangles with arbitrary but bounded aspect ratios. Second, we extend the input domains to be regions R 0 with arbitrary geometry and which might not be simply connected. Our algorithm halts as long as the curve has no singularities in the region, and intersects the boundary of R 0 transversally. Our algorithm is practical and easy to implement exactly. We report some very encouraging experimental results, showing that our algorithms can be much more efficient than the algorithms of Plantinga–Vegter and Snyder.

On the 16th Hilbert problem for limit cycles on nonsingular algebraic curves

We give an upper bound for the maximum number N of algebraic limit cycles that a planar polynomial vector field of degree n can exhibit if the vector field has exactly k nonsingular irreducible invariant algebraic curves. Additionally we provide sufficient conditions in order that all the algebraic limit cycles are hyperbolic. We also provide lower bounds for N.

Mild manifolds and a non-standard Riemann existence theorem

Let \({\mathcal{R}}\) be an o-minimal expansion of a real closed field R, and K be the algebraic closure of R. In earlier papers we investigated the notions of \({\mathcal{R}}\)-definable K-holomorphic maps, K-analytic manifolds and their K-analytic subsets. We call such a K-manifold mild if it eliminates quantifers after endowing it with all it K-analytic subsets. Examples are compact complex manifolds and non-singular algebraic curves over K.

Polynomial parametrization of nonsingular complete intersection curves

In this paper we give a new method to compute a polynomial parametrization, in case it exists, of an affine nonsingular complete intersection curve. Our method is based on the study of vector fields on nonsingular algebraic curves. In contrast to the existing methods, our algorithm does not use any projection, and no sample point on the curve is needed. It is also important to stress the fact that the algorithm produces a parametrization with coefficients in the ground field.

The Automorphism Group of Plane Algebraic Curves with Singer Automorphisms

The main result in Cossidente and Siciliano (J. Number Theory, Vol. 99 (2003) pp. 373---382) states that if a Singer subgroup of PGL(3,q) is an automorphism group of a projective, geometric irreducible, non-singular plane algebraic curve $$\mathcal{X}$$ then either $$\deg(\mathcal{X})=q+2$$ or $$\deg(\mathcal {X})\ge q^2+q+1$$ . In the former case $$\mathcal{X}$$ is projectively equivalent to the curve $$\mathcal{X}_q$$ with equation Xq+1Y+Yq+1+X=0 studied by Pellikaan. Furthermore, the curve $$\mathcal{X}_q$$ has a very nice property from Finite Geometry point of view: apart from the three distinguished points fixed by the Singer subgroup, the set of its $$\mathbb{F}_{{q}^{3}}$$ -rational points can be partitioned into finite projective planes $$P^{2}(\mathbb{F}_{q})$$ . In this paper, the full automorphism group of such curves is determined. It turns out that $$Aut(\mathcal {X}_q)$$ is the normalizer of a Singer group in $$PGL(3,\mathbb{F}_{q})$$ .

Real plane algebraic curves with asymptotically maximal number of even ovals

It has been known for a long time that a nonsingular real algebraic curve of degree 2k in the projective plane cannot have more than 7k2/4-9k/4+3/2 even ovals. We show here that this upper bound is asymptotically sharp; that is to say, we construct a family of curves of degree 2k such that p/k2→k→∞7/4, where p is the number of even ovals of the curves. We also show that the same kind of result is valid when dealing with odd ovals

Deformations of the generalised Picard bundle

Let X be a nonsingular algebraic curve of genus g ⩾ 3 , and let M ξ denote the moduli space of stable vector bundles of rank n ⩾ 2 and degree d with fixed determinant ξ over X such that n and d are coprime. We assume that if g = 3 then n ⩾ 4 and if g = 4 then n ⩾ 3 , and suppose further that n 0 , d 0 are integers such that n 0 ⩾ 1 and nd 0 + n 0 d > nn 0 ( 2 g - 2 ) . Let E be a semistable vector bundle over X of rank n 0 and degree d 0 . The generalised Picard bundle W ξ ( E ) is by definition the vector bundle over M ξ defined by the direct image p M ξ * ( U ξ ⊗ p X * E ) where U ξ is a universal vector bundle over X × M ξ . We obtain an inversion formula allowing us to recover E from W ξ ( E ) and show that the space of infinitesimal deformations of W ξ ( E ) is isomorphic to H 1 ( X , End ( E ) ) . This construction gives a locally complete family of vector bundles over M ξ parametrised by the moduli space M ( n 0 , d 0 ) of stable bundles of rank n 0 and degree d 0 over X. If ( n 0 , d 0 ) = 1 and W ξ ( E ) is stable for all E ∈ M ( n 0 , d 0 ) , the construction determines an isomorphism from M ( n 0 , d 0 ) to a connected component M 0 of a moduli space of stable sheaves over M ξ . This applies in particular when n 0 = 1 , in which case M 0 is isomorphic to the Jacobian J of X as a polarised variety. The paper as a whole is a generalisation of results of Kempf and Mukai on Picard bundles over J, and is also related to a paper of Tyurin on the geometry of moduli of vector bundles.

On the fundamental groups of the complements of Hurwitz curves

It is proved that the commutator subgroup of the fundamental group of the complement of any plane affine irreducible Hurwitz curve (or any plane affine irreducible pseudoholomorphic curve) is finitely presented. It is shown that there is a Hurwitz curve (resp. pseudoholomorphic curve) in such that the fundamental group of its complement is non-Hopfian and, therefore, this group is not residually finite. We also prove the existence of an irreducible non-singular algebraic curve and a bidisc such that the fundamental group is non-Hopfian.

On the number of rational points of a plane algebraic curve

The number of Fq -rational points of a plane non-singular algebraic curve \( \cal X \) defined over a finite field Fq is computed, provided that the generic point of \( \cal X \) is not an inflexion and that \( \cal X \) is Frobenius non-classical with respect to conics.

Symmetric Orientations of Dividing T-curves

In 1974, Rokhlim introduced complex orientations for nonsingular real algebraic plane projective curves of type I. Here we give a definition of symmetric orientations and of "type" for T-curves which are PL-curves constructed using a combinatorial method called T-construction. An important aspect of T-construction is that, under particular conditions, the constructed T-curve has the isotopy type of a nonsingular real algebraic plane projective curve. T-construction is in fact a particular case of the method of construction of real algebraic projective varieties due to O. Ya. Viro. We prove that if an algebraic curve is associated to a T-curve by the Viro process, then the type of the T-curve coincides with the type of the algebraic curve and its symmetric orientations are complex orientations as defined by Rokhlin. The main result of this paper is the classification theorem for T-curves of type I.