The Enumerative Geometry of Double Covers of Curves



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)
Additional Information: Permanent email address: jasonvanzelm@outlook.com
Uncontrolled Keywords: Algebraic Geometry, Moduli Spaces of Curves, Intersection Theory
Divisions: Fac of Science & Engineering > School of Mathematics
Depositing User: Symplectic Admin
Date Deposited: 15 Aug 2018 08:58
Last Modified: 09 Jan 2021 05:24
DOI: 10.17638/03022475
Supervisors:
  • Pagani, Nicola
URI: https://livrepository.liverpool.ac.uk/id/eprint/3022475