## Nagoya Mathematical Journal

### The one-dimensional stratum in the boundary of the moduli stack of stable curves

Jörg Zintl

#### Abstract

It is well-known that the moduli space $\overline{M}_{g}$ of Deligne-Mumford stable curves of genus $g$ admits a stratification by the loci of stable curves with a fixed number $i$ of nodes, where $0 \le i \le 3g-3$. There is an analogous stratification of the associated moduli stack $\overline{\mathcal{M}}_{g}$.

In this paper we are interested in that particular stratum of the moduli stack, which corresponds to stable curves with exactly $3g-4$ nodes. The irreducible components of this stratum are one-dimensional substacks of $\overline{\mathcal{M}}_{g}$. We show how these substacks can be related to simpler moduli stacks of (permutation classes of) pointed stable curves. Furthermore, we use this to construct all of the components of this boundary stratum generically in a new way as explicit quotient stacks.

#### Article information

Source
Nagoya Math. J., Volume 196 (2009), 27-66.

Dates
First available in Project Euclid: 15 January 2010

https://projecteuclid.org/euclid.nmj/1263564647

Mathematical Reviews number (MathSciNet)
MR2591090

Zentralblatt MATH identifier
1209.14025

Subjects
Primary: 14H10: Families, moduli (algebraic)
Secondary: 14H37: Automorphisms

#### Citation

Zintl, Jörg. The one-dimensional stratum in the boundary of the moduli stack of stable curves. Nagoya Math. J. 196 (2009), 27--66. https://projecteuclid.org/euclid.nmj/1263564647

#### References

• P. Deligne and D. Mumford, The irreducibility of the space of curves of a given genus, Publ. Math. I\.H\.E\.S., 36 (1969), 75--109.
• D. Edidin, Notes on the construction of the moduli space of curves, Recent progress in intersection theory (G. Ellingsrud, et al., eds.), 2000, pp. 85--113.
• C. Faber, Chow rings of moduli spaces of curves I: The Chow ring of $\overline\mathcalM_3$, Annals of Maths., 132 (1990), 331--419.
• D. Gieseker, Lectures on moduli of curves, Tata Institute Lecture Notes 69, 1982.
• J. Harris and I. Morrison, Moduli of Curves, Springer-Verlag, New York, 1998.
• F. Knudsen, The projectivity of the moduli space of stable curves, II: the stacks $M_g, n$, Math. Scand., 52 (1983), 161--199.
• J. Zintl, One-dimensional substacks of the moduli stack of Deligne-Mumford stable curves, Habilitationsschrift, Kaiserslautern, 2005, math.AG/0612802.
• J. Zintl, Moduli stacks of permutation classes of pointed stable curves, Milan j. math., 76 (2008), 401--418, math.AG/0611710.