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[MATH] (b) [MATH] has a basis given by [MATH] -classes. Proof. We consider the Leray spectral sequence for [MATH] [EQUATION] The combination of Lemma |
4.1 and Lemma 4.2 , shows that in the [MATH] -page the first two rows are zero, with the exception of [MATH] . It follows that [EQUATION] |
and that the edge homomorphism [EQUATION] is an isomorphism. Recall that in the Leray spectral sequence for a map [MATH] , the edge homomorphisms [MATH] on the [MATH] -axis are given by sending a cohomology class [MATH] to the section [MATH] of [MATH] . Since [MATH] is an orbifold [MATH] , associating to a finite-dimen... |
[EQUATION] Since the first Chern class [MATH] commutes with pullbacks, we have a commutative diagram [EQUATION] where [MATH] and [MATH] are given by restriction to [MATH] |
Let [MATH] be the restriction of the [MATH] -classes to the hyperelliptic locus. Recall that, by definition, [MATH] is the class of the line bundle [MATH] on [MATH] which to a family [MATH] in [MATH] with sections [MATH] associates the line bundle [MATH] . The restriction [MATH] of [MATH] to [MATH] is given by the pull... |
[EQUATION] Since [MATH] , the canonical class of [MATH] is a non-zero scalar multiple of the Poincaré dual of the class of a point. Pulling back to [MATH] , we see that [MATH] restricts to a scalar multiple of [MATH] . From the commutativity of the diagram, we see that [MATH] is a non-zero scalar multiple of [MATH] . B... |
Recall from 2.3 that [MATH] and its coarse space [MATH] have the same rational cohomology and rational Picard group. The exponential sequence for [MATH] (see , §2.8] ) gives |
[EQUATION] where [MATH] is given by the first Chern class. Recall that [MATH] is a full lattice in [MATH] . By 4.3 (a) and 2.3 (a), we have [MATH] , thus [MATH] . It follows that the rational Chern class [MATH] is injective. It is also surjective, since [MATH] is [MATH] -classes. |
5. Description of the boundary Proposition 5.1 The group [MATH] is [MATH] -classes and boundary divisors. Proof. By 4.3 [MATH] is [MATH] -classes. The conclusion immediately follows from the consideration of the excision exact sequence |
[EQUATION] where [MATH] are the codimension [MATH] components of the boundary of [MATH] ; see 35 , p. 614] , or 25 , Proposition 2.4.1] |
In order to complete the proof of 1.1 (b), we must study the geometry of the boundary of [MATH] and prove that [MATH] -classes and classes of boundary divisors are linearly independent. Our analysis requires some basic deformation theory, as explained, for example, in , Chapter XI] and |
Recall the product decomposition for an irreducible boundary divisor of [MATH] as the image of a finite morphism from a product of moduli of curves with smaller genus and number of markings; see |
or , §X.10, §XII.10] for detailed proofs. Informally, there is an irreducible boundary divisor [MATH] , which is the image of the finite map |
[EQUATION] that glues the last two sections to a node. Moreover, there is an irreducible boundary divisor [MATH] , which is the image of the finite morphism |
[EQUATION] that glues the two curves along the sections [MATH] and [MATH] , for every [MATH] and [MATH] such that the domain of [MATH] is non-empty. These divisors are the irreducible components of [MATH] , and the morphisms are called clutching morphisms. More generally, [MATH] can be stratified by topological type, a... |
We also recall the structure of the boundary of [MATH] . We refer the reader to , §X.3, §XIII.8] for a more thorough discussion. For a stable curve [MATH] , and nodes [MATH] of [MATH] , denote by [MATH] the partial normalization of [MATH] at [MATH] . The irreducible divisor [MATH] parametrizes stable hyperelliptic curv... |
Our purpose is now to generalize this description to the boundary of [MATH] , for a fixed integer [MATH] . We will do so by considering the inverse images of the boundary divisors of [MATH] under the forgetful morphism [MATH] . Recall from 2.2 that [MATH] is open. From now on, we denote by [MATH] a subset of [MATH] |
