Plane Cubics Research Articles

Quatroids and rational plane cubics

Quatroids and rational plane cubics

Arrangements of inflectional lines, hyperosculating conics, and plane cubics

Arrangements of inflectional lines, hyperosculating conics, and plane cubics

Stability of pencils of plane sextics and Halphen pencils of index two

We study the problem of classifying pencils of plane sextics up to projective equivalence via geometric invariant theory (GIT). In particular, we provide a complete description of the GIT stability of certain pencils of plane sextics called Halphen pencils of index two-classical geometric objects which were first introduced by G. Halphen in 1882. Inspired by the work of Miranda on pencils of plane cubics, we obtain explicit stability criteria in terms of the types of singular fibers appearing in their associated rational elliptic surfaces.

Dynamical invariants and intersection theory on the flex and gothic loci

The flex locus parameterizes plane cubics with three collinear cocritical points under a projection, and the gothic locus arises from quadratic differentials with zeros at a fiber of the projection and with poles at the cocritical points. The flex and gothic loci provide the first example of a primitive, totally geodesic subvariety of moduli space and new \({\text {SL}}_2({\mathbb {R}})\)-invariant varieties in Teichmüller dynamics, as discovered by McMullen–Mukamel–Wright. In this paper we determine the divisor class of the flex locus as well as various tautological intersection numbers on the gothic locus. For the case of the gothic locus our result confirms numerically a conjecture of Chen–Möller–Sauvaget about computing sums of Lyapunov exponents for \({\text {SL}}_2({\mathbb {R}})\)-invariant varieties via intersection theory.

Everywhere local solubility for hypersurfaces in products of projective spaces

We prove that a positive proportion of hypersurfaces in products of projective spaces over $${\mathbb {Q}}$$ are everywhere locally soluble, for almost all multidegrees and dimensions, as a generalization of a theorem of Poonen and Voloch [25]. We also study the specific case of genus 1 curves in $${\mathbb {P}}^1 \times {\mathbb {P}}^1$$ defined over $${\mathbb {Q}}$$ , represented as bidegree (2, 2)-forms, and show that the proportion of everywhere locally soluble such curves is approximately $$87.4\%$$ . As in the case of plane cubics [2], the proportion of these curves in $${\mathbb {P}}^1 \times {\mathbb {P}}^1$$ soluble over $${\mathbb {Q}}_p$$ is a rational function of p for each finite prime p. Finally, we include some experimental data on the Hasse principle for these curves.

Catalecticant intersections and confinement of decompositions of forms

We introduce the notion of confinement of decompositions for forms or vector of forms. The confinement, when it holds, lowers the number of parameters that one needs to consider, in order to find all the possible decompositions of a given set of data. With the technique of confinement, we obtain here two results. First, we give a new, shorter proof of a result by London (1890) that 3 general plane cubics have 2 simultaneous Waring decompositions of rank 6. Then we compute, with the software Bertini, that 4 general plane quartics have 18 different decompositions of rank 10 (a result which was not known before).

On the variety of the inflection points of plane cubic curves

In this paper we study properties of the nine-dimensional variety of the inflection points of plane cubics. We describe the local monodromy groups of the set of inflection points near singular cubic curves and give a detailed description of the normalizations of the surfaces of the inflection points of plane cubic curves belonging to general two-dimensional linear systems of cubics. We also prove the vanishing of the irregularity of a smooth manifold birationally isomorphic to the variety of the inflection points of plane cubics.

Rational differential forms on the variety of flexes of plane cubics

We prove that for every positive integer $d$, there are no nonzero regular differential $d$-forms on every smooth irreducible projective algebraic variety birationally isomorphic to the variety of flexes of plane cubics.

Rational differential forms on the variety of flexes of plane cubics

Rational differential forms on the variety of flexes of plane cubics

Minimum weight codewords in dual algebraic-geometric codes from the Giulietti-Korchmáros curve

In this paper we investigate the number of minimum weight codewords of some dual algebraic-geometric codes associated with the Giulietti–Korchmaros maximal curve, by computing the maximal number of intersections between the Giulietti–Korchmaros curve and lines, plane conics, and plane cubics.

The Chow Ring of the Stack of Smooth Plane Cubics

In this paper, we give an explicit presentation of the integral Chow ring of a stack of smooth plane cubics. We also determine some relations in the general case of hypersurfaces of any dimension and degree.

Waring loci and the Strassen conjecture

The Waring locus of a form F is the collection of the degree one forms appearing in some minimal sum of powers decomposition of F. In this paper, we give a complete description of Waring loci for several families of forms, such as quadrics, monomials, binary forms and plane cubics. We also introduce a Waring loci version of Strassen's Conjecture, which implies the original conjecture, and we prove it in many cases.

