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 number of hyperflexes in the moduli space M3 of curves of genus 3. The Weierstrass points of an algebraic curve, which can be defined over some extension of the base field, form a distinguished set of points of the curve having the property of being geometrically intrinsic. Curves of genus 0 or 1 have no Weierstrass points, and for hyperelliptic curves, the Weierstrass points can easily be characterized as the ramification points under the hyperelliptic involution, which generate the 2-torsion subgroup of the Jacobian. For nonhyperelliptic curves of genus 3, which admit a plane quartic model, the structure of the Weierstrass subgroup, i.e., the subgroup of the Jacobian generated by the images of the Weierstrass points under the Abel-Jacobi map, cannot be characterized so easily. This structure is known for curves with many automorphisms ([10], [12], [13]), in which cases the groups are finite. As with Weierstrass points, the Weierstrass subgroup is a geometric invariant of the curve, which we study in this paper. More precisely, we determine the structure of this group for all curves having either eight (Theorems 4.1 and 5.1) or nine hyperflexes (Theorem 3.1), i.e., points at which the tangent line to the curve meets the curve with multiplicity four. These are the first nontrivial cases of interest since there are no curves with either ten or eleven hyperflexes, and the two possible cases of curves with twelve hyperflexes have already been treated ([7], [13]). We show: Theorem 3.1. Let Ω1 be the plane quartic (X − Y Z) + (−3 + √ 7)Y Z(−2X + Y + Z)(−X + Y + Z) = 0 and let Ω2 be its conjugate. The group generated by the Weierstrass points of any of the two curves Ωi is W = (Z/4Z) × Z. Received by the editor March 6, 2003 and, in revised form, April 1, 2005. 2000 Mathematics Subject Classification. Primary 11G30, 14H55, 14Q05; Secondary 14H40.