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[EQUATION] Thus we can extract the mass of pseudoscalar particle by fitting Eq. ( 18 ) at zero temperature gbmass_fit1 eta_mass2015 myjob2017 |
We may also use Eq. ( 18 ) to extract the pseudoscalar glueball mass from TCDC at finite temperature in quenched lattice QCD, the mass [MATH] and amplitude are set to be two free parameters in the fitting procedure. The procedure has been applied to the two ensembles in Table . The effective smearing radius [MATH] of W... |
We find that when the starting point of the fitting range is fixed and the ending point is varied, once the error bar of the TCDC at the ending point touches the value zero, the fitting result is independent of the ending point. This phenomenon is also found in Ref. gbmass_fit1 . Therefore we fix the ending point that ... |
Both ensembles have the most stable plateau of the preliminary pseudoscalar glueball mass [MATH] at [MATH] Therefore we choose the data from [MATH] to extract [MATH] . The final fitting window is determined by the range that the plateaus of the preliminary pseudoscalar glueball mass overlap with plateaus nearby. In Fig... |
as Ref. gbmass_fit2 does. The fitting results are consistent with those from Ref. gbmass_fit2 . Noting that the final fitting window in the left panel is shorter than that in the right panel. It should be owing to the coarser lattice spacing [MATH] of the ensemble in the left panel, same thing has also been found in Re... |
[MATH] Summary In this paper we use Wilson flow to smear ensembles of quenched lattice QCD with lattice volume [MATH] at finite temperature. To study the topological structure of quenched QCD vacuum near [MATH] corresponding to the critical inverse coupling [MATH] , we have used HS caloron-like topological lumps and IP... |
Acknowledgments This work was mainly run on Tianhe-2 supercomputer at NSCC in Guangzhou. Supported in part by the National Natural Science Foundation of China (NSFC) under the project No.11335001, No.11275169. VI References |
# Source: arxiv 1806.05345 # Title: Rational Picard group of moduli of pointed hyperelliptic curves # Sections: all # Downloaded: 2026-03-03T01:47:31.642600+00:00 |
Rational Picard group of moduli of pointed hyperelliptic curves Abstract. We determine the rational divisor class group of the moduli spaces of smooth pointed hyperelliptic curves and of their Deligne-Mumford compactification, over the field of complex numbers. |
1. Introduction The moduli stack [MATH] of smooth curves of fixed genus [MATH] , together with its Deligne-Mumford compactification |
, are fundamental objects in algebraic geometry. In moduli theory, an important role is also played by the pointed versions [MATH] and [MATH] . Many geometric properties of these moduli stacks and of their coarse moduli spaces have been established. The irreducibility of [MATH] and [MATH] was proved in the seminal pape... |
, and the projectivity of the coarse moduli space of [MATH] by Knudsen in and Of great interest is the tautological Chow ring of [MATH] , whose study was initiated by Mumford |
. Relations between classes in this ring immediately inform the enumerative geometry of families of curves. In particular, over the field of complex numbers, the divisor class group (i.e. the Picard group) of [MATH] and [MATH] is well understood. When [MATH] , Harer proved that [MATH] is a free abelian group on [MATH] ... |
, and Mumford showed that [MATH] and [MATH] are free abelian groups in . In , Arbarello and Cornalba provided explicit bases for [MATH] and [MATH] , when [MATH] |
The stack [MATH] of smooth hyperelliptic curves of fixed genus [MATH] , its compactification [MATH] , and their pointed generalizations [MATH] and [MATH] are also basic objects of study in moduli theory, but less is known about them compared to [MATH] and [MATH] . In |
, Arsie and Vistoli described [MATH] as a moduli stack parametrizing double covers of the projective line. They used this description to show that [MATH] is finite cyclic, of order [MATH] for odd [MATH] , and [MATH] for even [MATH] . In |