Recall that we denote by [MATH] the moduli stack of hyperelliptic curves with rational tails. The complement [MATH] is a union of irreducible divisors [MATH] parametrizing curves having rational tails marked by [MATH] , for every [MATH] having at least two elements (so that the resulting pointed curves are stable); see... |
Let now [MATH] be an irreducible divisor of [MATH] mapping to the boundary of [MATH] . If [MATH] maps to some [MATH] then [MATH] is a component of [MATH] . The fiber above a point [MATH] contains the ordered configuration space [MATH] as a dense open subspace. Here [MATH] is defined in the same way as for smooth curves... |
We now claim that [MATH] has exactly [MATH] irreducible components. To prove this, it is enough to exhibit [MATH] disjoint open subsets of [MATH] and take their closures. For every [MATH] , denote by [MATH] the set of points of [MATH] corresponding to hyperelliptic curves [MATH] obtained by glueing two smooth hyperelli... |
Note that [MATH] for any choice of [MATH] , because the divisor [MATH] is defined only for [MATH] . Moreover, a point of [MATH] belongs to [MATH] if and only if it belongs to [MATH] and it does not belong to [MATH] for any [MATH] . The curves parametrized by [MATH] are all singular, hence they belong to at least one [M... |
[EQUATION] By 22 , Corollary 3.9] the clutching morphisms are finite, hence [MATH] is closed in [MATH] , and so [MATH] is open in [MATH] . Therefore, the [MATH] are [MATH] pairwise disjoint open subspaces of [MATH] . If we denote by [MATH] the closure of [MATH] , we have shown that |
[EQUATION] is the irredundant decomposition of [MATH] in irreducible components. For every [MATH] , the general fiber of [MATH] has dimension [MATH] , hence every [MATH] is an irreducible divisor of the boundary of [MATH] |
A similar reasoning shows that, for [MATH] [EQUATION] where every [MATH] is an divisor of [MATH] . A general point of [MATH] represents a pointed hyperelliptic curve [MATH] obtained by joining two smooth hyperelliptic curves [MATH] and [MATH] of genera [MATH] and [MATH] at two points that are conjugated under the invol... |
[EQUATION] is an irreducible divisor of [MATH] . We have obtained the following description of the boundary of [MATH] Proposition 5.2 |
The boundary of [MATH] is a divisor. Let [MATH] be an irreducible boundary divisor, and let [MATH] be a general curve parametrized by [MATH] . Then exactly one of the following holds. |
(i) We have [MATH] for some [MATH] and some [MATH] [MATH] is obtained by glueing two smooth hyperelliptic curves [MATH] and [MATH] of genera [MATH] and [MATH] at a Weierstrass point, and [MATH] if and only if [MATH] |
(ii) We have [MATH] for some [MATH] and some [MATH] [MATH] is obtained by glueing two smooth irreducible curves [MATH] and [MATH] of genera [MATH] and [MATH] at a pair of points switched by the involutions, and [MATH] if and only if [MATH] |
(iii) We have [MATH] [MATH] is irreducible and has exactly one node. If [MATH] is of the form [MATH] , a general point of [MATH] belongs to [MATH] . Otherwise, a general curve parametrized by [MATH] is stable even after removing the marked points. |
The purpose of this section is to determine the class of the restriction of each boundary divisor of [MATH] in [MATH] . If [MATH] is a stable [MATH] -pointed curve, we denote by [MATH] the ordered [MATH] -uple of markings, and we write [MATH] for [MATH] . We denote by [MATH] the forgetful morphism. |
Recall that if [MATH] is a family of pointed nodal curves, the locus of points in [MATH] having stable fiber is open; see , Lemma X.3.4] . It follows that there exists an open substack [MATH] of [MATH] parametrizing curves [MATH] such that [MATH] is stable as an unmarked curve. Clearly [MATH] is contained in [MATH] |
Lemma 5.3 The restriction of [MATH] to [MATH] is smooth. Proof. We must show that for every [MATH] such that [MATH] is stable as a curve without marked points, the differential [MATH] of [MATH] at [MATH] is surjective. |
We will use the theory of first order deformations of nodal curves; see e.g. , §XI.3] . We have the local-to-global [MATH] spectral sequences |
