Nonsingular Curve Research Articles

Galois subcovers of the Hermitian curve in characteristic p with respect to subgroups of order p2

A (projective, geometrically irreducible, non-singular) curve X defined over a finite field Fq2 is maximal if the number Nq2 of its Fq2-rational points attains the Hasse-Weil upper bound, that is Nq2=q2+2gq+1 where g is the genus of X. An important question, also motivated by applications to algebraic-geometry codes, is to find explicit equations for maximal curves. For a few curves which are Galois covered of the Hermitian curve, this has been done so far ad hoc, in particular in the cases where the Galois group has prime order. In this paper we obtain explicit equations of all Galois covers of the Hermitian curve with Galois group of order p2 where p is the characteristic of Fq2. Doing so we also determine the Fq2-isomorphism classes of such curves and describe their full Fq2-automorphism groups.

Brill–Noether Theory of Stable Vector Bundles on Ruled Surfaces

Let X be a ruled surface over a nonsingular curve C of genus g≥0.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g\\ge 0.$$\\end{document} Let MH:=MX,H(2;c1,c2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_H:=M_{X,H}(2;c_1,c_2)$$\\end{document} be the moduli space of H-stable rank 2 vector bundles E on X with fixed Chern classes ci:=ci(E)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c_i:=c_i(E)$$\\end{document} for i=1,2.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$i=1,2.$$\\end{document} The main goal of this paper is to contribute to a better understanding of the geometry of the moduli space MH\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_H$$\\end{document} in terms of its Brill–Noether locus WHk(2;c1,c2),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_H^k(2;c_1,c_2),$$\\end{document} whose points correspond to stable vector bundles in MH\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_H$$\\end{document} having at least k independent sections. We deal with the non-emptiness of this Brill–Noether locus, getting in most of the cases sharp bounds for the values of k such that WHk(2;c1,c2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_H^k(2;c_1,c_2)$$\\end{document} is non-empty.

Primes of the form $X^{3} + NY^{3}$ and a family of non-singular plane curves which violate the local-global principle

Primes of the form $X^{3} + NY^{3}$ and a family of non-singular plane curves which violate the local-global principle

Characterization of Lipschitz normally embedded complex curves

The main result of the paper states that a connected complex affine algebraic curve is LNE in Cn if and only if its germ at any singular point is a finite union of non-singular complex curve germs which are pairwise transverse, and its projective closure is in general position with the hyperplane at infinity. To this aim, we completely characterize complex analytic curves of a compact complex manifold which are LNE (regardless of the given Riemannian structure), and therefore we can relate when a projective algebraic curve is LNE in CPn with the property of its general affine traces being LNE in Cn. This allows us to deduce that Lipschitz classification of LNE curves is topological. We describe a complete invariant for the (outer and inner) Lipschitz equivalence of affine LNE curves and show that most such curves cannot be bi-Lipschitz homeomorphic to a plane one.

Operator index of a nonsingular algebraic curve

Operator index of a nonsingular algebraic curve

Optimal plane curves of degree q − 1 over a finite field

Let q≥5 be a prime power. In this note, we prove that if a plane curve X of degree q−1 defined over Fq without Fq-linear components attains the Sziklai upper bound (d−1)q+1=(q−1)2 for the number of its Fq-rational points, then X is projectively equivalent over Fq to the curve C(α,β,γ):αXq−1+βYq−1+γZq−1=0 for some α,β,γ∈Fq⁎ such that α+β+γ=0. This completes the classification of curves that are extremal with respect to the Sziklai bound. Also, since the Sziklai bound is equal to the Stöhr-Voloch's bound for plane curves of degree q−1, our main result classifies the Fq-Frobenius extremal classical nonsingular plane curves of degree q−1.

On the algebra of elliptic curves

AbstractIt is argued that a nonsingular elliptic curve admits a natural or fundamental abelian heap structure uniquely determined by the curve itself. It is shown that the set of complex analytic or rational functions from a nonsingular elliptic curve to itself is a truss arising from endomorphisms of this heap.

