Xuntao Hu | Searcharxiv

arXiv2018

General Variational Formulas for Abelian Differentials

We use the jump problem technique developed in a recent paper by Grushevsky, Krichever and the second author to compute the variational formula of any stable differential and its periods to arbitrary precision in plumbing coordinates. In particular, we give the explicit variational formula for the degeneration of the period matrices near an arbitrary stable curve, easily reproving the results of Yamada for nodal curves with one node. Concrete examples are included. We also apply the same technique to give an alternative proof of the sufficiency part of the theorem by Bainbridge, Chen, Gendron, Grushevsky and Möller on the closures of strata of differentials with prescribed multiplicities of zeroes and poles.

math.AG↗

arXiv2016

The Locus of Plane Quartics with A Hyperflex

Using the results of 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 in 1989. 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 by Ottaviani and Sernesi.

math.AG↗