## The locus of plane quartics with a hyperflex

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## 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 $\overline {\mathcal {M}_{3}}$, 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 $\overline {\mathcal {M}_{3}}$. 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.## References

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## Additional Information

**Xuntao Hu**- Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, New York 11794
- Email: xuntao.hu@stonybrook.edu
- Received by editor(s): January 26, 2016
- Received by editor(s) in revised form: May 20, 2016
- Published electronically: September 30, 2016
- Communicated by: Lev Borisov
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**145**(2017), 1399-1413 - MSC (2010): Primary 11F46, 14H50, 14J15, 14K25, 32G20; Secondary 14H15, 32G15, 14H45, 32G13
- DOI: https://doi.org/10.1090/proc/13314
- MathSciNet review: 3601534