The Chow ring of the stack of cyclic covers of the projective line

Fulghesu, Damiano; Viviani, Filippo

doi:10.5802/aif.2672

Cited by 11 publications

(19 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Bogomolov and Katsylo proved that the coarse space of H g is rational ( [29], [15]). The integral Chow groups of the moduli stack of hyperelliptic curves are known from a work of Fulghesu and Viviani [25], which builds on previous results of Arsie and Vistoli [7]. The latter works compute in particular the whole integral Chow groups (respectively, Picard groups) of the uniform components 1 of M 0,n (BG) when G is finite and cyclic.…”

Section: Nicola Paganimentioning

confidence: 97%

Moduli of abelian covers of elliptic curves

Pagani

2016

Journal of Pure and Applied Algebra

View full text Add to dashboard Cite

For any finite abelian group G, we study the moduli space of abelian Gcovers of elliptic curves, in particular identifying the irreducible components of the moduli space. We prove that, in the totally ramified case, the moduli space has trivial rational Picard group, and it is birational to the moduli space M1,n, where n is the number of branch points. In the particular case of moduli of bielliptic curves, we also prove that the boundary divisors are a basis of the rational Picard group of the admissible covers compactification of the moduli space. Our methods are entirely algebro-geometric. 1 2 NICOLA PAGANIfinite map is in fact an isomorphism), the latter is naturally a codimension-(g − 1) moduli subscheme of M g .We now summarize the main results of our work. In Theorem 1, we give a geometric description of the irreducible components of the moduli spaces of abelian G-covers of elliptic curves. From this description, we are able to deduce further results for those components that parametrize totally ramified covers. In fact, in Section 3.2, we prove that each component of the moduli space parametrizing totally ramified abelian G-covers is birational to M 1,n where n is the number of marked points (in general, a superset of the set of branch points). In Theorem 3, we exploit our description to prove the vanishing of the rational Picard group of such components of the moduli space, and to find the number of independent relations among the boundary divisors of the admissible covers compactification. In the last section we make our results more explicit in the case of the rational Picard group of the moduli spaces of admissible bielliptic curves, see Corollaries 6 and 7.The Picard number of the compact moduli of hyperelliptic curves H g is g [21]. In Corollary 6, we see that, when g > 2, it is 2g for the compact moduli B g of bielliptic curves (without ordering the branch). A simple analysis of the number of boundary divisors of the compact moduli of double covers of genus-h curves leads us to conjecture that the Picard number of the latter moduli space grows is (h + 1)g + g · o(1) for g → ∞.

show abstract

Section: Nicola Paganimentioning

confidence: 97%

Moduli of abelian covers of elliptic curves

Pagani

2016

Journal of Pure and Applied Algebra

View full text Add to dashboard Cite

show abstract

“…)) it suffices to find the chern classes of the P GL 2 representation (Sym n V )⊗(D ∨ ) ⊗n/2 regarded as a P GL 2 -equivariant vector bundle over a point. These chern classes are given in [15,Corollary 6.3]. The reader should also note that [15] contains mistakes elsewhere in the document (see [7,Introduction]).…”

Section: P Gl 2 and Gl 2 -Equivariant Chow Ringsmentioning

confidence: 99%

“…These chern classes are given in [15,Corollary 6.3]. The reader should also note that [15] contains mistakes elsewhere in the document (see [7,Introduction]). As a result, we have A…”

Section: P Gl 2 and Gl 2 -Equivariant Chow Ringsmentioning

confidence: 99%

$PGL_2$-equivariant strata of point configurations in $\mathbb{P}^1$

Spink¹,

Tseng²

2018

Preprint

View full text Add to dashboard Cite

We compute the integral Chow ring of the quotient stack [(P 1 ) n /P GL 2 ], which contains M 0,n as a dense open, and determine a natural Z-basis for the Chow ring in terms of certain ordered incidence strata. We further show that all Z-linear relations between the classes of ordered incidence strata arise from an analogue of the WDVV relations in A • (M 0,n ).Next we compute the classes of unordered incidence strata in the integral Chow ring of the quotient stack [Sym n P 1 /P GL 2 ] and classify all Z-linear relations between the strata via these analogues of WDVV relations.Finally, we compute the rational Chow rings of the complement of a union of unordered incidence strata.

show abstract

“…The first example that comes to mind is the stack M 1,1 of elliptic curves, which is a quotient [X/G m ], where X is the complement in A 2 of the discriminant hypersurface 4x 3 + 27y 2 = 0, and G m acts with weights 4 and 6. Very powerful techniques have been developed in equivariant intersection theory, after the landmark work of Edidin and Graham [10], and applications have flourished (e.g., [2,12,28]) in equivariant intersection theory.…”

Section: Introductionmentioning

confidence: 99%

Mapping class groups of trigonal loci

Bolognesi

Lönne²

2015

Sel. Math. New Ser.

View full text Add to dashboard Cite

In this paper, we study the topology of the stack T g of smooth trigonal curves of genus g over the complex field. We make use of a construction by the first named author and Vistoli, which describes T g as a quotient stack of the complement of the discriminant. This allows us to use techniques developed by the second named author to give presentations of the orbifold fundamental group of T g , and of its substrata with prescribed Maroni invariant, and describe their relation with the mapping class group Map g of Riemann surfaces of genus g.

show abstract

The Chow ring of the stack of cyclic covers of the projective line

Cited by 11 publications

References 15 publications

Moduli of abelian covers of elliptic curves

Moduli of abelian covers of elliptic curves

$PGL_2$-equivariant strata of point configurations in $\mathbb{P}^1$

Mapping class groups of trigonal loci

Contact Info

Product

Resources

About