scholarly journals Bielliptic smooth plane curves and quadratic points

2020 ◽  
pp. 1-20
Author(s):  
Eslam Badr ◽  
Francesc Bars
Keyword(s):  
Automorphism Group ◽  
Number Field ◽  
Plane Curve ◽  
Plane Curves ◽  
Extra Condition ◽  
Isomorphism Classes ◽  
Smooth Projective Plane ◽  
Quadratic Extensions ◽  
Smooth Plane ◽  
Quartic Curves

Let [Formula: see text] be a smooth plane curve of degree [Formula: see text] defined over a global field [Formula: see text] of characteristic [Formula: see text] or [Formula: see text] (up to an extra condition on [Formula: see text]). Unless the curve is bielliptic of degree four, we observe that it always admits finitely many quadratic points. We further show that there are only finitely many quadratic extensions [Formula: see text] when [Formula: see text] is a number field, in which we may have more points of [Formula: see text] than these over [Formula: see text]. In particular, we have this asymptotic phenomenon valid for Fermat’s and Klein’s equations. Second, we conjecture that there are two infinite sets [Formula: see text] and [Formula: see text] of isomorphism classes of smooth projective plane quartic curves over [Formula: see text] with a prescribed automorphism group, such that all members of [Formula: see text] (respectively [Formula: see text]) are bielliptic and have finitely (respectively infinitely) many quadratic points over a number field [Formula: see text]. We verify the conjecture over [Formula: see text] for [Formula: see text] and [Formula: see text]. The analog of the conjecture over global fields with [Formula: see text] is also considered.

scholarly journals The kernel of the monodromy of the universal family of degree d smooth plane curves

2020 ◽  
pp. 1-15
Author(s):  
Reid Monroe Harris
Keyword(s):  
Plane Curve ◽  
Teichmüller Space ◽  
Mapping Class ◽  
Plane Curves ◽  
Teichmuller Space ◽  
Infinite Rank ◽  
Universal Family ◽  
Smooth Plane ◽  
Quartic Curves ◽  
Countably Infinite

We consider the parameter space [Formula: see text] of smooth plane curves of degree [Formula: see text]. The universal smooth plane curve of degree [Formula: see text] is a fiber bundle [Formula: see text] with fiber diffeomorphic to a surface [Formula: see text]. This bundle gives rise to a monodromy homomorphism [Formula: see text], where [Formula: see text] is the mapping class group of [Formula: see text]. The main result of this paper is that the kernel of [Formula: see text] is isomorphic to [Formula: see text], where [Formula: see text] is a free group of countably infinite rank. In the process of proving this theorem, we show that the complement [Formula: see text] of the hyperelliptic locus [Formula: see text] in Teichmüller space [Formula: see text] has the homotopy type of an infinite wedge of spheres. As a corollary, we obtain that the moduli space of plane quartic curves is aspherical. The proofs use results from the Weil–Petersson geometry of Teichmüller space together with results from algebraic geometry.

Twists of non-hyperelliptic curves of genus 3

2018 ◽  
Vol 14 (06) ◽  
pp. 1785-1812 ◽  
Author(s):  
Elisa Lorenzo García
Keyword(s):  
Automorphism Group ◽  
Number Field ◽  
Hyperelliptic Curves ◽  
Starting Point ◽  
Smooth Plane ◽  
Genus Formula ◽  
Quartic Curves

In this paper, we compute explicit equations for the twists of all the smooth plane quartic curves defined over a number field [Formula: see text]. Since the plane quartic curves are non-hyperelliptic curves of genus [Formula: see text] we can apply the method developed by the author in a previous paper. The starting point is a classification due to Henn of the plane quartic curves with non-trivial automorphism group up to [Formula: see text]-isomorphism.

THE LOCUS OF SMOOTH PLANE CURVES WITH A SEXTACTIC POINT

2013 ◽  
Vol 13 (01) ◽  
pp. 1350079
Author(s):  
M. A. FARAHAT
Keyword(s):  
Moduli Space ◽  
Plane Curves ◽  
Smooth Curves ◽  
Isomorphism Classes ◽  
Smooth Plane ◽  
Quartic Curves ◽  
Appl Math

Let Mg be the moduli space of isomorphism classes of genus g smooth curves over ℂ. We show that the locus S2d-r ⊂ Mg whose general points represent smooth plane curves of degree d ≥ 4 with a sextactic point of sextactic order 2d - r, where r ∈ {0, 1, 2}, is an irreducible and rational subvariety of codimension d(d - 4) + 2 - r of Mg. These results generalize those results introduced by the author in case of quartic curves (see K. Alwaleed and M. Farahat, The locus of smooth quartic curves with a sextactic point, Appl. Math. Inf. Sci.7(2) (2013) 509–513).

scholarly journals Sharp Bertini Theorem for Plane Curves over Finite Fields

2019 ◽  
Vol 62 (02) ◽  
pp. 223-230 ◽  
Author(s):  
Shamil Asgarli
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Plane Curve ◽  
Plane Curves ◽  
Smooth Plane ◽  
Curves Over Finite Fields

