×

zbMATH — the first resource for mathematics

The Kodaira dimension of the moduli space of curves of genus \(\geq 23\). (English) Zbl 0631.14023
Here it is proved that the moduli scheme \(\bar M_ g\) of genus-g stable curves is of general type if \(g>23\) and has Kodaira dimension \(>0\) if \(g=23\), refining the results and simplifying the proofs of the fundamental paper by J. Harris and D. Mumford in Invent. Math. 67, 23-86 (1982; Zbl 0506.14016)]. The proofs here use the theory of limit linear series on stable curves with \(Pic^ 0\) compact introduced by the authors in Invent. Math. 85, 337-371 (1986; Zbl 0598.14003) and applied by the authors (and others) to obtain several interesting results. The proof uses the computation in \(Pic(M_ g)\otimes {\mathbb{Q}}\) of the class of the following divisors: \(D^ r_ s\) (when \(g+1\) is not prime, \(g+1=(r+1)(s-1))\), \(E_ s\) (when g is even, \(g=2(s-1))\); \(E_ s\) is the closure of the set of genus \(g\) curves with a linear series of degree \(s\) and dimension 1 violating Petri’s condition; \(D^ r_ s\) is the closure of the locus of curves with a \(g^ r_ d\) with \(d=rs-1\). Furthermore, when \(g=(r+1)(s-1)-1\), on a general curve C, \(W^ r_ d(C)\) is a smooth, irreducible curve; the authors need and compute its genus.
Reviewer: E.Ballico

MSC:
14H10 Families, moduli of curves (algebraic)
14C20 Divisors, linear systems, invertible sheaves
14D20 Algebraic moduli problems, moduli of vector bundles
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Arbarello, E., Cornalba, M., Griffiths, P., Harris, J.: The geometry of algebraic curves I. Berlin-Heidelberg-New York: Springer 1984 · Zbl 0559.14017
[2] Chang, M.-C., Ran, Z.: Unirationality of the moduli spaces of curves of genus 11, 13 (and 12). Invent. Math.76, 41-54 (1984) · Zbl 0541.14025 · doi:10.1007/BF01388490
[3] Comtet, L.: Advanced combinatorics. Boston: Reidel 1974 · Zbl 0283.05001
[4] Eisenbud, D., Harris, J.: Divisors on general curves and cuspidal rational curves. Invent. Math.74, 371-418 (1983a) · Zbl 0527.14022 · doi:10.1007/BF01394242
[5] Eisenbud, D., Harris, J.: On the Brill-Noether Theorem. Lect. Notes Math., Vol. 997, pp. 131-137. Berlin-Heidelberg-New York: Springer 1983a · Zbl 0512.14016
[6] Eisenbud, D., Harris, J.: Limit linear series: basic theory. Invent. Math.85, 337-371 (1986) · Zbl 0598.14003 · doi:10.1007/BF01389094
[7] Eisenbud, D., Harris, J.: The irreducibility of some families of linear series. Ann. Sci. Ec. Norm. Super., IV. Ser. (1987a) · Zbl 0625.14013
[8] Eisenbud, D., Harris, J.: Existence, decomposition, and limits of certain Weierstrass points. Invent. Math.87, 495-515 (1987b) · Zbl 0606.14014 · doi:10.1007/BF01389240
[9] Eisenbud, D., Harris, J.: When ramification points meet. Invent. Math.87, 485-493 (1987) · Zbl 0606.14008 · doi:10.1007/BF01389239
[10] Fulton, W.: Intersection theory. Berlin-Heidelberg-New York-Tokyo: Springer 1984 · Zbl 0541.14005
[11] Fulton, W.: Hurwitz schemes and the irreducibility of moduli of algebraic curves. Ann. Math.90, 542 (1969) · Zbl 0194.21901 · doi:10.2307/1970748
[12] Griffiths, P.A., Harris, J.: Principles of algebraic geometry. New York: John Wiley & Sons 1978 · Zbl 0408.14001
[13] Griffiths, P.A., Harris, J.: On the variety of special linear systems on a general algebraic curve. Duke Math. J.47, 233-272 (1980) · Zbl 0446.14011 · doi:10.1215/S0012-7094-80-04717-1
[14] Harer, J.: The second homology group of the mapping class group of an orientable surface. Invent. Math.72, 221-239 (1983) · Zbl 0533.57003 · doi:10.1007/BF01389321
[15] Harris, J.: On the Kodaira dimension of the moduli space of curves, II: The even genus case. Invent. Math.75, 437-466 (1984) · Zbl 0542.14014 · doi:10.1007/BF01388638
[16] Harris, J., Mumford, D.: On the Kodaira Dimension of the Moduli Space of Curves. Invent. Math.67, 23-86 (1982) · Zbl 0506.14016 · doi:10.1007/BF01393371
[17] Harris, J., Tu, L.: Chern numbers of kernel and cokernel bundles. [Appendix to Harris (1984)]. Invent. Math.75, 467-475 (1984) · Zbl 0542.14015 · doi:10.1007/BF01388639
[18] Hazewinkel, M., Martin, C.F.: Representations of the symmetric group, specialization order, systems and Grassmann manifolds. L’Ens. Math.29, 53-87 (1983) · Zbl 0536.20009
[19] Igusa, J.-I.: Arithmetic varieties of moduli for genus two. Ann. Math.72, 612-649 (1960) · Zbl 0122.39002 · doi:10.2307/1970233
[20] Kempf, G.: Curves ofg d 1 ’s. Compos. Math.55, 157-162 (1985) · Zbl 0601.14020
[21] Knudsen, F.: The projectivity of the moduli space of stable curves II: The stacksM g,n . Math. Scand.52, 161-199 (1983) · Zbl 0544.14020
[22] Koll?r, J., Schreyer, F.O.: The moduli of curves is stably rational forg?6. Duke Math. J.51, 239-242 (1984) · Zbl 0576.14027 · doi:10.1215/S0012-7094-84-05113-5
[23] Mumford, D.: Towards an enumerative geometry of the moduli space of curves. In: Arithmetic and geometry, Artin, M., Tate, J. (eds.), pp. 271-327. Boston: Birkh?user 1983 · Zbl 0554.14008
[24] Sernesi, E.: L’unirazionalit? della variet? dei moduli delli curve di genere dodici. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser.8, 405-439 (1981) · Zbl 0475.14024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.