Moduli of Curves of Genus One with Twisted Fields
Yi Hu, Jingchen Niu

TL;DR
This paper introduces a new stack for genus one stable curves with twisted fields, providing a blowup-free desingularization of the moduli space of genus one stable maps, advancing the theory of stacks with twisted fields.
Contribution
It constructs a smooth Artin stack for genus one curves with twisted fields and proves its isomorphism to a blowup stack, enabling a blowup-free resolution of the moduli space.
Findings
Constructed a smooth Artin stack for genus one curves with twisted fields.
Proved the stack is isomorphic to a blowup stack of existing moduli.
Provided a blowup-free desingularization of the moduli space of genus one stable maps.
Abstract
We construct a smooth Artin stack parameterizing the stable weighted curves of genus one with twisted fields and prove that it is isomorphic to the blowup stack of the moduli of genus one weighted curves studied by Hu and Li. This leads to a blowup-free construction of Vakil-Zinger's desingularization of the moduli of genus one stable maps to projective spaces. This construction provides the cornerstone of the theory of stacks with twisted fields, which is thoroughly studied in arXiv:2005.03384 and leads to a blowup-free resolution of the stable map moduli of genus two.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAlgebraic Geometry and Number Theory · Homotopy and Cohomology in Algebraic Topology · Advanced Algebra and Geometry
Moduli of Curves of Genus One with Twisted Fields
Yi Hu
Department of Mathematics, University of Arizona, USA.
and
Jingchen Niu
Department of Mathematics, University of Arizona, USA.
Abstract.
We construct a smooth Artin stack parameterizing the stable weighted curves of genus one with twisted fields and prove that it is isomorphic to the blowup stack of the moduli of genus one weighted curves studied by Hu and Li. This leads to a blowup-free construction of Vakil-Zinger’s desingularization of the moduli of genus one stable maps to projective spaces. This construction provides the cornerstone of the theory of stacks with twisted fields, which is thoroughly studied in [8] and leads to a blowup-free resolution of the stable map moduli of genus two.
1. Introduction
Moduli problems are of central importance in algebraic geometry. Many moduli spaces possess arbitrary singularities [12]. Among them, the moduli of degree stable maps from genus nodal curves into projective spaces are particularly important. We aim to resolve the singularities of , that is, to construct a new Deligne-Mumford stack that has smooth irreducible components and normal crossing boundaries and dominates properly and birationally onto the primary component (the component whose general points have smooth domain curves). The problem of resolution of singularities is arguably among the hardest ones in algebraic geometry [3, 4, 9, 10].
The stable map moduli are smooth if and singular if and . For , a resolution was constructed by Vakil and Zinger [13], followed by an algebraic approach of Hu and Li [5]. The latter is achieved by constructing a canonical smooth blowup of the Artin stack of weighted nodal curves of genus one. The method of [5] was further developed in [7] to finally establish a resolution in the case of . The resolution of [7] is achieved by constructing a canonical smooth blowup of the relative Picard stack of nodal curves of genus two.
In higher genus cases, the construction of a possible resolution of the stable map moduli may seem formidable. The constructions of the explicit resolutions in [13, 5, 7] rely on certain precise knowledge on the singularities of the moduli. For arbitrary genus, it calls for a more abstract and geometric approach. As advocated by the first author, every singular moduli space should admit a resolution which itself is also a moduli. Following this principle, we interpret the blowup stack of [5] as a smooth algebraic stack of stable weighted nodal curves of genus one with twisted fields, and consequently, the resolution of as a Deligne-Mumford stack of genus one stable maps with twisted fields. The results in this paper are the first step to tackle the arbitrary genus case.
The main theorem of this paper is the following:
Theorem 1.1**.**
There exits a smooth Artin stack parameterizing the weighted nodal curves of genus one with twisted fields, along with a universal family and a proper and birational forgetful morphism . Moreover, is isomorphic to the blowup stack .
We construct the strata of and the forgetful map in §2; see (2.13). We then glue the strata of together using smooth charts in §3 and conclude that is a smooth Artin stack and is birational to in Corollary 3.7. The universal family is described in Proposition 3.9. We finally show that is isomorphic to in Proposition 4.3, which implies the properness of . These results together establish Theorem 1.1.
We remark that a direct approach to the properness of (i.e. without the comparison with the blowup ) is provided in the proof of [8, Theorem 2.19()], in a more general setting.
We also point out that there should exist a groupoid, represented by , that sends any scheme to the set of the flat families of stable weighted nodal curves of genus 1 with twisted fields over as in (3.33); see Remark 3.10 for some details.
According to [5], the resolution of is given by
[TABLE]
where
[TABLE]
and is the canonical blowup. Analogously, we take
[TABLE]
where is the forgetful morphism aforementioned. Theorem 1.1 then leads to the following conclusion immediately.