AbstractWe prove that if $C$ is a reflexive smooth plane curve of degree $d$ defined over a finite field $\mathbb{F}_{q}$ with $d\leqslant q+1$ , then there is an $\mathbb{F}_{q}$ -line $L$ that intersects $C$ transversely. We also prove the same result for non-reflexive curves of degree $p+1$ and $2p+1$ when $q=p^{r}$ .

scholarly journals On plane curves given by separated polynomials and their automorphisms

2020 ◽  
Vol 20 (1) ◽  
pp. 61-70
Author(s):  
Matteo Bonini ◽  
Maria Montanucci ◽  
Giovanni Zini
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Algebraic Closure ◽  
Sufficient Conditions ◽  
Plane Curve ◽  
Plane Curves ◽  
Prime Field ◽  
Full Automorphism Group ◽  
Ag Codes ◽  
Finite Prime Field

AbstractLet 𝓒 be a plane curve defined over the algebraic closure K of a finite prime field 𝔽p by a separated polynomial, that is 𝓒 : A(Y) = B(X), where A(Y) is an additive polynomial of degree pn and the degree m of B(X) is coprime with p. Plane curves given by separated polynomials are widely studied; however, their automorphism groups are not completely determined. In this paper we compute the full automorphism group of 𝓒 when m ≢ 1 mod pn and B(X) = Xm. Moreover, some sufficient conditions for the automorphism group of 𝓒 to imply that B(X) = Xm are provided. Also, the full automorphism group of the norm-trace curve 𝓒 : X(qr – 1)/(q–1) = Yqr–1 + Yqr–2 + … + Y is computed. Finally, these results are used to show that certain one-point AG codes have many automorphisms.

CONSTRUCTION OF THE MULTI-WAVELETS ON SOME SMOOTH PLANE CURVES VIA LENGTH-PRESERVING PROJECTION

2013 ◽  
Vol 12 (01) ◽  
pp. 1450005 ◽  
Author(s):  
BAOQIN WANG ◽  
GANG WANG ◽  
XIAOHUI ZHOU
Keyword(s):  
Scaling Function ◽  
Plane Curve ◽  
Reconstruction Algorithm ◽  
Translation Operator ◽  
Plane Curves ◽  
Dilation Operator ◽  
Refinement Equations ◽  
Multi Resolution Analysis ◽  
Smooth Plane ◽  
Resolution Analysis

Based on the theory of the discrete multi-wavelets in the space L2(R), the theory of the discrete multi-wavelets in the space L2(C) is presented properly in this paper, where C denotes a smooth plane curve. Firstly, the length-preserving projection is constructed, and by the length-preserving projection, the multiplicity multi-resolution analysis in the space L2(C) is defined properly and we define the dilation operator and translation operator in the space L2(C). Then, the two-scale refinement equations of multi-scaling function and multi-wavelet in the space L2(C) is deduced by using length-preserving mapping, the orthogonality is discussed, and the decomposition and reconstruction algorithm is computed. Finally, the example is given.

scholarly journals Higher rank Clifford indices of curves on a K3 surface

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Soheyla Feyzbakhsh ◽  
Chunyi Li
Keyword(s):  
Vector Bundles ◽  
Plane Curve ◽  
Smooth Curve ◽  
Line Bundles ◽  
K3 Surface ◽  
Clifford Index ◽  
Smooth Plane ◽  
Higher Rank ◽  
Rank Vector ◽  
Semistable Vector Bundle

AbstractLet (X, H) be a polarized K3 surface with $$\mathrm {Pic}(X) = \mathbb {Z}H$$ Pic ( X ) = Z H , and let $$C\in |H|$$ C ∈ | H | be a smooth curve of genus g. We give an upper bound on the dimension of global sections of a semistable vector bundle on C. This allows us to compute the higher rank Clifford indices of C with high genus. In particular, when $$g\ge r^2\ge 4$$ g ≥ r 2 ≥ 4 , the rank r Clifford index of C can be computed by the restriction of Lazarsfeld–Mukai bundles on X corresponding to line bundles on the curve C. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher rank vector bundles. We also apply the same method to the projective plane and show that the rank r Clifford index of a degree $$d(\ge 5)$$ d ( ≥ 5 ) smooth plane curve is $$d-4$$ d - 4 , which is the same as the Clifford index of the curve.

scholarly journals Connected Numbers and the Embedded Topology of Plane Curves

2018 ◽  
Vol 61 (3) ◽  
pp. 650-658 ◽  
Author(s):  
Taketo Shirane
Keyword(s):  
Plane Curve ◽  
Smooth Curve ◽  
Plane Curves ◽  
Irreducible Curve ◽  
Galois Cover ◽  
Splitting Number

AbstractThe splitting number of a plane irreducible curve for a Galois cover is effective in distinguishing the embedded topology of plane curves. In this paper, we define the connected number of a plane curve (possibly reducible) for a Galois cover, which is similar to the splitting number. By using the connected number, we distinguish the embedded topology of Artal arrangements of degree b ≥ 4, where an Artal arrangement of degree b is a plane curve consisting of one smooth curve of degree b and three of its total inflectional tangents.

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