The locus of plane quartics with a hyperflex

Abstract

Using the results of a work by Dalla Piazza, Fiorentino and Salvati Manni, 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 ¯ \overline {\mathcal {M}_{3}} , first obtained by Cukierman. Moreover, the knowledge of such an explicit modular form also allows us to describe explicitly the boundary of the hyperflex locus in M 3 ¯ \overline {\mathcal {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 a work by Ottaviani and Sernesi.