Corollary 1.2**.**
* is a proper Deligne-Mumford stack and is isomorphic to .*
Via the above isomorphism and applying [5], one sees that the stack provides a resolution of . Nonetheless, without relating to , we can directly prove the resolution property of by investigating the local equations of in [5] and their pullbacks to ; see Remark 3.8.
The methods and ideas of this paper are essential to the development in [8] and forthcoming works. Based on the construction of , we introduce the theory of stacks with twisted fields (STF) in [8, Theorem 2.19]. To be somewhat more informative, we work on a smooth stack that has a stratification indexed by a set of graphs similar to (2.10); see [8, Definition 2.15]. The graphs in need not to come from the dual graphs as in (2.10), but the stratification of should resemble (3.2) locally. Moreover, need not to consist of trees, but it should contain necessary information on the notion of the (weighted) level trees in Definition 2.1 so that we can add the twisted fields to the strata of parallel to (2.12) and obtain a new stack ; see [8, Definition 2.17]. Such enjoys desirable properties as in Corollary 3.7 and Remark 3.8. As an application of the STF theory, in [8], we construct a smooth Artin stack of genus 2 nodal curves with line bundles and twisted fields, along with a proper and birational forgetful morphism , such that
[TABLE]
provides a resolution. Further, we expect that they can be extended to arbitrary genus, as far as the existence of moduli of nodal curves with twisted fields is concerned. This is the main motivation of the current article.
In a related work [11], D. Ranganathan, K. Santos-Parker, and J.Wise provide a different modular perspective of using logarithmic geometry.
Acknowledgments. We would like to thank Dawei Chen, Qile Chen, and Jack Hall for the valuable discussions.
Convention. The subscript “1” of the relevant stacks indicating the genus appears only in §1 and will be omitted starting §2, as we only deal with the genus 1 case in this paper. In particular, we will denote by
[TABLE]
the aforementioned stacks and , respectively.
2. Set-theoretic descriptions
In §2.1, we discuss the combinatorics of the dual graphs of nodal curves and introduce the notion of the weighted level trees. They will be used to define set-theoretically in §2.2.
2.1. Weighted level trees
Let be a rooted tree, i.e. a connected finite graph that contains no cycles, along with a special vertex , called the root. The sets of the vertices and the edges of are denoted by
[TABLE]
respectively. The set is endowed with a partial order, called the tree order, so that if and only if and belongs to a path between and . The root is thus the unique maximal element of with respect to the tree order.
For each , we denote by the endpoints of such that Then, every vertex corresponds to a unique
[TABLE]
The tree order on induces a partial order on , still called the tree order, so that
[TABLE]
We call a pair consisting of a rooted tree and a function
[TABLE]
a weighted tree. For such , we write and . The set of all the weighted trees is denoted by .
We call a map satisfying
[TABLE]
a level map. For each , let
[TABLE]
i.e. the level is right “above” the level ; see Figure 1. We remark that a rooted tree along with a level map is called a level graph with the root as the unique top level vertex in [1, §1.5].
Definition 2.1**.**
We call the tuple
[TABLE]
a weighted level tree if and is a level map.
For every weighted level tree as above, we write and . Set
[TABLE]
For any two levels , we write
[TABLE]
For every , let
[TABLE]
For each level , we set
[TABLE]
In other words, consists of all the edges crossing the gap between the levels and .
We remark that all the notions in the preceding paragraph depend on the weighted level tree , although we may hereafter omit in any of such notions when the context is clear.
Every weighted level tree determines a unique index set
[TABLE]
The set becomes empty if , i.e. the root is positively weighted. As mentioned before, we may simply write
[TABLE]
when the context is clear.
For each (possibly empty), let
[TABLE]
We construct a weighted level tree
[TABLE]
as follows:
- •
the rooted tree is obtained via the edge contraction
[TABLE]
such that the set of the contracted edges is
[TABLE]
- •
the weight function is given by
[TABLE]
- •
the level map is such that for any e\!\in\!\textnormal{Edg}\big{(}\gamma_{(I)}\big{)}~{}\big{(}\subset\!\textnormal{Edg}({\mathfrak{t}})\big{)},
[TABLE]
It is a direct check that is a weighted tree and satisfies the criteria of a level map, hence (2.5) gives a well defined weighted level tree.
The construction of implies
[TABLE]
Intuitively, the weighted level tree is obtained from by contracting all the edges labeled by , then lifting all the vertices with to the level , and finally contracting all the levels in . Such will be used to describe the local structure of the stack in §3.
Definition 2.2**.**
Two weighted level trees and are said to be equivalent, written as if
- (E1)
* as weighted trees;* 2. (E2)
for any satisfying , we have
[TABLE] 3. (E3)
for any satisfying and , we have
[TABLE]
It is a direct check that is an equivalence relation on the set of weighted level trees. Intuitively, this equivalence relation records the relative positions of the vertices above or in the level ; see Figure 1 for illustration.