A visual representation of the Steiner triple systems of order 13

Steiner triple systems (STSs) are a basic topic in combinatorics. In an STS the elements can be collected in threes in such a way that any pair of elements is contained in a unique triple. The two smallest nontrivial STSs, with 7 and 9 elements, arise in the context of finite geometry and nonsingular cubic curves, and have well-known pictorial representations. On the contrary, an STS with 13 elements does not have an intrinsic geometric nature, nor a natural pictorial illustration. In this paper we present a visual representation of the two non-isomorphic Steiner triple systems of order 13 by means of a regular hexagram. The thirteen points of each system are the vertices of the twelve equilateral triangles inscribed in the hexagram. In the case of the non-cyclic system, our representation allows one to visualize in a simple, elegant and highly symmetric way the twenty-six triples, the six automorphisms and their orbits, the eight quadrilaterals, the ten mitres, the thirteen grids, the four 3-colouring patterns, the block-colouring and some distinguished ovals. Our construction is based on a very simple idea (seeing the blocks as much as possible as equilateral triangles), which can be further extended to get new representations of the STSs of order 7 and 9, and of one of the STSs of order 15.

Maximality of moduli spaces of vector bundles on curves

We prove that moduli spaces of semistable vector bundles of coprime rank and degree over a non-singular real projective curve are maximal real algebraic varieties if and only if the base curve itself is maximal. This provides a new family of maximal varieties, with members of arbitrarily large dimension. We prove the result by comparing the Betti numbers of the real locus to the Hodge numbers of the complex locus and showing that moduli spaces of vector bundles over a maximal curve actually satisfy a property which is stronger than maximality and that we call Hodge-expressivity. We also give a brief account on other varieties for which this property was already known.

Arnol’d’s type theorem on a neighborhood of a cycle of rational curves

Arnol’d showed the uniqueness of the complex analytic structure of a small neighborhood of a non-singular elliptic curve embedded in a non-singular surface whose normal bundle satisfies a Diophantine condition in the Picard variety. We show an analogue of this theorem for a neighborhood of a cycle of rational curves.

Globally generated vector bundles on and the unirationality of

Chang and Ran proved, in 1984, the unirationality of the moduli spaces of curves of genus 11, 12, and 13. They consider a family U of nonsingular space curves. If U is non-empty, it dominates the moduli space. Considering certain vector bundles related to the curves, one shows that U is unirational. Finally, Chang and Ran prove that U is non-empty by constructing smoothable reducible curves. We prove, instead, the existence of (almost) globally generated vector bundles of the above kind. The dependency locus of (rank − 1) general global sections of such a bundle is a nonsingular curve from U.

Varieties of minimal rational tangents of unbendable rational curves subordinate to contact structures

A nonsingular rational curve $C$ in a complex manifold $X$ whose normal bundle is isomorphic to $\mathcal{O}_{\mathbb{P}^1}(1)^{\oplus p} \oplus \mathcal{O}_{\mathbb{P}^1}^{\oplus q}$ for some nonnegative integers $p$ and $q$ is called an unbendable rational curve on $X$. Associated with it is the variety of minimal rational tangents (VMRT) at a point $x \in C$, which is the germ of submanifolds $\mathcal{C}^C_{x} \subset \mathbb{P} T_{x} X$ consisting of tangent directions of small deformations of $C$ fixing $x$. Assuming that there exists a distribution $D \subset TX$ such that all small deformations of $C$ are tangent to $D$, one asks what kind of submanifolds of projective space can be realized as the VMRT $\mathcal{C}^{C}_{x} \subset \mathbb{P} D_{x}$. When $D \subset TX$ is a contact distribution, a well-known necessary condition is that $\mathcal{C}_{x}^{C}$ should be Legendrian with respect to the induced contact structure on $\mathbb{P} D_{x}$. We prove that this is also a sufficient condition: we construct a complex manifold $X$ with a contact structure $D \subset TX$ and an unbendable rational curve $C \subset X$ such that all small deformations of $C$ are tangent to $D$ and the VMRT $\mathcal{C}^C_{x} \subset \mathbb{P} D_{x}$ at some point $x \in C$ is projectively isomorphic to an arbitrarily given Legendrian submanifold. Our construction uses the geometry of contact lines on the Heisenberg group and a technical ingredient is the symplectic geometry of distributions the study of which has originated from geometric control theory.

Viro–Zvonilov Inequalities for Flexible Curves on an Almost Complex Four-Dimensional Manifold

The restrictions on the topology of nonsingular plane projective real algebraic curves of odd degree, obtained by O. Viro and the author in the paper published in the early 90s, are extended to flexible curves lying on an almost complex four-dimensional manifold. Some examples of real algebraic surfaces and real curves on them prove the sharpness of the obtained inequalities. In addition, it is proved that a compact Lie group smooth action can be lifted to a cyclic branched covering space $ \tilde{X} $ over a closed four-dimensional manifold, and a sufficient condition for $ H_1(\tilde{X})=0 $ was found.