, Gorchinskiy and Viviani produced geometrically meaningful generators for [MATH] . A presentation of [MATH] via generators and relations was given by Cornalba in |
. As a consequence of these results, [MATH] and [MATH] has a basis consisting of all the boundary divisors. In this article, we determine the rational divisor class group of the moduli of smooth [MATH] -pointed hyperelliptic curves [MATH] , and of its Deligne-Mumford compactification [MATH] , for every [MATH] (see Sect... |
for the definitions). We work over the field of complex numbers. Theorem 1.1 For every [MATH] and [MATH] , we have: (a) [MATH] has a basis given by [MATH] -classes; |
(b) [MATH] has a basis given by [MATH] -classes and all the boundary divisors. As we discuss in 2.2 , the stack [MATH] is smooth, hence [MATH] . We do not know if [MATH] is smooth, but we show in 5.4 (b) that it is smooth in codimension [MATH] . The question of the smoothness of [MATH] appears to be open. |
Recall the definition of [MATH] -classes [MATH] in [MATH] : denoted by [MATH] the line bundle on [MATH] which to a family [MATH] in [MATH] with sections [MATH] associates the line bundle [MATH] [MATH] is defined as the class of [MATH] in [MATH] ; see , §XIII.2] |
The strategy of the proof is as follows. Using the Leray spectral sequence for the forgetful morphism [MATH] together with results of Totaro |
on cohomology of configuration spaces, we show that [MATH] and that [MATH] has a basis consisting of all [MATH] -classes. The exponential sequence for [MATH] gives (a), from which it immediately follows that [MATH] is [MATH] -classes and irreducible boundary divisors. We then enumerate the irreducible boundary divisors... |
2. Preliminaries on moduli stacks of curves For every [MATH] such that [MATH] , we denote by [MATH] the moduli stack of smooth [MATH] -pointed projective curves of genus [MATH] , and by [MATH] its Deligne-Mumford compactification. We also let [MATH] be the moduli stack of stable curves with rational tails. By definitio... |
[EQUATION] Lemma 2.1 Let [MATH] such that [MATH] (a) The forgetful morphism [MATH] is proper, flat, and has connected fibers of dimension [MATH] |
(b) If [MATH] is a smooth curve of genus [MATH] , the fiber of [MATH] at [MATH] is isomorphic to [MATH] , and the fiber of [MATH] at [MATH] is isomorphic to the Fulton-MacPherson compactification of [MATH] |
(c) The forgetful morphism [MATH] is proper, smooth and has irreducible fibers. (d) The forgetful morphism [MATH] is smooth. Proof. |
Let [MATH] be the universal curve over [MATH] . By definition, a morphism [MATH] corresponds to a family of [MATH] -pointed stable curves over [MATH] , together with an extra section that is allowed to pass through the nodes; see 22 , Definition 1.2] . For every morphism [MATH] , the pullback of [MATH] along [MATH] is ... |
Knudsen constructed a contraction morphism [MATH] and a stabilization morphism [MATH] which are compatible with [MATH] and the morphism [MATH] which forgets the last section. He then proved that [MATH] and [MATH] are inverse to each other in 22 , Corollary 2.6] . The forgetful morphism [MATH] is defined as the composit... |
(a) We may factor the morphism [MATH] as a composition [EQUATION] By the previous discussion, every intermediate map is proper, flat, and has connected fibers of dimension [MATH] . It follows immediately that [MATH] is proper, flat and has connected fibers of dimension [MATH] |
(b) Let [MATH] be the [MATH] -fold fibered product of the forgetful morphism [MATH] . The fiber of [MATH] at [MATH] is isomorphic to [MATH] . We can view [MATH] as the stack parametrizing smooth curves of genus [MATH] with [MATH] sections that are allowed to cross each other. We have an open embedding [MATH] over [MATH... |
In 13 , p. 194-195] , the Fulton-MacPherson compactification of [MATH] is described as the moduli space of all configurations of [MATH] distinct smooth points on [MATH] with trees of [MATH] such that the resulting pointed nodal curve has finite automorphism group, up to projective equivalence on the rational curves. Th... |