We denote by the set of the equivalence classes of the weighted level trees. There is a natural forgetful map
[TABLE]
which is well defined by Condition (E1) of Definition 2.2.
If , then , and there exists a bijection
[TABLE]
where denotes the power set. The next lemma follows from direct check.
Lemma 2.3**.**
If , then for any .
2.2. Twisted fields
For every genus 1 nodal curve , its dual graph has either a unique vertex corresponding to the genus 1 irreducible component of or a unique loop. In the former case, can be considered as a rooted tree with the root ; in the latter case, we contract the loop to a single vertex and obtain a rooted tree with the root . Such defined rooted tree is denoted by and called the reduced dual tree of (c.f. [5, §3.4]). We call the minimal connected genus 1 subcurve of the core and denote it by . Other irreducible components of are smooth rational curves and denoted by , . For every incident pair , let
[TABLE]
be the nodal point corresponding to the edge .
Let be the Artin stack of genus 1 stable weighted curves introduced in [5, §2.1]. Here the subscript “1” indicating the genus is omitted as per our convention. The stack consists of the pairs of genus 1 nodal curves with non-negative weights , meaning that for all irreducible . Here is said to be stable if every rational irreducible component of weight [math] contains at least three nodal points. The weight of the core is defined as the sum of the weights of all irreducible components of the core.
Every uniquely determines a function
[TABLE]
which makes the pair a weighted tree, called the weighted dual tree. Thus, the stack can be stratified as
[TABLE]
If the sum of the weights of all vertices is fixed, the stability condition of then guarantees there are only finitely many so that is non-empty.
Given and , let
[TABLE]
be the line bundles whose fibers over a weighted curve are the tangent vectors of the irreducible components at the nodal points , respectively. We take
[TABLE]
For any direct sum of line bundles (over any base), we write
[TABLE]
For any morphisms , we write
[TABLE]
With notation as above, given and [{\mathfrak{t}}]\!=\!\big{[}{\tau},\ell\big{]}\!\in\!\mathscr{T}_{\mathsf{L}}^{\textnormal{wt}}, let
[TABLE]
where , , and are as in (LABEL:Eqn:bbI), (2.11), and (2.3), respectively. It is straightforward that both bundles in (2.12) are independent of the choice of the weighted level tree representing . Since
[TABLE]
we see that is a subset of . In addition, since each stratum is an algebraic stack, so are and .
Using (2.12) and (2.10), we define
[TABLE]
This is the set-theoretic definition of the proposed stack as well as the forgetful map in Theorem 1.1. For any , the points of the fiber \mathfrak{M}^{\textnormal{tf}}_{[{\mathfrak{t}}]}\big{|}_{x} are called the twisted fields over .
Remark 2.4*.*
By (2.13), in Corollary 1.2 consists of the tuples
[TABLE]
where are stable maps in , are the equivalence classes of weighted level trees satisfying \mathfrak{f}[{\mathfrak{t}}]\!=\!\big{(}\gamma_{C},c_{1}(\mathbf{u}^{*}\mathscr{O}_{\mathbb{P}^{n}}(1))\big{)}, and are twisted fields over \big{(}C,c_{1}(\mathbf{u}^{*}\mathscr{O}_{\mathbb{P}^{n}}(1))\big{)}.
3. The stack structure of
In §3, we show is naturally a smooth Artin stack and describe its universal family.
3.1. Twisted charts
We first fix [{\mathfrak{t}}]\!=\!\big{[}\gamma,\mathbf{w},\ell\big{]}\!\in\!\mathscr{T}_{\mathsf{L}}^{\textnormal{wt}} and , and write
[TABLE]
Since is smooth, we take an affine smooth chart
[TABLE]
containing .
As in [5, §4.3] and [7, §2.5], there exists a set of regular functions
[TABLE]
called the modular parameters, so that for each , the locus
[TABLE]
is the irreducible smooth Cartier divisor on where the node labeled by is not smoothed. For any , let
[TABLE]
Then, is an open subset of . Shrinking if necessary, we see that
[TABLE]
where denotes the set of the connected components. In particular, can be considered as a smooth chart of the stratum . Rigorously, the sets and depend on the choice of the weighted level tree representing , however, and are independent of such choice; see Lemma 2.3.
Given a set of the modular parameters as in (3.1), we may extend it to a set of local parameters on centered at :
[TABLE]
where is a finite set. We do not impose other conditions on .
For each , we set
[TABLE]
Lemma 3.1**.**
For every and every , the restriction of to is a nowhere vanishing section of the restriction of the line bundle in (2.11) to .
Proof.
