The locus of plane quartics with a hyperflex

Abstract

Using the results of [DPFSM14], we determine an explicit modular form defining the locus of plane quartics with a hyperflex among all plane quartics. As a result, we provide a direct way to compute the divisor class of the locus of plane quartics with a hyperflex within M 3 , first obtained in [Cuk89]. Moreover, the knowledge of such an explicit modular form also allows us to describe explicitly the boundary of the hyperflex locus in M 3 . As an example we show that the locus of banana curves (two irreducible components intersecting at two nodes) is contained in the closure of the hyperflex locus. We also identify an explicit modular form defining the locus of Clebsch quartics and use it to recompute the class of this divisor, first obtained in [OS11].

Compactification of strata of Abelian differentials

“…In [FP16] Farkas and Pandharipande also studied the Deligne-Mumford compactification of PΩM 3,1 (4) for several cases of stable curves with a fixed dual graph. In [Hu17] Hu obtained an explicit modular form defining the locus of hyperflexes in M 3 , that is, the image of PΩM 3 (4) odd in M 3 .…”