[EQUATION] [EQUATION] The inclusion [MATH] induces a homomorphism from the first spectral sequence to the second. Since [MATH] is a curve, |
[EQUATION] Considering the associated five-term short exact sequences, we obtain by functoriality a commutative diagram with exact rows: |
[MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] Here the vertical maps are canonically induced from the inclusion [MATH] |
Let [MATH] be the normalization of [MATH] [MATH] , and [MATH] be the inverse image of the nodes of [MATH] . Since [MATH] is stable as an unmarked curve, [MATH] maps to [MATH] in [MATH] . An elementary computation shows that [MATH] and [MATH] ; see , p. 182, p. 186] . The sheaves [MATH] and [MATH] are isomorphic and con... |
[MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] where [MATH] is induced from the natural inclusion [MATH] . The rows of this diagram appear in , XI.3 (3.14)] By the snake lemma, to prove the surjectivity of [MATH] it is enough to show that [MATH] is surjective. Consider the following... |
[EQUATION] Since [MATH] is supported on a zero-dimensional locus, the associated cohomology long exact sequence [EQUATION] shows the surjectivity of [MATH] |
Let [MATH] , and define [MATH] as the union of [MATH] and [MATH] . The stack [MATH] is open in [MATH] , and it contains [MATH] . By 5.2 [MATH] is dense in every boundary divisor, hence its complement has codimension at least [MATH] |
Proposition 5.4 (a) The morphism [MATH] is smooth. (b) The stack [MATH] is smooth. In particular, [MATH] is smooth in codimension one. |
Proof. (a) By 2.2 , the morphism [MATH] is smooth. The morphism [MATH] is the base change of the morphism [MATH] along the inclusion [MATH] , hence it is smooth by Lemma |
5.3 . As smoothness is a local property on the domain, the proof is complete. (b) This follows from (a) and the smoothness of [MATH] |
Let [MATH] be a morphism of smooth Deligne-Mumford stacks over [MATH] , let [MATH] be a smooth locally closed substack, and let [MATH] . We say that [MATH] and [MATH] are transverse at [MATH] if for every point [MATH] in [MATH] we have [MATH] ; c.f. 11 , p. 18] and |
. By 5.4 [MATH] is smooth. The next two lemmas will show that [MATH] is transverse at every point to all the clutching morphisms [MATH] , where [MATH] is a boundary divisor of [MATH] |
Lemma 5.5 Let [MATH] , and let [MATH] be an irreducible boundary divisor of [MATH] containing [MATH] . Then the clutching map [MATH] is transverse to [MATH] at [MATH] |
Proof. If [MATH] , it follows from Lemma 5.3 that the forgetful morphism [MATH] is an orbifold submersion at [MATH] . If [MATH] , by Lemma |
2.1 (c) the forgetful morphism [MATH] is again an orbifold submersion around [MATH] . Since [MATH] (set-theoretically), [MATH] is locally the inverse image of [MATH] . Therefore, in each case we obtain |
[EQUATION] By 22 , Corollary 3.9] [MATH] is unramified, so if [MATH] is a point of the domain of [MATH] mapping to [MATH] [MATH] is injective and [MATH] has codimension one in [MATH] . Therefore, to prove transversality of [MATH] it suffices to show that there is a vector [MATH] that does not belong to the image of [MA... |
5.3 , we may lift [MATH] to a vector [MATH] . Since not all nodes may be smoothed inside [MATH] , the vector [MATH] does not belong to [MATH] |
Let [MATH] be the natural inclusion, and let [MATH] and [MATH] denote the composition [MATH] . Recall that since we do not know if [MATH] is smooth, we do not know if [MATH] is an isomorphism, or even injective. |
Proposition 5.6 We have the following identities in [MATH] [EQUATION] Proof. Since the complement of [MATH] has codimension [MATH] in [MATH] , the restriction map |
[EQUATION] is an isomorphism. It thus suffices to prove the relations in [MATH] Let [MATH] , let [MATH] be the nodes of [MATH] , and let [MATH] be the dimension of [MATH] . We recall some standard facts on universal deformations of [MATH] -pointed stable curves; good references on the topic are 10 , p. 82] (when [MATH]... |
[EQUATION] such that for every [MATH] we have [EQUATION] for suitable [MATH] . Informally speaking, [MATH] is the locus where [MATH] remains a node. |
