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Chris Hall : Sequences of curves with growing gonality

Given a smooth irreducible complex curve $C$, there are several isomorphism invariants one can attach to $C$. One invariant is the genus of $C$, that is, the number of handles in the corresponding Riemann surface. A subtler invariant is the gonality of $C$, that is, the minimal degree of a dominant map from $C$ of $\mathbb{P}^1$. A lower bound for either invariant has diophantine consequences when $C$ can be defined over a number field, but the ability to give non-trivial lower bounds depends on how $C$ is presented. In this talk we will consider a sequence $C_1,C_2,\ldots$ of finite unramified covers of $C$ and give spectral criteria for the gonality of $C_n$ to tend to infinity.

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