Several problems on moduli stacks of hyperelliptic curves
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The main objects to investigate in this thesis are the moduli stacks of smooth hyperelliptic curves of genus g, denoted by Hg, and its Deligne-Mumford compactification Hg in Mg. A result by Kleiman and Lonsted says that the universal hyperelliptic curve Hg,1 contains a smooth Cartier divisor Hg,w, which parametrizes families of smooth hyperelliptic curves with a Weierstrass point. The stacks Hwg,n parametrizing families of smooth hyperelliptic curves with n marked Weierstrass points can be constructed recursively. In the first part of the thesis, we compute the integral Chow rings of the stacks Hwg,n, and prove that the integral Chow ring of each of these stacks is generated as an algebra by any of the [psi]-classes and that all relations live in degree one. The moduli stacks of pointed genus g stable hyperelliptic curves contain some naturally occurring divisors. In particular, the universal stable hyperelliptic curve Hg,1 has a Weierstrass divisor Hg,w, and the stack Hg,2 contains the universal g12 divisor Hg,g12. In the second part, we calculate the Chow classes of divisors Hg,w and Hg,g12 using the method of test curves, and express the results in terms of a basis for the divisor class groups Cl(Hg,1)Q and Cl(Hg,2)Q computed by Scavia. The Picard group of Hg is fully understood by the work of Cornalba. Finally, we study the map Sym2 Pic(Hg)Q -> A2(Hg)Q, and prove that its kernel is generated by a single relation depending on the parity of the genus with a simple recursive formula.
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Ph. D.