Since the restriction of to () is identically zero, we observe that the restriction of to is a nowhere vanishing section of the normal bundle of . It is a well-known fact of the moduli of curves that the normal bundle of is ; see [2, Proposition 3.31]. ∎
For each level , we choose a special vertex
[TABLE]
We then denote by , , and respectively the edges and the vertex satisfying
[TABLE]
Each determines a strictly increasing sequence
[TABLE]
We would like to remark that and in (2.1) need not to be the same; see Figure 1 for illustration. This sequence is finite, as there is a unique step satisfying .
By Lemma 3.1, there exist with so that the fixed over can uniquely be written as
[TABLE]
Let
[TABLE]
be an open subset containing the point
[TABLE]
The coordinates on are denoted by
[TABLE]
For any , we set
[TABLE]
This gives rise to a stratification
[TABLE]
We remark that neither nor its stratification (3.10) depends on the choice of the weighted level tree representing , even though the sets and depend on such choice. We also notice that is an open subset of , but the strata are not open unless .
For each , we take
[TABLE]
By (3.7), shrinking if necessary, we have
[TABLE]
With the local parameters and as in (3.3), we construct a morphism
[TABLE]
given by
[TABLE]
The numerator and the denominator in the first line of (LABEL:Eqn:theta_x) are both finite products, because (3.6) is always a finite sequence.
For any , it follows from (3.11) and (LABEL:Eqn:theta_x) that
[TABLE]
where and are described before (3.2).
Fix ( may be empty). With
[TABLE]
as in (2.5), as in (2.13), and the chart as in (3.2), let
[TABLE]
be the morphism so that for any ,
[TABLE]
where
[TABLE]
for all and . Similar to (LABEL:Eqn:theta_x), the products in the first pair of parentheses above are both finite products.
By (3.13), the description of above (3.2), and (2.6), we see that
[TABLE]
Moreover, by (3.11),
[TABLE]
This, along with (3.13), Lemma 3.1, and (2.12), implies is well-defined.
The morphisms , , together determine
[TABLE]
We remark that and are also independent of the choice of the weighted level tree representing . Moreover, we observe that
[TABLE]
where is given in (3.8).
A priori is just a map, for the set-theoretic definition (2.13) of does not describe its stack structure, although each is a stack. In §3.2, we will show such patch together to endow with a smooth stack structure. Each will hereafter be called a twisted chart centered at (lying over ), although rigorously it becomes a chart of only after Corollary 3.7 is established.
Lemma 3.2**.**
For every , of (3.14) is an isomorphism to an open subset of .
Proof.
For any , notice that every edge in of the weighted level tree is not contracted in the construction of (c.f. (2.6)). Thus,
[TABLE]
In particular, the edges , , can be used as the special edges of . For conciseness, let
[TABLE]
see (2.7) for notation.
Let be a set of the local parameters on centered at as in (3.3). By the definition of above (3.2),
[TABLE]
is a set of local parameters of .
Recall that there exist , , such that
[TABLE]
as in (3.7). Let be an open subset of such that
[TABLE]
The coordinates of are denoted by
[TABLE]
In addition, we set
[TABLE]
Thus, the function is defined for all , and is nowhere vanishing on for all .
The smooth chart in (3.2) induces a smooth chart
[TABLE]
given by
[TABLE]
We will construct a morphism
[TABLE]
such that and are both the identity morphisms, which will then establish Lemma 3.2.
Given \big{(}\mathfrak{z},(\mu_{e})_{e\in\mathbf{E}({\mathfrak{t}}_{(I)})}\big{)}\!\in\!\mathcal{U}^{\prime}_{x;{\mathfrak{t}}_{(I)}}, we denote its image by
[TABLE]
which is to be constructed. With the coordinates on as in (3.9), we set
[TABLE]
By (3.2), we see that
[TABLE]
The rest of the coordinates of are much more complicated; we describe them by induction over the levels in . More precisely, we will show that * with and with are all rational functions in and , satisfying *
[TABLE]
In particular, (3.21) and (3.20) imply , i.e. is well-defined.
The base case of the induction is for the level
[TABLE]
We take
[TABLE]
By (3.2), we see that satisfies (3.21). We take
[TABLE]
For any with , we set
[TABLE]
If , then by (3.17) and (3.2), we have if and only if ( and) . If , then if and only if ( and) . We thus conclude that satisfies (3.21) for all with Moreover, such and are obviously rational functions in and . Hence, the base case is complete.
Next, for any , assume that all with and all with and have been expressed as rational functions in and , satisfying (3.21).
For the level , we first construct . The construction is subdivided into three cases.
Case 1. If , then set
[TABLE]
Obviously this satisfies (3.21).
Case 2. If and , then , hence . Let
[TABLE]
Intuitively, is the level containing the image of in . Thus,
[TABLE]
Let be given by
[TABLE]
The inductive assumption implies that all with , as well as all and with , are non-zero and are rational functions in and . By (3.17) and (2.6), we also see that all and in (3.24) are non-zero. Therefore, such defined is a rational function in and , satisfying (3.21).