(c) The morphism [MATH] is flat because it is the pullback of [MATH] along the open embedding [MATH] . By (b) its fibers are the Fulton-MacPherson compactifications, which are smooth. |
(d) This follows immediately from (c). Let [MATH] be an integer. A hyperelliptic curve [MATH] of genus [MATH] is a smooth complete curve admitting a morphism [MATH] of degree [MATH] . Equivalently, there exists an involution [MATH] of [MATH] with quotient the projective line. It is a classical fact that such [MATH] is ... |
Let [MATH] be the closed substack of [MATH] , with reduced structure, parametrizing hyperelliptic curves of genus [MATH] . Both [MATH] and its Zariski closure [MATH] in [MATH] are smooth and irreducible of dimension [MATH] ; see , Lemma XI.6.15, Exercise XII.C-1] , and |
[Corollary 4.7] for a presentation of [MATH] as a quotient stack. A stable curve [MATH] is defined to be hyperelliptic if it admits an involution [MATH] with only isolated fixed points and such that the quotient [MATH] is a nodal curve of genus [MATH] ; see , p. 101] . Such an involution is again unique and is called t... |
We denote by [MATH] the hyperelliptic mapping class group. It is a standard fact that [MATH] may be constructed as an orbifold quotient of a contractible analytic subspace of the Teichmüller space [MATH] by the action of [MATH] ; see |
or 24 , §1, §3.2] . Therefore, [MATH] is an Eilenberg-Maclane space [MATH] , in the sense of orbifolds. The stack [MATH] parametrizes smooth hyperelliptic curves of genus [MATH] , together with [MATH] distinct marked points. We denote by [MATH] the coarse moduli space of [MATH] . We define the stack [MATH] as the inver... |
Let [MATH] be the moduli stack of hyperelliptic curves with rational tails. It is the inverse image of [MATH] under the forgetful morphism [MATH] |
Proposition 2.2 Let [MATH] and [MATH] (a) The forgetful morphism [MATH] is universally open and has connected fibers of dimension [MATH] |
(b) If [MATH] is a smooth hyperelliptic curve of genus [MATH] , the fiber of [MATH] at [MATH] is isomorphic to [MATH] , and the fiber of [MATH] at [MATH] is the Fulton-MacPherson compactification of [MATH] |
(c) The forgetful morphism [MATH] is proper, smooth, and has irreducible fibers. (d) The forgetful morphism [MATH] is smooth. (e) |
The stacks [MATH] and [MATH] are smooth and irreducible. (f) The stack [MATH] is irreducible, and it is the closure of [MATH] inside [MATH] |
Proof. (a) By Lemma 2.1 (a), the forgetful morphism [MATH] is flat, hence universally open. It follows that [MATH] is open (openness is a topological property, so it holds even if we have given [MATH] the reduced structure), and has connected fibers of dimension [MATH] |
(b) This follows immediately from Lemma 2.1 (b). (c) The morphism [MATH] is the base change of [MATH] along the inclusion [MATH] , hence it is smooth by Lemma |
2.1 (c). (d) This follows immediately from (c). (e) As [MATH] is open in [MATH] , it suffices to prove the claim for [MATH] . We know that [MATH] is smooth and irreducible. By (a), the forgetful morphism [MATH] is open, and by (b) the fibers of [MATH] are irreducible. The conclusion follows from 31 , 004Z] |
(f) By (e) it is enough to show that [MATH] is the dense in [MATH] . Let [MATH] be a non-empty open substack. By (a), the image of [MATH] under [MATH] is open and non-empty, hence it intersects [MATH] . It follows that [MATH] intersects [MATH] . By (e), [MATH] is irreducible. Since [MATH] is open in [MATH] , we conclud... |
If [MATH] is a topological stack, the singular (Betti) homology and cohomology of [MATH] are defined; see , Definition 33] We briefly sketch the definition, referring the reader to , p. 22, p. 27-28] for the details. One fixes a groupoid presentation of [MATH] , takes the associated simplicial nerve, and constructs a d... |
