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Hurwitz stacks appear basically as correspondences between moduli stacks of pointed curves. Please log in to get access to this content Log in Register for free. The singular points on the moduli spaces for smooth curves. Basics on families of hyperelliptic curves. Mestrano N. Reine Angew. Mumford, D. Oort F. Singularities of the moduli scheme for curves of genus three. Popp H. The singularities of the moduli schemes of curves.
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Rauch H. The singularities of the modulus space. Vistoli A. Steklov Mathematical Institute Moscow Russia 2. Personalised recommendations. Cite article How to cite? So, essentially, the shift group G S must be a subgroup of the group of points of order 2 in Y, acting by translations. This r is called the rank of G.
They fall into 35 types. Most of these are known to be in distinct families, but as explained in [DW09] and reviewed in our appendix, there are a few gaps where the possible existence of isomorphisms is still not known. In fact, one pair from the list in [DW09] has since been shown to be isomorphic, cf. After explaining the required notation, we recall that classification and comment on its current state in the Appendix, in Table 3.
In this work we focus on ten of the families in this table, the ones whose moduli spaces are three dimensional. The best known example, denoted in [DW09], was originally studied by [VW95]. In that case the group action has 48 fixed points, leading to singularities of X that need to be resolved by a crepant resolution. There are four cases where the group action is free, leading to smooth quotients of Hodge numbers 3, 3. In [DW09] these are denoted , , , and The notation r-i means that the group G S has rank r, and that this is the i-th case listed in [DW09] with that given r.
For the four cases with Hodge numbers 3, 3 , the holonomy is a proper subgroup of SU 3 , in fact a finite subgroup. In doing so, we will also describe a connected component M X isomorphic to M X of the moduli space of complex structures on a particular resolution X. What we will find is that M X is isomorphic to a global quotient of H 3, where H is the complex upper half plane. We will prove that the Picard group of M X is a finitely generated abelian group, and calculate the rank of its free abelian part in each case. Despite the fact that Pic M X will often be infinite, we will prove that the Hodge bundle has finite order in each of these cases.
In fact,. The moduli spaces 2. Notation and Terminology. Its dimension is h2,1 X. This is the space we care about! Here are some remarks regarding our terminology. Given a complex algebraic variety X, there is a moduli stack M alg X parameterizing deformations of X over pointed schemes.
Note that we do not require the data of a polarization on our varieties that is, a fixed ample invertible sheaf L on X , so standard results about the existence of a quasi-projective coarse moduli scheme [Vie95] would not be available. In the cases of this paper, X will be Calabi-Yau with singularities in at worst codimension 2, in which case a result of Ran [Ra93] implies that M alg X is still smooth. The stack M alg X determines an underlying analytic stack see [BN05], section 3, for basic definitions regarding complex analytic stacks , which we will refer to as M X for the rest of this paper.
Our goal is to provide a similar global description of M X and ultimately M X, for X a smooth crepant resolution of X More details: the spaces and the maps between them. H denotes the complex upper half plane. Recall that an elliptic curve E, 0 , often abbreviated to just E, is a smooth genus 1 curve E with a marked point 0 E.
If M denotes the moduli stack of elliptic curves, the family U defines a map H M. Note that over the space H 3 we have the family U 3 H 3 of abelian varieties, which will be cruical for later developments. G acts naturally on the family U 3 H 3 acting trivially on the base. Finally, we let X denote any Calabi-Yau resolution of the possibly singular X, and let M X be the moduli space of complex structures on X.
We will construct a discrete group H acting on H 3. Automorphisms of f X 3. Group actions. In this section we will identify a group H acting on both f X and H 3. It is this action that will induce the map f mentioned in section 2. There is also an S 3 action on U 3 by permutation of the three factors, as well as an S 3 action on A 3, so we form the extension 1 A 3 A 3 S 3 1. So A 3 acts on U 3, combining all of the previous actions. Since the group G acts on U 3 by translations of points of order 2 and elliptic inversions, it embeds naturally into A 3.
Since H normalizes G, its action on U 3 descends to an action on f X.
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We remark that since G acts trivially on H 3 and f X, H acts on both of these spaces. Proposition 3. Let g G. For hgh 1 to belong to G, its T 6 component must be a point of order 2. As we mentioned, this element must have order 2.
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The previous proposition shows that H H max in all cases, hence the name. Furthermore, A 2 3 is always contained in H. Firstly, we must verify that A 2 is a subgroup of A. Therefore, A 2 is indeed closed under the product.
Since A 2 3 is clearly preserved by permutation, it follows that A 2 3 H max, as claimed. In particular, A 2 3 H. The technique is straightforward. Since L is by definition the image of H under the quotient map H max L max, we need to determine which elements l L max admit a lift l H max which normalizes G.
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Note that if l and l are two distinct lifts of l, then l normalizes G if and only if l normalizes G. This result follows since l 1 l A 2 3, which commutes with G. Since L max and G are finite, this algorithm can be very easily implemented on a computer.
Related Picard groups of moduli problems
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