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Moduli of Hyperelliptic Curves and Invariants of Binary Forms
 Taylor, David Wihr
 Advisor(s): Varadarajan, Veeravalli S.
Abstract
For families of elliptic and genus 2 hyperelliptic curves over an algebraically closed field k of characteristic not equal to 2, we derive smooth local normal forms parameterized by families of binary nforms over k with distinct linear factors. We also prove that the invariant theory of the general linear group of twobytwo matrices acting on binary nforms is isomorphic to the invariant theory of the permutation group of n elements acting on an affine kvariety. We show that this ring of permutation group invariants is purely inseparable over the tensor product of k with a polynomial ring defined over the integers. Next we show that the GL_2(k) invariant theory of binary sextic forms in positive characteristic is purely inseparable over a ring given by the classical invariant theory of binary sextic forms taken modulo p for all but finitely many primes p. When the characteristic of k is not 2 these results imply that for all but finitely many primes, p, a power of the Frobenius endomorphism induces a morphism between the coarse moduli scheme of genus 2 curves and the spectrum of a ring constructed using the classical invariant theory of binary sextics taken modulo p. Moreover, this morphism is a bijection at the level of closed points.
We also give, via a combinatorial argument, explicit formulae for the PicardFuch equations satisfied by the periods of hyperelliptic curves of arbitrary genera defined over the complex numbers. This allows us to conclude directly that the PicardFuchs equations are regular singular.
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