Case 3. If (hence ), then we see ; c.f. (2.6). Intuitively, this means is contracted in the construction of . By the description of above (3.2), we see that . Let be given by
[TABLE]
Mimicking the argument in Case 2, we conclude that is a rational function in and , satisfying (3.21).
Next, we construct for with . Set
[TABLE]
For , the construction is subdivided into two cases.
Case A. If , then
[TABLE]
hence exists, and if and only if . In Case A, since , (2.7) further implies
[TABLE]
Let
[TABLE]
Since , we have
[TABLE]
Let be given by
[TABLE]
We observe that if , then and hence ; if , then the previous construction of , along with the inductive assumption, guarantees . Mimicking the argument of Case 2 of the construction of and taking (3.26) into account, we see that determined by (3.27) is a rational function in and , and it satisfies (3.21).
Case B. If , then (2.6) gives
[TABLE]
Let be given by
[TABLE]
Once again, mimicking the argument of Case A of the construction of , and taking (3.28) as well as the description of right before (3.2) into account, we see that determined by (3.29) is a rational function in and , and it satisfies (3.21).
The cases 1-3, A, and B together complete the inductive construction of . Moreover, comparing
(3.19) with the second line of (LABEL:Eqn:theta_x), 2.
(3.22), the second case of (3.23), (3.25), and (3.29) with the first line (LABEL:Eqn:theta_x), 3.
the first case of (3.23), (3.24), and (3.27) with the expressions of right after (3.14),
we observe that is the inverse of . ∎
Corollary 3.3**.**
* is injective.*
Proof.
This follows from Lemma 3.2 and the stratification (3.10) and (2.13) directly. ∎
3.2. Stack structure
In this subsection, we will show the twisted charts patch together to endow with a smooth stack structure; c.f. Proposition 3.6 and Corollary 3.7. Note that a priori, depends on the choices of the special vertices (3.4) and of the local parameters (3.3). Nonetheless, Lemmas 3.4 and 3.5 below will guarantee that such choices do not affect the proposed stack structure of , hence will make the proof of Proposition 3.6 more concise.
Let be another set of the special vertices satisfying (3.4), and , , and be the analogues of , , and in (3.5), respectively. As in (3.6), each level similarly determines a finite sequence
[TABLE]
We take an open subset
[TABLE]
with the coordinates
[TABLE]
as in (3.9), and then construct
[TABLE]
parallel to (LABEL:Eqn:theta_x) and (3.14). Let .
Lemma 3.4**.**
The transition map
[TABLE]
is an isomorphism.
Proof.
Let be the isomorphism given by
[TABLE]
The fact that is an isomorphism can be shown by constructing its inverse explicitly, which is similar to the proof of Lemma 3.2, but is simpler. The key point of the construction is that
[TABLE]
and each is a product of and a rational function of with .
It is a direct check that the isomorphism satisfies
[TABLE]
Thus, and hence is an isomorphism. ∎
Let
[TABLE]
be another set of extended modular parameters centered at on the same chart ; see (3.3). We use this set of local parameters to construct another twisted chart ; in particular, we have and \widehat{\mu}_{e;i;I}\!\in\!\Gamma\big{(}\mathscr{O}_{\widehat{\mathfrak{U}}_{x;(I)}^{\circ}}\big{)} as in (LABEL:Eqn:theta_x) and (3.14), respectively. Parallel to (3.9), the coordinates on are denoted by
[TABLE]
Let .
Lemma 3.5**.**
The transition map
[TABLE]
is an isomorphism.
Proof.
By Lemma 3.4, it suffices to use the same set of the special vertices for both and . For any , the local parameters and defines the same locus , hence there exists such that
[TABLE]
Therefore, we have
[TABLE]
Let be the isomorphism given by
[TABLE]
The explicit expression of above implies it is invertible; see the parallel argument in the proof of Lemma 3.4.
It is a direct check that and
[TABLE]
Taking (3.30) into account, we conclude that and hence is an isomorphism. ∎
Given and x^{\prime}\!\in\!\Phi_{x}\big{(}\mathfrak{U}_{x;[{\mathfrak{t}}_{(I)}]}\big{)}, let be a chart containing and be a twisted chart centered at over . Let
Proposition 3.6**.**
The transition map
[TABLE]
is an isomorphism.
Proof.
Since x^{\prime}\!\in\!\Phi_{x}\big{(}\mathfrak{U}_{x;{\mathfrak{t}}_{(I)}}\big{)}\!\subset\!\Phi_{x}(\mathfrak{U}_{x}), its underlying weighted curve satisfies
[TABLE]
Thus, replacing by if necessary, we may assume
[TABLE]
Moreover, the following modular parameters on :
[TABLE]
also serve as modular parameters on . Thus by Lemma 3.5, we may assume is constructed using the local parameters on :
[TABLE]
as the analogue of (3.3).