Assume now that [MATH] . Since [MATH] is locally closed in [MATH] , the completed stalk [MATH] of [MATH] at [MATH] is a quotient of [MATH] . By 5.4 (b), [MATH] is smooth, hence [MATH] is also a power series ring. If [MATH] , we denote by [MATH] its reduction in [MATH] . We also let [MATH] be the maximal ideal of [MATH]... |
Let [MATH] be a boundary divisor through [MATH] , let [MATH] , let [MATH] be a boundary divisor of [MATH] which appears as an irreducible component of [MATH] , and assume that [MATH] is a general point. As [MATH] and [MATH] are smooth, [MATH] and [MATH] are Cartier divisors in [MATH] and [MATH] , respectively. Since [M... |
Since [MATH] is a Cartier divisor, we may compute the divisor class [MATH] by computing the multiplicities of [MATH] at its irreducible components; see 31 , Tags 0AZA, 0B05, 0DR4] . The multiplicity of [MATH] along [MATH] equals the length of the ring [MATH] , that is, the natural number [MATH] such that [MATH] (note t... |
Assume first that [MATH] . Since [MATH] does not contain the general point of any boundary divisor of [MATH] other than [MATH] , we need only consider the case when [MATH] . Then [MATH] , and the image of [MATH] at [MATH] is the hyperplane [MATH] (recall that clutching maps are unramified by 22 , Corollary 3.9] ). By L... |
5.5 , we know that [MATH] intersects [MATH] transversely at [MATH] . This means exactly that [MATH] , and so we may complete [MATH] to a system of regular parameters for [MATH] . In particular, [MATH] is a prime ideal, hence [MATH] . This proves the first equality. |
Assume now that [MATH] . Set-theoretically, [MATH] coincides with the union of all the [MATH] and [MATH] . If [MATH] , then [MATH] and [MATH] , the proof being identical to that of the previous case. |
Consider the case when [MATH] . Then [MATH] , and the image of [MATH] at [MATH] is the union of the hyperplanes [MATH] and [MATH] . By Lemma |
5.5 , the clutching map [MATH] is transverse to [MATH] at [MATH] , hence both hyperplanes are transverse to [MATH] at [MATH] . As before, this means that [MATH] , and so [MATH] and [MATH] are prime ideals. It follows that [MATH] , hence [MATH] and [MATH] . This proves the second equality. |
6. Proof of Theorem 1.1 (b) To complete the proof of 1.1 , we will use the method of test curves, as explained in 17 , Lemma 3.94] , in conjunction with 5.6 |
Proof of 1.1 (b). By 5.1 , it suffices to prove that [MATH] -classes and irreducible boundary divisors are linearly independent in [MATH] . Since [MATH] is smooth and its complement has codimension [MATH] , we have [MATH] . Let |
[EQUATION] and assume that [MATH] . We may assume that all the coefficients are integers. We will show that each coefficient is zero by computing the degree of [MATH] on suitable families of test curves. In order to compute degrees, we need to make sure that our test families are entirely contained in [MATH] |
By 1.1 (a), the [MATH] -classes are independent in [MATH] , hence restriction to [MATH] shows that [MATH] for each [MATH] . We split the rest of the proof in several lemmas. |
Lemma 6.1 We have [MATH] if [MATH] Proof. Let [MATH] be a smooth hyperelliptic curve of genus [MATH] , and let [MATH] be distinct points of [MATH] . Denote by [MATH] the blow-up of [MATH] at [MATH] . For [MATH] , let [MATH] be the proper transform of [MATH] in [MATH] , and let [MATH] be the proper transform of the diag... |
[EQUATION] and all other divisors are trivial. We obtain the relation [EQUATION] The points [MATH] were arbitrarily chosen, therefore [MATH] for [MATH] . Changing order of the sections, we obtain [MATH] for every [MATH] of cardinality [MATH] , as desired. |
Lemma 6.2 We have [MATH] for every [MATH] such that [MATH] Proof. Recall that in order for [MATH] to exist, we must have [MATH] . Assume that [MATH] does not contain [MATH] . Consider a smooth hyperelliptic curve [MATH] of genus [MATH] , a smooth rational curve [MATH] , and let [MATH] and [MATH] . Consider distinct poi... |
