Abstract
We show that the canonical bundle of the Hurwitz stack classifying covers of genus g>1 and degree k>2 of the projective line is big. We show that all coarse moduli spaces of trigonal curves of genus g>1 are of general type.
Highlights
Mg where b associates to an admissible cover its set of branch points, whereas σ assigns to an admissible cover the stable model of its source curve
The symmetric group Sb operates on Hkg by permuting the branch points of each admissible cover and we set Hg,k := Hkg/Sb
Hkg corresponds to a twisted stable map [ f : C → R, p1, . . . , pb], where C is a nodal curve of arithmetic genus g, the target curve R is a tree of smooth rational curves, f is a finite map of degree k satisfying f −1(Rsing) = Csing, and p1, . . . , pb ∈ Rreg denote the branch points of f
Summary
Hkg corresponds to a twisted stable map [ f : C → R, p1, . Pb], where C is a nodal curve of arithmetic genus g, the target curve R is a tree of smooth rational curves, f is a finite map of degree k satisfying f −1(Rsing) = Csing, and p1, . Let Bi be the boundary divisor of M0,b defined as the closure of the locus of unions of two smooth rational curves meeting at one point, such that precisely i of the marked points lie on one component. Ei :μ be the boundary divisor on Hkg given as the closure of the locus of covers f : C → R, p1, . Pb] ∈ B|I | ⊆ M0,b, with f −1( p) having partition type μ, and exactly i of the branch points p1, .
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