Let the special vertices and edges of be respectively as in (3.4) and (3.5). By Lemma 3.4, we may further assume that the special vertices and edges of are respectively
[TABLE]
For any and , let , , and be the analogues of , , and , respectively, for the weighted level tree instead of ; see (2.1), (3.5), and (3.6) for notation.
Recall that \mathbb{I}_{-}\big{(}{\mathfrak{t}}_{(I)}\big{)}\!=\!\mathbb{I}_{-}\backslash I_{-}\!\sqcup\!\big{\{}e\!\in\!\mathbb{I}_{\mathbf{m}}\backslash I_{\mathbf{m}}\!:\ell(v_{e}^{+})\!\leq\!{\mathbf{m}}\big{(}{\mathfrak{t}}_{(I)}\big{)}\big{\}}. We denote by
[TABLE]
the coordinates on parallel to (3.9), and construct
[TABLE]
parallel to
[TABLE]
of (LABEL:Eqn:theta_x) and (3.14), respectively. In this way, is constructed analogously to .
Let be the isomorphism given by
[TABLE]
and
[TABLE]
To see is well defined, notice that implies that
[TABLE]
Thus, every above can be considered as a function on . By (3.31) and (3.13), the function with is nowhere vanishing on whenever . Taking (3.11) and (3.15) into account, we conclude that is well defined.
Once again, the explicit expression of implies it is invertible; see the parallel argument in the proof of Lemma 3.4. Moreover, it is a direct check that and
[TABLE]
Thus, and hence is an isomorphism. ∎
Corollary 3.7**.**
* is a smooth Artin stack that is birational to , with as smooth charts. Moreover, the structure of the stratification (2.13) is locally identical to the one induced by (3.10). Furthermore, for any , any chart containing , and any twisted chart centered at lying over , we have*
\mathfrak{U}_{x}$$\mathfrak{M}^{\textnormal{tf}}$$\mathcal{V}$$\mathfrak{M}^{\textnormal{wt}}$$\Phi_{x}$$\varpi$$\theta_{x}
where is the forgetful morphism as in (2.13) and is as in (LABEL:Eqn:theta_x).
Proof.
The first statement follows from Proposition 3.6, (3.16), and the fact that restricts to the identity map on the preimage of the open subset
[TABLE]
Lemma 3.2 then implies for every , the stack structure of is the same as that induced from the inclusion ; i.e. the second statement of Corollary 3.7 holds. The last statement follows from (3.14). ∎
Remark 3.8*.*
By (LABEL:Eqn:theta_x) and Corollary 3.7, one sees that on an arbitrary twisted chart of ,
[TABLE]
This, along with the local equations of in [5, §5.2], implies that the primary component of is smooth and contains at worst normal crossing singularities. This observation should be useful for the cases of higher genera.
3.3. A simple example
Let be given by the leftmost diagram in Figure 2. Then,
[TABLE]
Each of the four distinct subsets of determines a weighted level tree ; see Figure 2. Let be a weighted curve of genus 1 with twisted fields over . The core and the nodes of are labeled by and by , respectively.
Let be an affine smooth chart containing , with a set of local parameters
[TABLE]
centered at , where are the modular parameters. There then exist non-zero and such that
[TABLE]
We choose the special edges (3.5) of to be and . Let
[TABLE]
be an open subset containing the point
[TABLE]
The coordinates of are denoted by
[TABLE]
We may take .
By Corollary 3.7 and (LABEL:Eqn:theta_x), the forgetful morphism can locally be written as such that
[TABLE]
Considering all possible subsets of in (3.14), we obtain a twisted chart centered at over so that for any
[TABLE]
if , then
[TABLE] 2.
if and , then
[TABLE] 3.
if and , then
[TABLE] 4.
if and , then
[TABLE]
With the expressions of as above, it is straightforward to check
[TABLE]
as well as to verify the statements of Lemmas 3.2, 3.4, 3.5, and Proposition 3.6 in this situation.
3.4. Universal family
Let be the universal weighted nodal curves of genus 1. The stratification (2.10) gives rise to a stratification
[TABLE]
Parallel to (2.12) and (2.13), we set
[TABLE]
Mimicking the construction of the stack structure of in §3.1 and §3.2, we can endow with a stack structure analogously. Furthermore, the projection induces a unique projection
[TABLE]
It is straightforward that
[TABLE]
For any scheme , a flat family of stable weighted nodal curves of genus 1 with twisted fields corresponds to a morphism such that is the pullback of (3.32):
[TABLE]
This leads to the following statement.
