Square-tiled surfaces and rigid curves on moduli spaces

By Dawei Chen. 2010

48 citations recorded by the source corpus.

This work is motivated by two central questions in the birational geometry of moduli spaces of curves -- Fulton's conjecture and the effective cone of $\bar M_g$. We study the algebro-geometric aspect of Teichmuller curves parameterizing square-tiled surfaces with two applications: (a) there exist infinitely many rigid curves on the moduli space of hyperelliptic curves, they span the same extremal ray of the cone of moving curves and their union is Zariski dense, hence they yield infinitely many rigid curves with the same properties on the moduli space of stable n-pointed rational curves for even n; (b) the limit of slopes of Teichmuller curves and the sum of Lyapunov exponents of the Hodge bundle determine each other, by which we can have a better understanding for the cone of effective divisors on the moduli space of curves.

Subfields: Computational Mechanics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/be2b790d4d5a236ed598b361ac11bddc18fd9b6f

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In Computational Mechanics

In Geometry and Topology

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Citations in the corpus

Quasimodularity and large genus limits of Siegel-Veech constants (2016, background): “It appeared for arithmetic Teichmüller curves in the appendix of [9], and we give a self-contained statement and proof in Proposition 17.”
Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves (2019, background): “ gain the new height parameter hi (correspondingly 2hi) responsible for the level of the cut. As a corollary of the above theorem, D. Chen and A. Eskin proved the following Theorem (see Appendix A in [C]): Theorem (D. Chen, A. Eskin). For any connected component of any stratum of Abelian diﬀerential”
A criterion for the simplicity of the Lyapunov spectrum of square-tiled surfaces (2013, background): “The only general result is due to Eskin: in the appendix of [11], he proves that, as N goes to +∞, the average of the sum of nonnegative Lyapunov exponents over all N square origamis in a component of some moduli space converges to the sum of Lyapunov exponents w.”
Alternate Compactifications of Moduli Spaces of Curves (2010, background): “Somewhat surprisingly, the resulting asymptotic lower bound on sg in [Che10b] turns out to be 576 5g as well, which led Chen to conjecture that sg ∼ 576 5g for g →∞.”
Quadratic differentials in low genus: exceptional and non-varying (2012, background): “where κ = 1 12 ∑k i=1 mi(mi+2) m1+1 , see [9] and [6].”