Plane Quartic Curve Research Articles

A Tropical Count of Real Bitangents to Plane Quartic Curves

A smooth tropical plane quartic curve has seven tropical bitangent classes. Their shapes can vary within the same combinatorial type of curve. We study deformations of these shapes and we show that the conditions determined by Cueto and Markwig for lifting them to real bitangent lines are independent of the deformations. From this we deduce a tropical proof of Plücker and Zeuthen's count of the number of real bitangents to smooth plane quartic curves.

Counting points on smooth plane quartics

We present efficient algorithms for counting points on a smooth plane quartic curve X modulo a prime p. We address both the case where X is defined over {mathbb {F}}_p and the case where X is defined over {mathbb {Q}} and p is a prime of good reduction. We consider two approaches for computing #X({mathbb {F}}_p), one which runs in O(plog plog log p) time using O(log p) space and one which runs in O(p^{1/2}log ^2p) time using O(p^{1/2}log p) space. Both approaches yield algorithms that are faster in practice than existing methods. We also present average polynomial-time algorithms for X/{mathbb {Q}} that compute #X({mathbb {F}}_p) for good primes pleqslant N in O(Nlog ^3 N) time using O(N) space. These are the first practical implementations of average polynomial-time algorithms for curves that are not cyclic covers of {mathbb {P}}^1, which in combination with previous results addresses all curves of genus gleqslant 3. Our algorithms also compute Cartier–Manin/Hasse–Witt matrices that may be of independent interest.

New sizes of complete (k, 4)-arcs in PG(2,17)

In this paper, the packing problem for complete ( 4)-arcs in is partially solved. The minimum and the maximum sizes of complete ( 4)-arcs in are obtained. The idea that has been used to do this classification is based on using the algorithm introduced in Section 3 in this paper. Also, this paper establishes the connection between the projective geometry in terms of a complete ( , 4)-arc in and the algebraic characteristics of a plane quartic curve over the field represented by the number of its rational points and inflexion points. In addition, some sizes of complete ( 6)-arcs in the projective plane of order thirteen are established, namely for = 53, 54, 55, 56.

An enriched count of the bitangents to a smooth plane quartic curve

Recent work of Kass–Wickelgren gives an enriched count of the 27 lines on a smooth cubic surface over arbitrary fields. Their approach using $$\mathbb {A}^1$$ -enumerative geometry suggests that other classical enumerative problems should have similar enrichments, when the answer is computed as the degree of the Euler class of a relatively orientable vector bundle. Here, we consider the closely related problem of the 28 bitangents to a smooth plane quartic. However, it turns out that the relevant vector bundle is not relatively orientable and new ideas are needed to produce enriched counts. We introduce a fixed “line at infinity,” which leads to enriched counts of bitangents that depend on their geometry relative to the quartic and this distinguished line.

Bielliptic smooth plane curves and quadratic points

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.

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

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.

Weierstrass points of order three on smooth quartic curves

In this paper, we study the [Formula: see text]-Weierstrass points on smooth projective plane quartic curves and investigate their geometry. Moreover, we use a technique to determine in a very precise way the distribution of such points on any smooth projective plane quartic curve. We also give a variety of examples that illustrate and enrich the subject.

Reconstructing Plane Quartics from Their Invariants

We present an explicit method that, given a generic tuple of Dixmier–Ohno invariants, reconstructs a corresponding plane quartic curve.

Twists of non-hyperelliptic curves of genus 3

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.

𝐾₂ of certain families of plane quartic curves

We construct three elements in the kernel of the tame symbol on families of quartic curves. We show that these elements are integral under certain conditions on the parameters. Moreover, we prove that these elements are in general linearly independent by calculating the limit of the regulator.

Hyperbolic geometry and moduli of real curves of genus three

The moduli space of smooth real plane quartic curves consists of six connected components. We prove that each of these components admits a real hyperbolic structure. These connected components correspond to the six real forms of a certain hyperbolic lattice over the Gaussian integers. We will study this Gaussian lattice in detail. For the connected component that corresponds to maximal real quartic curves, we obtain a more explicit description. We construct a Coxeter diagram that encodes the geometry of this component.