The glueing of the proper transform of [MATH] and [MATH] corresponds to a node [MATH] in each fiber, such that [MATH] belongs to the component of genus [MATH] through [MATH] if and only if [MATH] ; there are no isolated nodes with the same property. The self-intersection of [MATH] and [MATH] is zero, and blowing up dec... |
[EQUATION] For every [MATH] [MATH] , there is exactly one fiber admitting a node [MATH] , such that [MATH] and [MATH] belong to the component of genus [MATH] through [MATH] , and no other section does. Finally, there is one fiber consisting of a curve of genus [MATH] and a tree with two rational components, such that [... |
[EQUATION] and the other divisors have degree zero. By Lemma 6.1 [MATH] , it follows that [MATH] . Permuting sections and letting [MATH] vary, we deduce that [MATH] is independent of [MATH] . By Lemma |
6.1 , we conclude that [MATH] for every [MATH] In , Theorem XIII.8.4] , where the independence of boundary divisors is shown for [MATH] (see , §4.(b)] for the original proof), one-parameter families [MATH] of unmarked stable hyperelliptic curves of genus [MATH] are constructed, such that the general fiber is smooth and... |
(i) any singular fiber of [MATH] has one node of type [MATH] , and no other node; (ii) for [MATH] , any singular fiber of [MATH] either has a pair of nodes of type [MATH] , and no other node, or has a node of type [MATH] , and no other node; |
(iii) for [MATH] , any singular fiber of [MATH] either has a node of type [MATH] , and no other node, or has a node of type [MATH] , and no other node. |
We now endow suitably modified versions of the [MATH] with [MATH] sections, and use them as test curves. Lemma 6.3 We have [MATH] |
Proof. We start by recalling the construction of [MATH] ; see , Theorem XIII.8.4] for more details. In [MATH] , consider general divisors [MATH] of bidegree [MATH] , general divisors [MATH] of bidegree [MATH] , and set [MATH] . Then [MATH] has bidegree [MATH] and its singular locus [MATH] consists of [MATH] nodes. If [... |
Let now [MATH] , and consider [MATH] points [MATH] in [MATH] . If we require the [MATH] to be sufficiently general, then the [MATH] are not contained in [MATH] , and do not intersect [MATH] . Let [MATH] be the inverse image of the strict transform of [MATH] . It is a ramified double cover of [MATH] , possibly split. No... |
Lemma 6.4 We have [MATH] for every [MATH] and every [MATH] Proof. We first recall the construction of [MATH] . Let [MATH] be a point, and choose divisors [MATH] of bidegree [MATH] , generic among those passing through [MATH] , and general divisors [MATH] of bidegree [MATH] . Set [MATH] . The singular locus [MATH] of [M... |
Let now [MATH] , and consider [MATH] general points [MATH] and [MATH] divisors [MATH] of bidegree [MATH] passing through [MATH] . By choosing the configuration to be sufficiently general, we may assume that all intersections between the [MATH] , the [MATH] , and the [MATH] are transverse, that all intersection points o... |
For [MATH] , denote by [MATH] the inverse image of the proper transform of [MATH] , and for [MATH] let [MATH] be the inverse image of the proper transform of [MATH] . The [MATH] are ramified double covers of [MATH] . As in the proof of Lemma |
6.3 , using the [MATH] we construct a ramified cover [MATH] such that the pullback of [MATH] along [MATH] splits as the disjoint union of two sections [MATH] and [MATH] . We denote by [MATH] the base change of [MATH] along [MATH] |
Let [MATH] be the fiber of [MATH] at [MATH] . Then [MATH] is a stable nodal curve of genus [MATH] with a pair of nodes of type [MATH] , and no other nodes. It is the only fiber of [MATH] with a pair of nodes of type [MATH] . Moreover, the [MATH] intersect [MATH] in the genus [MATH] component when [MATH] , and in the ge... |
By our genericity requirement, all the intersections between the [MATH] are transverse, and the morphism [MATH] is unramified at the images of the intersection points. Therefore, the intersections between the [MATH] belong to a smooth fiber of [MATH] and are transverse. Blowing up the intersection points of the [MATH] ... |