In this paper, we will be exclusively interested in cohomology with rational coefficients of [MATH] , which we denote by [MATH] Proposition 2.3 |
Let [MATH] be a Deligne-Mumford stack with coarse moduli space [MATH] (a) The induced map [MATH] is an isomorphism. (b) If [MATH] is pure-dimensional, the induced map [MATH] is an isomorphism. |
(c) Let [MATH] be one of [MATH] . Then the induced maps [MATH] , are isomorphisms. Proof. (a) See , Proposition 36] (b) This is a particular case of 35 , Proposition 6.1] |
(c) See , Lemma XIII.6.6] for a proof in the case of [MATH] and [MATH] , the case of [MATH] being entirely analogous (another reference is 17 , Proposition 3.88] ). |
3. Cohomology of fibers of the forgetful morphism We will understand the low dimensional cohomology of [MATH] by relating it to the cohomology of [MATH] via the Leray spectral sequence for the forgetful morphism [MATH] . In order to do so, we must first understand the cohomology of the fibers of [MATH] . By 2.2 , the m... |
Let [MATH] and [MATH] be an oriented manifold of dimension [MATH] . Let [MATH] be the projection onto the [MATH] -th factor, and [MATH] be the projection onto the [MATH] -th and [MATH] -th factors. Denote by [MATH] the class of the Poincaré dual of the diagonal in [MATH] . Let [MATH] be the differential bigraded commut... |
[MATH] [MATH] [MATH] [MATH] [MATH] for any [MATH] and for any [MATH] . The differential is defined by [MATH] and [MATH] for [MATH] |
Proposition 3.1 (Totaro) Let [MATH] be an oriented manifold of dimension [MATH] . Then the differential bigraded algebra [MATH] is isomorphic to the [MATH] -page of the Leray spectral sequence for [MATH] . The [MATH] -page is the first page of the spectral sequence with non-trivial differentials. |
Proof. 34 , Theorem 1, Theorem 2] Proposition 3.2 (Totaro) If [MATH] is a smooth projective variety over the complex numbers, every differential in the Leray spectral sequence for [MATH] except [MATH] vanishes, and the rational cohomology of [MATH] is the cohomology of [MATH] |
Proof. 34 , Theorem 3] We will apply these results to the case where [MATH] is a smooth hyperelliptic curve of genus [MATH] (so [MATH] ). By the Künneth Formula: |
[EQUATION] Fix a point [MATH] of [MATH] . For [MATH] , let [MATH] , where [MATH] is the Poincaré dual of the class of [MATH] in [MATH] . In ( 3.1 ), the homomorphisms [MATH] are induced by the projections [MATH] . It follows that under the isomorphism ( 3.1 ), [MATH] is a generator for the [MATH] -th summand [MATH] ins... |
Denote by [MATH] the projection to [MATH] of [MATH] with respect to the Künneth decomposition [EQUATION] and set [MATH] The inclusions [MATH] factor as |
[EQUATION] where the maps on the left are cross products, and the maps on the right are induced by the projections [MATH] ; see 18 , Appendix 3.3.B] |
Lemma 3.3 (a) The [MATH] and the [MATH] generate the same subspace as the [MATH] and [MATH] (b) The [MATH] and the [MATH] (for [MATH] ) are linearly independent. |
Proof. (a) With respect to ( 3.2 ) we have [MATH] , therefore [MATH] is a linear combination of [MATH] and [MATH] (b) By what we have said above, under the isomorphism ( 3.1 ) each summand [MATH] contains exactly one of the [MATH] , and each [MATH] contains exactly one of the [MATH] , therefore the [MATH] and [MATH] ar... |
Lemma 3.4 Let [MATH] be a smooth curve, and consider the Leray spectral sequence for [MATH] (a) A basis for the image of [MATH] is given by the classes [MATH] in [MATH] , where [MATH] |
(b) The differentials [MATH] and [MATH] are injective. Proof. By 3.1 , we may identify the [MATH] -page of the Leray spectral sequence for [MATH] with [MATH] |
By definition, [MATH] is the [MATH] -vector space with basis [MATH] , where [MATH] (recall that [MATH] , since [MATH] is even). Moreover, [MATH] . The [MATH] (for [MATH] ) form a basis of [MATH] , and by Lemma |
3.3 (b) the [MATH] are linearly independent (for [MATH] ). We deduce that the differential [MATH] is injective and that the [MATH] form a basis of the image of [MATH] . By 3.1 , this proves (a) and the injectivity of [MATH] |
