Plane quartics with at least 8 hyperinflection points

Rational Points on Quartics

The group of Weierstrass points of a plane quartic with at least eight hyperflexes

The group generated by the Weierstrass points of a smooth curve in its Jacobian is an intrinsic invariant of the curve. We determine this group for all smooth quartics with eight hyperflexes or more. Since Weierstrass points are closely related to moduli spaces of curves, as an application, we get bounds on both the rank and the torsion part of this group for a generic quartic having a fixed nu...

On semidefinite representations of plane quartics

This note focuses on the problem of representing convex sets as projections of the cone of positive semidefinite matrices, in the particular case of sets generated by bivariate polynomials of degree four. Conditions are given for the convex hull of a plane quartic to be exactly semidefinite representable with at most 12 lifting variables. If the quartic is rationally parametrizable, an exact se...

Plane Quartics with Jacobians Isomorphic to a Hyperelliptic Jacobian

We show how for every integer n one can explicitly construct n distinct plane quartics and one hyperelliptic curve over C all of whose Jacobians are isomorphic to one another as abelian varieties without polarization. When we say that the curves can be constructed “explicitly”, we mean that the coefficients of the defining equations of the curves are simple rational expressions in algebraic num...

Moduli of Plane Quartics, Göpel Invariants and Borcherds Products

ABSTRACT. It is known that the moduli space of plane quartic curves is birational to an arithmetic quotient of a 6-dimensional complex ball ([Kon1]). In this paper, we shall show that there exists a 15-dimensional space of meromorphic automorphic forms on the complex ball which gives a birational embedding of the moduli space of plane quartics with level 2 structure into P. This map coincides w...