the general fiber is smooth; a singular fiber either admits a pair of nodes of type [MATH] and no other node, or exactly one node of type [MATH] , or is the union of a smooth curve of genus [MATH] and a smooth rational curve with exactly two markings; |
for every fiber admitting a pair of nodes of type [MATH] , a section [MATH] marks the component of genus [MATH] if and only if [MATH] |
In particular, [MATH] is entirely contained in [MATH] By 5.6 and 17 , Lemma 3.94] [MATH] . By Lemma 6.1 and Lemma 6.3 , we have [MATH] , hence [MATH] , as desired. |
Lemma 6.5 We have [MATH] for every [MATH] and every [MATH] Proof. The proof is essentially the same as that of Lemma 6.4 , using the families [MATH] of , Theorem XIII.8.4] instead of the [MATH] |
Combining all the steps, we get [MATH] in [MATH] . This completes the proof of 1.1 Acknowledgments I would like to thank Mattia Talpo and Ben Williams for useful discussions and for reading a first version of the article, Madhav Nori for a very nice conversation on this topic, my advisor Zinovy Reichstein for his guida... |
F. Scavia, Department of Mathematics, University of British Columbia, Vancouver, British Columbia, V6T 1Z4 E-mail address scavia@math.ubc.ca |
# Source: arxiv 1806.05348 # Title: Elastically Collective Nonlinear Langevin Equation Theory of Dynamics in Glass-Forming Liquids: Transient Localization, Thermodynamic Mapping and Cooperativity # Sections: all # Downloaded: 2026-03-03T05:14:00.828055+00:00 |
Elastically Collective Nonlinear Langevin Equation Theory of Dynamics in Glass-Forming Liquids: Transient Localization, Thermodynamic Mapping and Cooperativity |
Abstract We analyze multiple new issues concerning activated relaxation in glassy hard sphere fluids and molecular and polymer liquids based on the Elastically Collective Nonlinear Langevin Equation (ECNLE) theory. By invoking a high temperature reference state, a near universality of the apparent dynamic localization ... |
Introduction The construction of a quantitative, predictive, force-level theory of activated glassy structural relaxation at the level of atoms or molecules remains a grand challenge in statistical mechanics . Recently, Mirigian and Schweizer formulated and applied a force-based dynamical theory that relates thermodyna... |
The initial formulation of ECNLE theory for rigid molecules is based on a quasi-universal mapping, is devoid of fit parameters, has no divergences at finite temperature or below random close packing, and accurately captures the alpha relaxation time over 14 decades . Extension to polymer liquids is based on a disconnec... |
ECNLE theory has also been extended and applied to other problems: spatially heterogeneous relaxation in free standing thin films , segmental relaxation in polymer nanocomposites , attractive glass and gel formation in dense sticky colloidal suspensions , the effect of random pinning in dense liquids , penetrant diffus... |
In this article, we revisit the basics of ECNLE theory of 1-component liquids to further establish it physical picture and address new questions. After a brief review of key technical aspects in section II, new numerical studies are presented in section III that explore a possible universality of the dynamic transient ... |
II ENCLE Theory and Chemical Mapping As relevant background, the present state of bulk liquid ECNLE theory is briefly reviewed. All aspects have been discussed in great detail in prior papers [4-8]. |
II.1 Quasi-Universal ECNLE Theory of Spherical Particle Liquids ECNLE theory describes the activated relaxation of a tagged particle as a mixed local-nonlocal rare hopping event . Figure 1 shows a cartoon of the key physical elements. The foundational quantity for a tagged spherical particle (diameter, [MATH] ) liquid ... |
Key local lengths (see Fig. ) are the minimum and maximum of the dynamic free energy ( [MATH] and [MATH] , respectively), and jump distance [MATH] ; key energies are the local cage barrier height, [MATH] , and harmonic curvature at the dynamic free energy minimum, [MATH] . The precise nature of the elastic fluctuation ... |
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