We now prove that [MATH] is injective. The vector space [MATH] has a basis consisting of the [MATH] , where [MATH] , and [MATH] , together with the [MATH] , where [MATH] and [MATH] . We identify [MATH] with [MATH] . For every [MATH] , we have |
[EQUATION] Consider an element [MATH] , where the indices satisfy the conditions above. Fix [MATH] , and consider the projection [MATH] of [MATH] to the summand [MATH] . We have |
[EQUATION] where [MATH] . By Lemma 3.3 (b), the [MATH] (for [MATH] ) are linearly independent. In the first sum of ( 3.3 ) we are considering pairs [MATH] such that [MATH] , and in the second sum [MATH] . Thus, ( 3.3 ) is an irredundant linear combination of the [MATH] (for [MATH] ). Assume now that [MATH] . Then [MATH... |
A presentation of the hyperelliptic mapping class group [MATH] was given by Birman and Hilden in . There are [MATH] generators [MATH] . We refer the reader to 32 , §2.2] for the statement of the theorem. For the convenience of the reader, we transcribe the description of the images [MATH] of the generators under the mo... |
[EQUATION] The homomorphism [MATH] is given by considering the action of [MATH] on [MATH] , in the coordinates given by the standard symplectic basis [MATH] for [MATH] Let |
[EQUATION] Denote by [MATH] the [MATH] identity matrix. Using the standard symplectic basis [MATH] for [MATH] , we can write [EQUATION] |
The group of orientation-preserving diffeomorphisms of [MATH] naturally acts on [MATH] , so it acts on [MATH] and on the fat diagonal of [MATH] , hence on [MATH] and therefore on [MATH] . The subgroup of diffeomorphisms that are isotopic to the identity of [MATH] acts trivially on [MATH] , hence we obtain an induced ac... |
Lemma 3.5 Let [MATH] be the projection of the Poincaré dual of the class of the diagonal in [MATH] to [MATH] under the decomposition ( 3.2 ). Then the [MATH] -invariant subspace of [MATH] is exactly the one-dimensional subspace [MATH] |
Proof. Using the standard symplectic basis [MATH] for [MATH] , we may represent an element [MATH] of [MATH] as a square matrix [MATH] of size [MATH] . In these coordinates, if [MATH] is [MATH] -invariant, then [MATH] satisfies |
[EQUATION] In particular, by considering [MATH] , for [MATH] , we see that [EQUATION] Let [MATH] be the [MATH] matrix [EQUATION] |
We now impose the condition [MATH] for each [MATH] . In principle, it suffices to impose the conditions [MATH] , however the computations become trickier. |
[MATH] [MATH] is a block diagonal matrix, with [MATH] blocks of size [MATH] [MATH] the first block of [MATH] is a multiple of [MATH] |
[MATH] the last block of [MATH] is a multiple of [MATH] [MATH] every block is a multiple of [MATH] , and with the same coefficient. |
Therefore, the invariants form a one-dimensional vector space, [EQUATION] By 26 , Theorem 11.11] , this is exactly [MATH] We are ready to prove the main results of this section. They concern the [MATH] -invariants and the [MATH] -invariants in [MATH] |
Proposition 3.6 Let [MATH] be a smooth hyperelliptic curve, let [MATH] , and denote by [MATH] the natural open embedding. (a) The pullback [MATH] is an isomorphism. Moreover, the natural action of [MATH] on [MATH] has no non-zero invariants. |
(b) The vector space [MATH] has a basis given by [MATH] and [MATH] (for [MATH] ). (c) The pullback [MATH] induces a surjective map [MATH] . Moreover, [MATH] is a basis for [MATH] |
Proof. By 3.1 , we may identify the Leray spectral sequence for the inclusion [MATH] with [MATH] . By Lemma 3.4 , after applying the differential [MATH] , the spectral sequence becomes as in Figure 1. |
In the figure, [MATH] is a subspace of [MATH] , and [EQUATION] Recall that in the Leray spectral sequence for a map [MATH] the edge homomorphisms [MATH] on the [MATH] -axis coincide with the usual pullback in cohomology [MATH] . By 3.2 , we deduce that [MATH] is an isomorphism, and we obtain a short exact sequence |
