scholarly journals On fibrations approaching the Arakelov equality

2021 ◽  
Author(s):  
Maximilian Bieri
Keyword(s):  
Lyapunov Exponents ◽  
Euler Characteristic ◽  
Small Base ◽  
Teichmuller Curves ◽  
Teichmüller Curves

AbstractThe sum of Lyapunov exponents $$L_f$$ L f of a semi-stable fibration is the ratio of the degree of the Hodge bundle by the Euler characteristic of the base. This ratio is bounded from above by the Arakelov inequality. Sheng-Li Tan showed that for fiber genus $$g\ge 2$$ g ≥ 2 the Arakelov equality is never attained. We investigate whether there are sequences of fibrations approaching asymptotically the Arakelov bound. The answer turns out to be no, if the fibration is smooth, or non-hyperelliptic, or has a small base genus. Moreover, we construct examples of semi-stable fibrations showing that Teichmüller curves are not attaining the maximal possible value of $$L_f$$ L f .

scholarly journals ORBIFOLD POINTS ON PRYM–TEICHMÜLLER CURVES IN GENUS 

2017 ◽  
Vol 18 (4) ◽  
pp. 673-706 ◽  
Author(s):  
David Torres-Teigell ◽  
Jonathan Zachhuber
Keyword(s):  
Elliptic Curves ◽  
Explicit Construction ◽  
Side Product ◽  
Geometric Description ◽  
Flat Surfaces ◽  
Teichmuller Curves ◽  
Teichmüller Curves

For each discriminant $D>1$, McMullen constructed the Prym–Teichmüller curves $W_{D}(4)$ and $W_{D}(6)$ in ${\mathcal{M}}_{3}$ and ${\mathcal{M}}_{4}$, which constitute one of the few known infinite families of geometrically primitive Teichmüller curves. In the present paper, we determine for each $D$ the number and type of orbifold points on $W_{D}(6)$. These results, together with a previous result of the two authors in the genus $3$ case and with results of Lanneau–Nguyen and Möller, complete the topological characterisation of all Prym–Teichmüller curves and determine their genus. The study of orbifold points relies on the analysis of intersections of $W_{D}(6)$ with certain families of genus $4$ curves with extra automorphisms. As a side product of this study, we give an explicit construction of such families and describe their Prym–Torelli images, which turn out to be isomorphic to certain products of elliptic curves. We also give a geometric description of the flat surfaces associated to these families and describe the asymptotics of the genus of $W_{D}(6)$ for large $D$.

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