Modular invariants and singularity indices of hyperelliptic fibrations

We prove that a Shimura curve in the Siegel modular variety is not generically contained in the open Torelli locus as long as the rank of unitary part in its canonical Higgs bundle satisfies a numerical upper bound. As an application we show that the Coleman-Oort conjecture holds for Shimura curves associated to partial corestriction upon a suitable choice of parameters, which generalizes a construction due to Mumford.

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“…Moreover, similar to [Liu16], one proves that Lemma 3.6. Let f : S → B be a double cover fibration as above.…”

Section: S Ssupporting

confidence: 66%

The Oort Conjecture for Shimura Curves of Small Unitary Rank

Chen

Zuo

2018

Commun. Math. Stat.

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“…If g = 1, then δ(f ) is the number of poles of the J-function of the family (see [Li16] for generalization). When g ≥ 2, it is shown ( [Ta94,Ta96]) that λ(f ) = 0 if and only if κ(f ) = 0 if and only if f is an isotrivial family.…”

Section: Preliminaries 21 Modular Invariantsmentioning

confidence: 99%

Fractional Dehn twists and modular invariants

Liu

2020

Sci. China Math.

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In this paper, we aim to give explicit lower bounds of modular invariants of families of curves of genus 2 and 3. According to the relation between fractional Dehn twists and modular invariants, we give the sharp lower bounds of fractional Dehn twist coefficients and classify pseudo-periodic maps with minimal coefficients firstly. Then we give explicit lower bounds of modular invariants, which are sharp for genus 2. We also obtain equivalent conditions for families of curves with these bounds. As an application, we give a better uniform lower bounds for the effective Bogomolov conjecture for genus 2 and 3.

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“…In particular, if f : X → Y is isotrivial, then δ(f ) = 0. Modular invariants are basic in the study of fibrations of algebraic surfaces and moduli spaces of algebraic curves, see [Ta10,LT13,Li16,No07]. In arithmetic algebraic geometry, modular invariants are some heights of algebraic curves, and can be used to give uniformity properties of curves, see [LT17].…”

Section: Introductionmentioning

confidence: 99%