About the Fricke–Macbeath curve

A closed Riemann surface of genus $$g \geqslant 2$$ is called a Hurwitz curve if its group of conformal automorphisms has order $$84(g-1)$$ . In 1895, Wiman noticed that there is no Hurwitz curve of genus $$g=2,4,5,6$$ and, up to isomorphisms, there is a unique Hurwitz curve of genus $$g=3$$ ; this being Klein’s plane quartic curve. Later, in 1965, Macbeath proved the existence, up to isomorphisms, of a unique Hurwitz curve of genus $$g=7$$ ; this known as the Fricke–Macbeath curve. Equations were also provided; that being the fiber product of suitable three elliptic curves. In the same year, Edge constructed such a genus seven Hurwitz curve by elementary projective geometry. Such a construction was provided by first constructing a 4-dimensional family of genus seven closed Riemann surfaces $$S_{\mu }$$ admitting a group of conformal automorphisms so that $$S_{\mu }/G_{\mu }$$ has genus zero. In this paper we discuss the above curves in terms of fiber products of classical Fermat curves and we provide a geometrical explanation of the three elliptic curves in Macbeath’s description. We also observe that the jacobian variety of each $$S_{\mu }$$ is isogenous to the product of seven elliptic curves (explicitly given) and, for the particular Fricke–Macbeath curve, we obtain the well-known fact that its jacobian variety is isogenous to $$E^{7}$$ for a suitable elliptic curve E.

Arithmetic invariant theory and 2-descent for plane quartic curves

This research was partially conducted during the period the author served as a Clay Research Fellow [main article]. This research is supported in part by NSF grant DMS-1161489 [appendix].

Bitangents of tropical plane quartic curves

We study smooth tropical plane quartic curves and show that they satisfy certain properties analogous to (but also different from) smooth plane quartics in algebraic geometry. For example, we show that every such curve admits either infinitely many or exactly 7 bitangent lines. We also prove that a smooth tropical plane quartic curve cannot be hyperelliptic.

Explicit computations of invariants of plane quartic curves

We establish a complete set of invariants for ternary quartic forms. Further, we express four classical invariants in terms of these generators.

Fibrations by curves with more than one nonsmooth point

Bertini’s theorem on variable singular pointsmay fail in positive characteristic, as was discovered by Zariski in 1944. In fact, he found fibrations by nonsmooth curves. In this work we continue to classify this phenomenon in characteristic three by constructing the first example, rising in the literature, of fibrations with more than one nonsmooth point. Our approach has been motivated by the close relation between it and the theory of regular but nonsmooth curves, or equivalently, nonconservative function fields in one variable. In analogy to the Kodaira-Neron classification of special fibers of minimal fibrations by elliptic curves, we also construct the minimal proper regular model of some fibrations by nonsmooth projective plane quartic curves and determine the structure of the bad fibers.

On the moduli space of smooth plane quartic curves with a sextactic point

Let Mg be the moduli space of smooth algebraic curves of genus g over C: In this paper, we prove that the set Sr M 3 of moduli points of smooth plane quartic curves (nonhyperelliptic curves of genus 3) having at least one sextactic point of sextact multiplicity r; where r 2 f1;2;3g; is an irreducible, closed and rational subvariety of codimensional r 1 of M3 H 3 (where H3 M 3 is the hyperelliptic locus ).

A computational approach to Lüroth quartics

A plane quartic curve is called Lüroth if it contains the ten vertices of a complete pentalateral. White and Miller constructed in 1909 a covariant quartic fourfold, associated to any plane quartic. We review their construction and we show how it gives a computational tool to detect if a plane quartic is Lüroth. As a byproduct, we show that the 28 bitangents of a general plane quartic correspond to 28 singular points of the associated White–Miller quartic fourfold.

Membership tests for images of algebraic sets by linear projections

Given a witness set for an irreducible variety V and a linear map π, we describe membership tests for both the constructible algebraic set π(V) and the algebraic set π(V)¯. We also provide applications and examples of these new tests including computing the codimension one components of π(V)¯⧹π(V)¯. Additionally, we also describe computing the geometric genus of a curve section of an irreducible component of the solution set of a polynomial system and a test for deciding whether a plane quartic curve is a Lüroth quartic.

Galois Lines for Normal Elliptic Space Curves, II

For each linearly normal elliptic curve C ⊂ ℙ3, we determine Galois lines and their arrangement. We prove that the curve C has exactly six V4-lines. In case j(C) = 1, it has eight Z4-lines in addition. The V4-lines form the edges of a tetrahedron. In case j(C) = 1, for each vertex of the tetrahedron, there exist exactly two Z4-lines passing through it. As a corollary we obtain that each plane quartic curve of genus 1 does not have more than one Galois point.