Proposition 3.9**.**
* in (3.32) gives the universal family of .*
Remark 3.10*.*
One may establish a moduli interpretation of as follows: for any scheme , every flat family of stable weighted nodal curves of genus 1 with twisted fields can be constructed directly as follows. A priori, should be over a flat family of stable weighted curves, thus by the universality of the moduli , there exists a morphism
[TABLE]
such that is the pullback of via . This induces a stratification of the scheme :
[TABLE]
We take
[TABLE]
For any chart , shrinking it if necessary, we see there exists a (smooth) chart such that
\mathcal{V}_{S}$$\mathcal{V}$$S$$\mathfrak{M}^{\textnormal{wt}}
commutes. The modular parameters on pull back to regular functions on , which are denoted by . By (3.34), we have
[TABLE]
Mimicking the construction in §3.1 and §3.2, we can thus endow with a scheme structure.
We say is a flat family of stable weighted nodal curves of genus 1 with twisted fields if and only if there exists a section of such that
[TABLE]
This construction is consistent with (3.33). One can check that the groupoid sending any scheme to the set of all such defined flat families is represented by .
We would like to remark that a more succinct definition of a flat family of stable weighted nodal curves of genus 1 with twisted fields should be desirable.
4. Comparison with Hu-Li’s blowup stack
Let be the sequential blowup constructed in [5, §2.2]. Since is a smooth Artin stack and the blowup centers are all smooth, so is . As per the convention of this paper, we omit the subscript indicating the genus. In Proposition 4.3, we show that is isomorphic to . Lemma 4.2 is rather technical; it is only used in the proof of Proposition 4.3.
We briefly recall the notion of the locally tree compatible blowups described in [7, §3]. Let be a smooth stack, be a rooted tree, and be an affine smooth chart of . If there exists a set of local parameters on so that a subset of which can be written as
[TABLE]
then the set is called a -labeled subset of local parameters on . For example, if and is a chart centered at a weighted curve whose reduced dual tree is , then the set of the modular parameters is a -labeled subset of local parameters.
Let be the set of the minimal vertices of with respect to the tree order. We call a subset of a traverse section if for any , the path between and contains exactly one element of . For example, the subsets of as in (2.3) are traverse sections. Let be the set of the traverse sections. The tree order on induces a partial order on such that
[TABLE]
We remark that the tree order on and the induced order on in this paper are both opposite to those in [7], in order to be consistent with the order of the levels of the weighted level trees.
Let be the sequential blowup of successively along the proper transforms of the closed substacks of .
Definition 4.1**.**
[7, Definitions 3.2.4 & 3.2.1]** The blowup above is said to be locally tree-compatible if there exists an étale cover of such that for each , there exist a rooted tree , a partition of :
[TABLE]
and a -labeled subset of local parameters on such that
- •
for every ,
[TABLE]
- •
if , , and , then .
If a sequential blowup is locally tree-compatible, then the blowup procedure is finite on each , because the set is finite.
Lemma 4.2**.**
If the blowup successively along the proper transforms of the closed substacks of is locally tree-compatible, then the blowup successively along the total transforms of
[TABLE]
yields the same space, i.e. .
Proof.
We prove the statement by induction. For each , we will show that after the -th step, the blowup stacks of along the total transforms of is the same as the blowup of along the proper transforms of .
The base case of the induction is trivial. Suppose the blowup . We will show that for any and any lift of after the -th step, the blowup along the total transform of has the same effect as that along the proper transform of near . Since and are arbitrary, this will establish the -th step of the induction.
W.l.o.g. we may assume (otherwise we simply omit the loci not containing and change the indices of and accordingly). The blowup is locally tree-compatible, hence there exist a rooted tree , an affine smooth chart containing , and a -labeled subset of local parameters , on such that
[TABLE]
As shown in [7, Lemma 3.3.2], there exist traverse sections and (c.f. Definition 4.1), an affine smooth chart , and a subset of local parameters
[TABLE]
on so that is locally given by
[TABLE]
Moreover, by [7, (3.13)], the total transform of each with is locally given by . Thus, is locally given by
[TABLE]
That is, on the chart , and are defined by the ideals
[TABLE]
respectively. Therefore, blowing up along has the same effect on as that along . ∎
Proposition 4.3**.**
* is isomorphic to . In particular, is proper.*
Proof.
Our goal is to construct two morphisms and between and so that the following diagram
\widetilde{\mathfrak{M}}^{\textnormal{wt}}$$\mathfrak{M}^{\textnormal{tf}}$$\mathfrak{M}^{\textnormal{wt}}$$\psi_{2}$$\psi_{1}$$\pi$$\varpi
commutes. Since and restrict to the identity map on the preimages of the open subset
[TABLE]
respectively, we see that and are the identity maps. This then implies the former statement of Propoistion 4.3. The latter statement follows from the former as well as the properness of the blowup .