Counterexamples to the local-global principle for non-singular plane curves and a cubic analogue of Ankeny-Artin-Chowla-Mordell conjecture

In this article, we introduce a systematic and uniform construction of non-singular plane curves of odd degrees n ≥ 5 n \geq 5 which violate the local-global principle. Our construction works unconditionally for n n divisible by p 2 p^{2} for some odd prime number p p . Moreover, our construction also works for n n divisible by some p ≥ 5 p \geq 5 which satisfies a conjecture on a p p -adic property of the fundamental unit of Q ( p 1 / 3 ) \mathbb {Q}(p^{1/3}) and Q ( ( 2 p ) 1 / 3 ) \mathbb {Q}((2p)^{1/3}) . This conjecture is a natural cubic analogue of the classical Ankeny-Artin-Chowla-Mordell conjecture for Q ( p 1 / 2 ) \mathbb {Q}(p^{1/2}) and easily verified numerically.

Quot schemes of curves and surfaces: virtual classes, integrals, Euler characteristics

We compute tautological integrals over Quot schemes on curves and surfaces. After obtaining several explicit formulas over Quot schemes of dimension 0 quotients on curves (and finding a new symmetry), we apply the results to tautological integrals against the virtual fundamental classes of Quot schemes of dimension 0 and 1 quotients on surfaces (using also universality, torus localization, and cosection localization). The virtual Euler characteristics of Quot schemes of surfaces, a new theory parallel to the Vafa-Witten Euler characteristics of the moduli of bundles, is defined and studied. Complete formulas for the virtual Euler characteristics are found in the case of dimension 0 quotients on surfaces. Dimension 1 quotients are studied on K3 surfaces and surfaces of general type with connections to the Kawai-Yoshioka formula and the Seiberg-Witten invariants respectively. The dimension 1 theory is completely solved for minimal surfaces of general type admitting a nonsingular canonical curve. Along the way, we find a new connection between weighted tree counting and multivariate Fuss-Catalan numbers which is of independent interest.

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 }.

Decomposed Richelot isogenies of Jacobian varieties of curves of genus 3

For a non-singular projective curve C of genus 3 defined over an algebraically closed field of characteristic p≠2, we give a necessary and sufficient condition that the Jacobian variety J(C) has a decomposed Richelot isogeny outgoing from it and we determine the structures of decomposed ones.

Chow group of 1-cycles of the moduli of parabolic bundles over a curve

We study the Chow group of 1-cycles of the moduli space of semistable parabolic vector bundles of fixed rank, determinant and a generic weight over a nonsingular projective curve over \({\mathbb {C}}\) of genus at least 3. We show that, the Chow group of 1-cycles remains isomorphic as we vary the generic weight. As a consequence, we can give an explicit description of the Chow group in the case of rank 2 and determinant \({\mathcal {O}}(x)\), where \(x\in X\) is a fixed point, which extends the earlier result of Choe and Hwang (Math. Z. 253 (2006) 253–281, Main theorem).

Projective plane curves whose automorphism groups are simple and primitive

We study complex projective plane curves with a given group of automorphisms. Let $G$ be a simple primitive subgroup of $\mathrm{PGL}(3, \mathbf{C})$, which is isomorphic to $\mathfrak{A}_{6}$, $\mathfrak{A}_{5}$ or $\mathrm{PSL}(2, \mathbf{F}_{7})$. We obtain a necessary and sufficient condition on $d$ for the existence of a nonsingular projective plane curve of degree $d$ invariant under $G$. We also study an analogous problem on integral curves.

Algebraically unrealizable complex orientations of plane real pseudoholomorphic curves

We prove two inequalities for the complex orientations of a separating non-singular real algebraic curve in \({\mathbb {RP}}^2\) of any odd degree. We also construct a separating non-singular real (i.e., invariant under the complex conjugation) pseudoholomorphic curve in \({\mathbb {CP}}^2\) of any degree congruent to 9 mod 12 which does not satisfy one of these inequalities. Therefore the oriented isotopy type of the real locus of each of these curves is algebraically unrealizable.