[EQUATION] The composition [MATH] is given by pullback in cohomology. If [MATH] is a group acting on [MATH] through homeomorphisms, then [MATH] acts diagonally on [MATH] and on [MATH] , therefore it acts on the cohomology of [MATH] and [MATH] . By the functoriality of the Leray spectral sequence, [MATH] acts on the spe... |
(a) We have already shown that [MATH] is an [MATH] -equivariant isomorphism. To show that [MATH] , it suffices to show that [MATH] . By the Künneth formula, we have an [MATH] -equivariant isomorphism [MATH] , hence it is enough to show that [MATH] . Let [MATH] be the (algebraic) hyperelliptic involution of [MATH] . It ... |
[EQUATION] Here [MATH] is a quotient stack, and the second isomorphism comes from 2.3 (a). (b) The decomposition ( 3.1 ) is [MATH] -equivariant. The summands [MATH] are clearly [MATH] -invariant, and [MATH] by Lemma |
3.5 . Looking back at the definitions of [MATH] and [MATH] given after ( 3.1 ), we deduce that [MATH] is [MATH] and the [MATH] [MATH] . By Lemma |
3.3 , we deduce that the [MATH] and the [MATH] (for [MATH] ) are a basis of [MATH] (c) The group of orientation-preserving diffeomorphisms of [MATH] acts on [MATH] , hence on ( 3.4 ). Diffeomorphisms that are isotopic to the identity act trivially on ( 3.4 ), so we obtain an action of [MATH] . Taking [MATH] -invariants... |
[EQUATION] Denote by [MATH] the hyperelliptic involution of [MATH] , in the sense of topology. It is an element of [MATH] that acts on [MATH] by [MATH] ; see 12 , p. 215-216] . In particular, [MATH] , and so [MATH] . As [MATH] is a subspace of [MATH] , we deduce that [MATH] . Combining this with ( 3.5 ), we deduce that... |
[EQUATION] which defines [MATH] . The first term of ( 3.6 ) is a trivial [MATH] -module. By 19 , Corollary 3.3] [MATH] , so ( 3.6 ) stays exact after taking [MATH] -invariants. By (b) [MATH] has a basis consisting of [MATH] and [MATH] (for [MATH] ), and by Lemma |
3.4 (a) the image of [MATH] is [MATH] . Therefore, the surjective linear map [MATH] sends [MATH] to a basis of [MATH] , as desired. Since [MATH] is an isomorphism, we conclude that the composition |
[EQUATION] is surjective and sends [MATH] to a basis of [MATH] , as desired. 4. Proof of Theorem 1.1 (a) Let [MATH] be the natural forgetful morphism. We intend to compute [MATH] and [MATH] using the Leray spectral sequence for [MATH] |
Recall that if [MATH] is a (topologically) locally trivial orbifold fibration, then for any [MATH] the sheaf [MATH] is a local system on [MATH] ; see 29 , §2] . The morphism [MATH] is a locally trivial orbifold fibration: it is enough to show this for [MATH] , in which case the result follows from embedding [MATH] in t... |
[EQUATION] for every point [MATH] Lemma 4.1 We have [MATH] . Furthermore [EQUATION] Proof. For every point [MATH] , the fiber of the local system [MATH] at [MATH] is given by [MATH] . This is canonically isomorphic to [MATH] because [MATH] is connected. Therefore, the local system [MATH] corresponds to the trivial one-... |
By 20 , Theorem 2.13] [MATH] has the rational cohomology of a point, so the result follows. Lemma 4.2 We have [MATH] Proof. Let [MATH] . Denote by [MATH] the structure morphism to the coarse moduli space. We wish to compute [MATH] using the Leray spectral sequence for [MATH] and [MATH] |
[EQUATION] It is enough to show that all terms in the [MATH] -page of the spectral sequence are zero. To prove this, it suffices in turn to show that [MATH] for every [MATH] Let [MATH] be a hyperelliptic curve. The fiber of [MATH] above [MATH] is the classifying space [MATH] . This implies that the fiber of [MATH] at [... |
[EQUATION] On the right hand side we are considering group cohomology, where [MATH] acts diagonally on [MATH] . Since the coefficient module is a [MATH] -vector space and [MATH] is finite by Hurwitz’s theorem , Exercise 1.F] , group cohomology vanishes for [MATH] . Therefore, [MATH] for every [MATH] . By 3.6 (a), |
[EQUATION] so [MATH] as well. We now come to the main result of this section, which implies 1.1 (a). Proposition 4.3 We have: (a) |
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