We first construct . For each , let be the closed locus whose general point is obtained by attaching smooth positively-weighted rational curves to the smooth 0-weighted elliptic core at pairwise distinct points. By Lemma 4.2, the blowup successively along the proper transforms of can be identified with the blowup of successively along the total tranforms of
[TABLE]
We observe that for each , the pullback to is a Cartier divisor. In fact, for any and , let be a twisted chart centered at , lying over a chart . In [5], the blowup locally on is proved to be compatible with the weighted tree obtained by contracting all the edges of as long as there exists satisfying . Let be a set of modular parameters on as in (3.1) and
[TABLE]
be the subset of the parameters (3.9) on . We claim that
[TABLE]
To show (4.1), we first notice that by Corollary 3.7. Every irreducible component of can be written in the form
[TABLE]
For each irreducible component of , the local expression of as in (LABEL:Eqn:theta_x) implies the pullback can be written as
[TABLE]
Since \mathfrak{S}\!\cap\!\big{(}\widehat{\textnormal{Edg}}({\mathfrak{t}})\backslash\mathbb{I}_{{\mathbf{m}}}\big{)}\!\neq\!\emptyset and , we can always find e\!\in\!\mathfrak{S}\!\cap\!\big{(}\widehat{\textnormal{Edg}}({\mathfrak{t}})\backslash\mathbb{I}_{{\mathbf{m}}}\big{)} such that for all . By (4.1) and (LABEL:Eqn:Yk_pullback), the pullback is thus a sub-locus of the right-hand side of (4.1). Moreover, it is a direct check that
[TABLE]
Therefore, (4.1) holds.
Since every is a Cartier divisor of , by the universality of the blowup , we obtain a unique morphism
[TABLE]
that factors through.
We next construct explicitly. For any , let be its image in . As shown in [7, §3.3], there exists a unique maximal sequence of exceptional divisors
[TABLE]
containing . Each is obtained from blowing up along the proper transform of . Note that is possibly 0, which means is not in the blowup loci. The weighted dual tree , along with the exceptional divisors , uniquely determines a weighted level tree such that
[TABLE]
In particular, .
With the line bundles , , as in (2.11), the notation and as in (2.2), and the notation as in (3.6), the line bundles
[TABLE]
can be constructed inductively over . Then, we take
[TABLE]
In particular, . For each , (4.3) and (4.4) imply
[TABLE]
Hence for each ,
[TABLE]
For , let be the image of in the exceptional divisor of the -th step. Given , The proper transform of after the first steps of the blowup may have several connected components; see [7, Lemma 3.3.2]. The normal bundle of the component containing is the pullback , where is the blowup after the -th step.
Notice that the non-zero entries of exactly correspond to the edges satisfying . Therefore,
[TABLE]
Then, with together determine a unique
[TABLE]
The last equality above follows from (4.5). We then set
[TABLE]
Obviously, this implies .
It remains to verify such defined is a morphism. Let be a smooth chart containing , and be a set of local parameters centered at as in (3.3). As shown in [7, §3.1&§3.3], there exists a chart containing with local parameters
[TABLE]
All are nowhere vanishing on . Moreover, with denoting the blowup, we have
[TABLE]
For , we set . Then,
[TABLE]
Let be a twisted chart centered at , lying over . The parameters on are as in (3.3). It is a direct check that the point-wise defined can locally be written as
[TABLE]
such that
[TABLE]
This shows is a morphism. ∎
Remark 4.4*.*
In [6], another resolution , called the derived resolution of , is constructed for the purpose of diagonalizing certain direct image sheaves. That resolution is “smaller” in that the resolution of [5] factors through . Mimicking the approach of §3, we may construct a moduli stack
[TABLE]
This moduli should be isomorphic to .
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] M. Bainbridge, D. Chen, Q. Gendron, S. Grushevsky, and M. Möller, Compactification of strata of Abelian differentials , Duke Math. J. 167 (2018), no. 12, 2347–2416.
- 2[2] J. Harris and I. Morrison, Moduli of curves , Graduate Texts in Math. vol. 187, Springer-Verlag, New York, 1998.
- 3[3] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I , Ann. of Math., 2, 79 (1), (1964) 109–203.
- 4[4] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. II , Ann. of Math., 2, 79 (1), (1964) 205–326.
- 5[5] Y. Hu and J. Li, Genus-one stable maps, local equations and Vakil-Zinger’s desingularization , Math. Ann. 348 (2010), no. 4, 929–963.
- 6[6] Y. Hu and J. Li, Derived resolution property for stacks, Euler classes and applications , Math. Res. Lett. 18 (2011), no. 04, 677–690.
- 7[7] Y. Hu, J. Li, and J. Niu, Genus two stable maps, local equations and modular resolutions , math/1201.2427 v 3
- 8[8] Y. Hu and J. Niu, A theory of stacks with twisted fields and resolution of moduli of genus two stable maps , math/2005.03384.
