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 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 Parametrizing the moduli space of curves and applications to smooth plane quartics over finite fields

2014 ◽  
Vol 17 (A) ◽  
pp. 128-147 ◽  
Author(s):  
Reynald Lercier ◽  
Christophe Ritzenthaler ◽  
Florent Rovetta ◽  
Jeroen Sijsling
Keyword(s):  
Finite Fields ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Moduli Space Of Curves ◽  
Isomorphism Classes ◽  
Families Of Curves ◽  
Plane Quartics ◽  
Smooth Plane

AbstractWe study new families of curves that are suitable for efficiently parametrizing their moduli spaces. We explicitly construct such families for smooth plane quartics in order to determine unique representatives for the isomorphism classes of smooth plane quartics over finite fields. In this way, we can visualize the distributions of their traces of Frobenius. This leads to new observations on fluctuations with respect to the limiting symmetry imposed by the theory of Katz and Sarnak.

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.

scholarly journals The Characteristic Numbers of Quartic Plane Curves

1999 ◽  
Vol 51 (5) ◽  
pp. 1089-1120 ◽  
Author(s):  
Ravi Vakil
Keyword(s):  
Moduli Space ◽  
Intersection Theory ◽  
Plane Curves ◽  
Stable Maps ◽  
Characteristic Numbers ◽  
Plane Quartics ◽  
Smooth Plane

AbstractThe characteristic numbers of smooth plane quartics are computed using intersection theory on a component of the moduli space of stable maps. This completes the verification of Zeuthen’s prediction of characteristic numbers of smooth plane curves. A short sketch of a computation of the characteristic numbers of plane cubics is also given as an illustration.

Random Dieudonné modules, random -divisible groups, and random curves over finite fields

2013 ◽  
Vol 12 (3) ◽  
pp. 651-676 ◽  
Author(s):  
Bryden Cais ◽  
Jordan S. Ellenberg ◽  
David Zureick-Brown
Keyword(s):  
Probability Distribution ◽  
Finite Field ◽  
Random Matrix ◽  
Hyperelliptic Curve ◽  
Hyperelliptic Curves ◽  
Plane Curves ◽  
Random Curves ◽  
Isomorphism Classes ◽  
Curves Over Finite Fields ◽  
Divisible Groups

AbstractWe describe a probability distribution on isomorphism classes of principally quasi-polarized $p$-divisible groups over a finite field $k$ of characteristic $p$ which can reasonably be thought of as a ‘uniform distribution’, and we compute the distribution of various statistics ($p$-corank, $a$-number, etc.) of $p$-divisible groups drawn from this distribution. It is then natural to ask to what extent the $p$-divisible groups attached to a randomly chosen hyperelliptic curve (respectively, curve; respectively, abelian variety) over $k$ are uniformly distributed in this sense. This heuristic is analogous to conjectures of Cohen–Lenstra type for $\text{char~} k\not = p$, in which case the random $p$-divisible group is defined by a random matrix recording the action of Frobenius. Extensive numerical investigation reveals some cases of agreement with the heuristic and some interesting discrepancies. For example, plane curves over ${\mathbf{F} }_{3} $ appear substantially less likely to be ordinary than hyperelliptic curves over ${\mathbf{F} }_{3} $.

scholarly journals A proof of Aronhold's theorem upon quartic curves

1940 ◽  
Vol 6 (3) ◽  
pp. 190-191
Author(s):  
H. W. Richmond
Keyword(s):  
Finite Number ◽  
Simple Proof ◽  
Plane Curves ◽  
Cubic Surface ◽  
Quartic Curves ◽  
Order Four ◽  
The Given

It is to be expected that a finite number of plane curves of order four should have seven given lines as bitangents, because the number of conditions imposed is equal to the number of effective free constants in the equation of such a curve, viz., 14. Aronhold made the interesting discovery that one curve could be determined in which no three of the given lines have their six points of contact on a conic. The method, due to Geiser, of obtaining the bitangents as projections of the lines of a cubic surface leads to a simple proof of the existence of this quartic.

scholarly journals Commuting elements in central products of special unitary groups

2012 ◽  
Vol 56 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Alejandro Adem ◽  
F. R. Cohen ◽  
José Manuel Gómez
Keyword(s):  
Prime Number ◽  
Moduli Space ◽  
Unitary Group ◽  
Unitary Groups ◽  
Connected Components ◽  
Flat Connections ◽  
Isomorphism Classes ◽  
Central Product ◽  
Central Products ◽  
Path Connected

AbstractWe study the space of commuting elements in the central product Gm,p of m copies of the special unitary group SU(p), where p is a prime number. In particular, a computation for the number of path-connected components of these spaces is given and the geometry of the moduli space Rep(ℤn, Gm,p) of isomorphism classes of flat connections on principal Gm,p-bundles over the n-torus is completely described for all values of n, m and p.

scholarly journals Factorization of Generalized Theta Functions Revisited

2017 ◽  
Vol 24 (01) ◽  
pp. 1-52
Author(s):  
Xiaotao Sun
Keyword(s):  
Line Bundle ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Theta Functions ◽  
Vanishing Theorems ◽  
Smooth Curves ◽  
Canonical Line Bundle

This survey is based on my lectures given in the last few years. As a reference, constructions of moduli spaces of parabolic sheaves and generalized parabolic sheaves are provided. By a refinement of the proof of vanishing theorems, we show, without using vanishing theorems, a new observation that [Formula: see text] is independent of all of the choices for any smooth curves. The estimate of various codimensions and computation of canonical line bundle of moduli space of generalized parabolic sheaves on a reducible curve are provided in Section 6, which is completely new.

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