Plane Sextic Curves Research Articles

Representation of non-special curves of genus 5 as plane sextic curves and its application to finding curves with many rational points

In algebraic geometry, it is important to provide effective parametrizations for families of curves, both in theory and in practice. In this paper, we present such an effective parametrization for the moduli of genus-5 curves that are neither hyperelliptic nor trigonal. Subsequently, we construct an algorithm for a complete enumeration of non-special genus-5 curves having more rational points than a specified bound, where “non-special curve” means that the curve is non-hyperelliptic and non-trigonal with mild singularities of the associated sextic model that we propose. As a practical application, we implement this algorithm using the computer algebra system MAGMA, specifically for curves over the prime field of characteristic 3.

Tritangents to smooth sextic curves

We prove that a smooth plane sextic curve can have at most 72 tritangents, whereas a smooth real sextic may have at most 66 real tritangents.

Quadric surface bundles over surfaces and stable rationality

We prove a general specialization theorem which implies stable irrationality for a wide class of quadric surface bundles over rational surfaces. As an application, we solve with the exception of two cases, the stable rationality problem for any very general complex projective quadric surface bundle over the projective plane, given by a symmetric matrix of homogeneous polynomials. Both exceptions degenerate over a plane sextic curve and the corresponding double cover is a K3 surface.

Geography of irreducible plane sextics

We complete the equisingular deformation classification of irreducible singular plane sextic curves. As a by-product, we also compute the fundamental groups of the complement of all but a few maximizing sextics.

Parametric equations of plane sextic curves with a maximal set of double points

We give explicit parametric equations for all irreducible plane projective sextic curves which have at most double points and whose total Milnor number is maximal (is equal to 19). In each case we find a parametrization over a number field of the minimal possible degree and try to choose coordinates so that the coefficients are as small as we can do.

On nodal prime Fano threefolds of degree 10

We study the geometry and the period map of nodal complex prime Fano threefolds with index 1 and degree 10. We show that these threefolds are birationally isomorphic to Verra threefolds, i.e., hypersurfaces of bidegree (2, 2) in P 2 × P 2. Using Verra’s results on the period map for these threefolds and on the Prym map for double étale covers of plane sextic curves, we prove that the fiber of the period map for our nodal threefolds is the union of two disjoint surfaces, for which we give several descriptions. This result is the analog in the nodal case of a result of Debarre O, Iliev A, Manivel L (arXiv: 0812.3670) in the smooth case.

Lattice Zariski k-ples of plane sextic curves and Z-splitting curves for double plane sextics

A simple sextic is a reduced complex projective plane curve of degree 6 with only simple singularities. We introduce a notion of Z-splitting curves for the double covering of the projective plane branching along a simple sextic, and investigate lattice Zariski k-ples of simple sextics by Z-splitting curves. We define specialization of lattice types, and classify all lattice types of Z-splitting curves of degree less than or equal to 3 up to specializations.

The Plane Sextic Curve Fixed by A6

Plane curves with non-trivial collineation groups are rare: those of low order are thus interesting, and often exhibit special geometric features. The largest primitive plane group is A6. It is known by standard algebraic means that this fixes a sextic curve Ω. The present paper constructs Ω geometrically, and speedily obtains its most significant geometric property: its 72 inflexions lie by pairs on 36 biflexional tangents. There is no standard technique for determining the multiplicities of bitangents of plane curves. For Ω we show that each biflexional tangent counts 4-fold as a bitangent, and identify the other 180 ordinary bitangents. A brief comparison of the geometric properties of Ω with those of Klein’s quartic curve is given.

A Plane Sextic Curve of Genus 4 with A 5 for Collineation Group

A Plane Sextic Curve of Genus 4 with <i>A</i> ₅ for Collineation Group

Combinations of rational singularities on plane sextic curves with the sum of Milnor numbers less than sixteen

Combinations of rational singularities on plane sextic curves with the sum of Milnor numbers less than sixteen

Projective description of some plane sextic curves derived from conics as base curves

Introduction. Comparatively few of the plane sextic curves are known. None of the papers written about them gives a complete treatment or a classification of these curves. The sextics appear more or less isolated in the different treatises. Apparently the methods employed produce only a very limited number of sextics. The projective method of deriving higher plane curves described by Dr. H. P. Pett i t in his Projective description of some higher plane curves allows a more systematical study of sextics and especially a direct construction point by point. I t is the purpose of this paper to give a projective description of some of the sextics that can be derived by this method, if conies are the base curves. The number of types of these curves which could be generated by this method is here limited to 354 by giving the triangle of reference special positions with respect to the base conies.

On the circuits of a plane sextic curve

On the circuits of a plane sextic curve

The Ovals of the Plane Sextic Curve

The Ovals of the Plane Sextic Curve

The Ovals of the Plane Sextic Curve

The Ovals of the Plane Sextic Curve