A New Orbit Closure in Genus 8?

Abstract

We provide numerical evidence that the ${\text{GL}}_2(\mathbb{R})$-orbit closure of the unfolding of the (3, 4, 13)-triangle in the moduli space of Abelian differentials is a previously unknown 4-dimensional variety. Furthermore, we report on an exhaustive exploration that supports the idea that this would be the last example of exceptional triangle and quadrilateral unfoldings.

KEYWORDS:

• translation surfaces
• moduli space
• geometry
• dynamical systems

Acknowledgments

We thank Alex Eskin for assisting in the development of the program discussed in Section 3, and for helpful conversations. We thank Alex Wright for continuous support and for many insights on how one may approach Conjecture 1.1. We did hope to have found an explanation for this mysterious orbit closure all together [13]. But after further discussions with Paul Apisa and Johannes Schwab it turned out that our Lemma 5.4 part (4) is wrong. This article contains the experimental part of [13].

Notes

1 This is not the same as the field of definition of $\mathcal{N}$ in the sense of algebraic geometry.

2 Note that taking the double cover of a quadratic differential (X, Q) does not specify the points ${\tilde{x}}_1$, …, ${\tilde{x}}_{n\prime }$ in the cover. In our situation we only consider coverings of quadratic differentials in the stratum ${\mathcal{Q}}_4(11,1)$. The double cover construction is a Riemann surface of genus 8 endowed with an Abelian differential with two zeros of order, respectively, 12 and 2. We label them in the same order as in the base surface.

Additional information

Funding

Julian Rüth was supported by the Simons Foundation Simons Investigator grant of Alex Eskin.