Van Zelm, J
(2018)
The Enumerative Geometry of Double Covers of Curves.
PhD thesis, University of Liverpool.
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Abstract
Let Adm(g, h)_2m be the space of admissible double covers C → D of curves of genus g and h, with all the ramification and branch points of C and D marked, and where the covering involution permutes an extra set of 2m marked points of C pairwise. For each 0 ≤ n ≤ 2g +2−4h there is a natural map φ_n : Adm(g, h)_2m → Mbar_g,n+2m mapping the admissible cover C → D to the stabilization of the source curve C together with the 2m points and the first n ramification points. In this thesis we will study classes of the form [φ_n (Adm(g, h)_2m )] in the Chow ring of Mbar_g,n+2m . We will derive a formula for the intersection of any such class with the class of any decorated stratum class of Mbar_g,n+2m in Chapter 2. In Chapter 3 we will use this formula to compute the class [φ_n(Adm(g, h)_2m)] in terms of bases of decorated stratum classes for low values of g, h, n and m. In particular we give explicit expression in terms of decorated stratum classes of the class [φ 0 (Adm(4, 1))] of the locus of bielliptic curves of genus 4 and the class [φ 0 (Adm(5, 0))] of the locus of hyperelliptic curves of genus 5. In Chapter 4 we will prove that for g + 2m ≥ 12 and g ≥ 2 the class [φ_n(Adm(g, 1)_2m)] is not contained in the tautological ring. For g + 2m = 12 and g ≥ 2 we will show that the same result holds on the moduli space M_g,n+2m of smooth curves.
Item Type: | Thesis (PhD) |
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Uncontrolled Keywords: | Algebraic Geometry, Intersection Theory, Moduli Spaces of Curves |
Divisions: | Faculty of Science and Engineering > School of Physical Sciences |
Depositing User: | Symplectic Admin |
Date Deposited: | 15 Aug 2018 08:58 |
Last Modified: | 16 Jan 2024 17:21 |
DOI: | 10.17638/03022475 |
Supervisors: |
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URI: | https://livrepository.liverpool.ac.uk/id/eprint/3022475 |