The zero section of the universal semiabelian variety and the double ramification cycle. (English) Zbl 1302.14039
The main result of the present paper is the computation of a certain natural divisor class on the universal family over a partial compactification of \(\mathcal{A}_g\), the moduli space of principally polarized abelian varieties of dimension \(g\). Namely, denote by \(\mathcal{A}_g'\) Mumford’s partial compactification of \(\mathcal{A}_g\) obtained by adding semi-abelic varieties of torus rank one, i.e., compactifications of \(\mathbb{C}^*\)-extensions of abelian varieties of dimension \((g-1)\). The boundary \(\mathcal{A}_g' \setminus \mathcal{A}_g\) is isomorphic to \(\mathcal{X}_{g-1}\), the universal family over \(\mathcal{A}_{g-1}\) (the isomorphism is explicitly described in the paper). Furthermore it has the property that it is contained in every toroidal compactification of \(\mathcal{A}_g\). It is thus a natural starting point for the study of the Chow ring and cohomology groups of any compactification of \(\mathcal{A}_g\).
The universal family \(\mathcal{X}_g \to \mathcal{A}_g\) admits a zero section \(z_g : \mathcal{A}_g \to \mathcal{X}_g\), which set-theoretically assigns to any abelian variety \(A\) with origin \(0 \in A\) the moduli point \((A, 0)\) in \(\mathcal{X}_g\). This setup can be extended to a universal family \(\mathcal{X}_g'\) and a zero section \(z_g': \mathcal{A}_g' \to \mathcal{X}_g'\) over the partial compactification. The main result is then the computation of the class of the image of this section, expressed as a polynomial in certain geometrically defined classes of codimension \(1\) and \(2\).
This result is then used to compute the class of the closure of the double ramification cycle on a partial compactification of \(\mathcal{M}_{g,n}\), the moduli space of \(n\)-pointed curves of genus \(g\). Given an \(n\)-tuple \(\underline{d} = (d_1, \dots, d_n) \in \mathbb{Z}^n\) of integers summing to zero, the latter is defined as the locus of pointed curves \((C; p_1, \dots, p_n) \in \mathcal{M}_{g,n}\) such that \(\mathcal{O}_C(\sum d_i p_i) = 0 \in \mathrm{Jac}(C)\). This locus can be expressed as the pull-back of the zero section discussed above under the Abel-Jacobi map \(s_{\underline{d}}: \mathcal{M}_{g,n} \to \mathcal{X}_g\) defined set-theoretically by \(s_{\underline{d}}(C; p_1, \dots, p_n) = (\mathrm{Jac}(C), \mathcal{O}_C(\sum d_i p_i))\). It can also be interpreted as the Hurwitz locus of curves admitting a map to \(\mathbb{P}^1\) with prescribed preimages and ramification over two points (hence the name).
This construction of the double ramification cycle is not restricted to the case of smooth curves: It can be extended to curves of compact type, i.e., stable curves with no non-separating nodes. The class of the closure of the double ramification cycle on \(\mathcal{M}_{g,n}^{\mathrm{ct}}\) was computed by R. M. Hain [Math. Sci. Res. Inst. Publ. 28, 97–143 (1995; Zbl 0868.14006)]. Here the authors take this result one step further by extending the computation to \(\mathcal{M}_{g,n}^o\), the moduli spaces of stable curves with at most one non-separating node. Moreover they show that the Abel-Jacobi map does not extend to the locus of curves having more than one non-separating node.
The universal family \(\mathcal{X}_g \to \mathcal{A}_g\) admits a zero section \(z_g : \mathcal{A}_g \to \mathcal{X}_g\), which set-theoretically assigns to any abelian variety \(A\) with origin \(0 \in A\) the moduli point \((A, 0)\) in \(\mathcal{X}_g\). This setup can be extended to a universal family \(\mathcal{X}_g'\) and a zero section \(z_g': \mathcal{A}_g' \to \mathcal{X}_g'\) over the partial compactification. The main result is then the computation of the class of the image of this section, expressed as a polynomial in certain geometrically defined classes of codimension \(1\) and \(2\).
This result is then used to compute the class of the closure of the double ramification cycle on a partial compactification of \(\mathcal{M}_{g,n}\), the moduli space of \(n\)-pointed curves of genus \(g\). Given an \(n\)-tuple \(\underline{d} = (d_1, \dots, d_n) \in \mathbb{Z}^n\) of integers summing to zero, the latter is defined as the locus of pointed curves \((C; p_1, \dots, p_n) \in \mathcal{M}_{g,n}\) such that \(\mathcal{O}_C(\sum d_i p_i) = 0 \in \mathrm{Jac}(C)\). This locus can be expressed as the pull-back of the zero section discussed above under the Abel-Jacobi map \(s_{\underline{d}}: \mathcal{M}_{g,n} \to \mathcal{X}_g\) defined set-theoretically by \(s_{\underline{d}}(C; p_1, \dots, p_n) = (\mathrm{Jac}(C), \mathcal{O}_C(\sum d_i p_i))\). It can also be interpreted as the Hurwitz locus of curves admitting a map to \(\mathbb{P}^1\) with prescribed preimages and ramification over two points (hence the name).
This construction of the double ramification cycle is not restricted to the case of smooth curves: It can be extended to curves of compact type, i.e., stable curves with no non-separating nodes. The class of the closure of the double ramification cycle on \(\mathcal{M}_{g,n}^{\mathrm{ct}}\) was computed by R. M. Hain [Math. Sci. Res. Inst. Publ. 28, 97–143 (1995; Zbl 0868.14006)]. Here the authors take this result one step further by extending the computation to \(\mathcal{M}_{g,n}^o\), the moduli spaces of stable curves with at most one non-separating node. Moreover they show that the Abel-Jacobi map does not extend to the locus of curves having more than one non-separating node.
Reviewer: Fabian Müller (Berlin)
MSC:
| 14K10 | Algebraic moduli of abelian varieties, classification |
| 14H10 | Families, moduli of curves (algebraic) |
Keywords:
principally polarized abelian varieties; moduli space; toroidal compactification, semi-abelic varieties, double ramification cycleCitations:
Zbl 0868.14006Software:
MapleReferences:
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