A few examples of points of finite order on some smooth plane cubics

In this expository article we give a self-contained presentation on the rational points of finite order on smooth cubics. We mainly construct examples of points of order between four and twelve, except the eleven case of course. And, in some cases, we offer families of cubics having such kinds of points.

Frobenius nonclassicality of Fermat curves with respect to cubics

For Fermat curves F: aX n + bY n = Z n defined over F q , we establish necessary and sufficient conditions for F to be F q -Frobenius nonclassical with respect to the linear system of plane cubics. In the new F q -Frobenius nonclassical cases, we determine explicit formulas for the number N q (F) of F q -rational points on F. For the remaining Fermat curves, nice upper bounds for N q (F) are immediately given by the Stohr–Voloch Theory.

Classification of Elliptic Cubic Curves Over The Finite Field of Order Nineteen

Plane cubics curves may be classified up to isomorphism or projective equivalence. In this paper, the inequivalent elliptic cubic curves which are non-singular plane cubic curves have been classified projectively over the finite field of order nineteen, and determined if they are complete or incomplete as arcs of degree three. Also, the maximum size of a complete elliptic curve that can be constructed from each incomplete elliptic curve are given.

An algorithm to obtain linear determinantal representations of smooth plane cubics over finite fields

We give a brief report on our computations of linear determinantal representations of smooth plane cubics over finite fields. After recalling a classical interpretation of linear determinantal representations as rational points on the affine part of Jacobian varieties, we give an algorithm to obtain all linear determinantal representations up to equivalence. We also report our recent study on computations of linear determinantal representations of twisted Fermat cubics defined over the field of rational numbers. This paper is a summary of the author's talk at the JSIAM JANT workshop on algorithmic number theory in March, 2016. Details will appear elsewhere.

Classification of Elliptic Cubic Curves Over The Finite Field of Order Nineteen

Plane cubics curves may be classified up to isomorphism or projective equivalence. In this paper, the inequivalent elliptic cubic curves which are non-singular plane cubic curves have been classified projectively over the finite field of order nineteen, and determined if they are complete or incomplete as arcs of degree three. Also, the maximum size of a complete elliptic curve that can be constructed from each incomplete elliptic curve are given.

3-Nets realizing a group in a projective plane

In a projective plane $\mathit{PG}(2,\mathbb{K})$ defined over an algebraically closed field $\mathbb{K}$ of characteristic 0, we give a complete classification of 3-nets realizing a finite group. An infinite family, due to Yuzvinsky (Compos. Math. 140:1614---1624, 2004), arises from plane cubics and comprises 3-nets realizing cyclic and direct products of two cyclic groups. Another known infinite family, due to Pereira and Yuzvinsky (Adv. Math. 219:672---688, 2008), comprises 3-nets realizing dihedral groups. We prove that there is no further infinite family. Urzua's 3-nets (Adv. Geom. 10:287---310, 2010) realizing the quaternion group of order 8 are the unique sporadic examples. If p is larger than the order of the group, the above classification holds in characteristic p>0 apart from three possible exceptions $\rm{Alt}_{4}$ , $\rm{Sym}_{4}$ , and $\rm{Alt}_{5}$ . Motivation for the study of finite 3-nets in the complex plane comes from the study of complex line arrangements and from resonance theory; see (Falk and Yuzvinsky in Compos. Math. 143:1069---1088, 2007; Miguel and Buzunariz in Graphs Comb. 25:469---488, 2009; Pereira and Yuzvinsky in Adv. Math. 219:672---688, 2008; Yuzvinsky in Compos. Math. 140:1614---1624, 2004; Yuzvinsky in Proc. Am. Math. Soc. 137:1641---1648, 2009).

Equivariant and invariant theory of nets of conics with an application to Thom polynomials

Two parameter families of plane conics are called nets of conics. There is a natural group action on the vector space of nets of conics, namely the product of the group reparametrizing the underlying plane, and the group reparametrizing the parameter space of the family. We calculate equivariant fundamental classes of orbit closures. Based on this calculation we develop the invariant theory of nets of conics. As an application we determine Thom polynomials of contact singularities of type (3,3). We also show how enumerative problems in particular the intersection multiplicities of the determinant map from nets of conics to plane cubics can be solved studying equivariant classes of orbit closures.

Topology of real cubic fourfolds

A solution of the problem of topological classification of real cubic fourfolds is given. It is proven that the real locus of a real non-singular cubic fourfold is diffeomorphic either to a connected sum ℝP4 #i(S2 × S2)#j(S1 × S3) or to a disjoint union ℝP4 ⊔ S4. La voie la plus courte et la meilleure entre deux vérités de domaine réel passe souvent par le domaine imaginaire.J. S. Hadamard, Essai sur la psychologie de l'invention dans le domainemathématique, Edition Gaithier-Villars, 1975, p. 114.