Abstract
We find for g ≤ 5 a stratification of depth g − 2 of the moduli space of curves \({\mathcal M_g}\) with the property that its strata are affine and the classes of their closures provide a \({\mathbb{Q}}\)-basis for the Chow ring of \({\mathcal M_g}\). The first property confirms a conjecture of one of us. The way we establish the second property yields new (and simpler) proofs of theorems of Faber and Izadi which, taken together, amount to the statement that in this range the Chow ring is generated by the λ-class.
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Accola, R.D.M.: Some loci of Teichmüller space for genus five defined by vanishing theta nulls. In: Contributions to Analysis (A Collection of Papers Dedicated to Lipman Bers), pp. 11–18. Academic Press, New York (1974)
Arbarello E.: Weierstrass points and moduli of curves. Compos. Math. 29, 325–342 (1974)
Arbarello E.: On subvarieties of the moduli space of curves of genus g defined in terms of Weierstrass points. Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia (8) 15(1), 3–20 (1978)
Arbarello E., Cornalba M.: Calculating cohomology groups of moduli spaces of curves via algebraic geometry. Inst. Hautes Études Sci. Publ. Math. 88, 97–127 (1998)
Arbarello, E., Cornalba, M., Griffiths, P.A., Harris, J.: Geometry of algebraic curves, vol. I. Grundlehren der Mathematischen Wissenschaften 267, xvi+386 pp. Springer-Verlag, New York (1985)
Cornalba M., Harris J.: Divisor classes associated to families of stable varieties, with applications to the moduli space of curves. Ann. Sci. Ec. Norm. Sup., 4 s., t. 21, 455–475 (1988)
Faber, C.: Chow rings of moduli spaces of curves. I, II. Ann. Math. 132, 331–419, ibid. 421–449 (1990)
Faber, C.: A conjectural description of the tautological ring of the moduli space of curves. In: Faber, C., Looijenga, E. (eds.) Moduli of Curves and Abelian Varieties (the Dutch Intercity Seminar on Moduli). Aspects of Mathematics E33, pp. 109–129. Vieweg, Wiesbaden (1999)
Fontanari C.: Moduli of curves via algebraic geometry. Liaison and related topics (Turin, 2001). Rend. Sem. Mat. Univ. Politec. Torino 59(2), 137–139 (2003)
Fulton W.: Hurwitz schemes and irreducibility of moduli of algebraic curves. Ann. Math. 90, 542–575 (1969)
Hain R., Looijenga E.: Mapping class groups and moduli spaces of curves. Proc. Symp. Pure Math., AMS 62, 97–142 (1998)
Harer J.: The virtual cohomological dimension of the mapping class group of an orientable surface. Invent. Math. 84, 157–176 (1986)
Izadi, E.: The Chow ring of the moduli space of curves of genus 5. In: The Moduli Space of Curves (Texel Island, 1994). Progr. Math., vol. 129, pp. 267–304. Birkhaüser Boston, Boston, MA (1995)
Looijenga E.: On the tautological ring of M g . Invent. Math. 121, 411–419 (1995)
Mumford, D.: Towards an enumerative geometry of the moduli space of curves. In: Arithmetic and Geometry, vol. II. Progr. Math. 36, pp. 271–328. Birkhäuser Boston, Boston, MA (1983)
Ran Z.: Curvilinear enumerative geometry. Acta Math. 155, 81–101 (1985)
Saint-Donat B.: On Petri’s analysis of the linear system of quadrics through a canonical curve. Math. Ann. 206, 157–175 (1973)
Teixidor i Bigas M.: The divisor of curves with a vanishing theta-null. Comp. Math. 66(1), 15–22 (1988)
Vakil, R., Roth, M.: The affine stratification number and the moduli space of curves. In: CRM Proceedings and Lecture Notes Université de Montréal, vol. 38, pp. 213–227 (2004)
Vistoli A.: Chow groups of quotient varieties. J. Algebra 107, 410–424 (1987)
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Fontanari, C., Looijenga, E. A perfect stratification of \({\mathcal M_g}\) for g ≤ 5. Geom Dedicata 136, 133–143 (2008). https://doi.org/10.1007/s10711-008-9280-y
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DOI: https://doi.org/10.1007/s10711-008-9280-y