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The locus of plane quartics with a hyperflex


Author: Xuntao Hu
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
Published electronically: September 30, 2016
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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.


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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

DOI: https://doi.org/10.1090/proc/13314
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
Article copyright: © Copyright 2016 American Mathematical Society