The Kodaira dimension of moduli spaces of curves with marked points
By Adam Logan. 2001
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The purpose of this paper is to generalize the methods and results due to Eisenbud, Harris, and Mumford on the moduli space of curves of genus g to the moduli space of curves of genus g with n marked points. The main result is the determination, for each g with 4 ≤ g ≤ 23, of an n such that the moduli space is of general type. 1. Introduction. In the early 1980s, Eisenbud, Harris, and Mumford stud- ied Mg, the moduli space of curves of a given genus, and succeeded in proving that if the genus is at least 24, the moduli space is of general type. This was contrary to many people's expectations, as it had been known for some time that moduli spaces of curves of small genus were rational or unirational. Briefly, what is involved in their proof is the computation of the canonical divisor of Mg, the exhibition of certain effective divisors and the determination of their classes, and the verification that the canonical divisor is the sum of an ample and an effective divisor. In the other direction, it is known that Mg has Kodaira dimension −∞ when g ≤ 13 or g = 15. One can generalize the definition of Mg by asking for a parametrization of the curves of genus g together with n distinct nonsingular points on them. This amounts to studying, instead of families of curves, families of curves with n disjoint sections. As before, it turns out that one obtains a noncompact moduli space, which is compactified by replacing "nonsingular curve" by "stable curve" in the definitions. In order to be considered stable, an n-pointed curve is required only to have finitely many automorphisms that fix all the marked points. The space can still be compact even when the marked points are required to be distinct and nonsingular, because coincident or singular marked points can be blown up on the total space of the family.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/160ffb757689c37e2d44ca029f0b3d2c89760814
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Citations in the corpus
- The geometry of the moduli space of odd spin curves (2010, background): “12 13 . Thisimpliesthefollowing upperboundfortheslopes(M12)ofthemodulispace: Corollary 6.2. 6+ 10 12 ≤s(M12) := infD∈Eff(M 12) s(D) ≤ 4415 642 = 6 + 10 12 + 14 321 . Another immediate application, via [Log], [F1], concerns the birational type of Mg,n: Theorem 6.3. The moduli space of n-pointed curves”
- The classification of universal Jacobians over the moduli space of curves (2010, methodology): “rked points), are varieties of general type, except a finite number of casesoccurring for relatively small genus,whenthesevarieties tend to be unirational, or at least uniruled, see [HM], [EH1], [FL], [Log], [V] for illustrations of this fact. In all known cases, the transition from uniruledness to b”
- Quasimodularity and large genus limits of Siegel-Veech constants (2016, unknown): “t. Lemma 4.7. Let ωπn+1 be the first Chern class of the relative dualizing sheaf associated to πn+1. Then hπn+1∗(ω2π n+1 )ψ2 ···ψni1,n = 0. SIEGEL-VEECH CONSTANTS 27 Proof. From the relation (see e.g. [34]) ψn+1 = ωπn+1 + Xn i=1 δ0,{i,n+1} in the tautological ring, we deduce πn+1∗(ψ 2 n+1) = πn+1∗ ψn”
- Alternate Compactifications of Moduli Spaces of Curves (2010, background): “s of i’s in the sum bn=X2c i=1 a+ b i(n i) n 1 i are positive. From this, one sees that the restricted eective cone of M0;n is spanned by the two rays indicated in Table3.2. Finally, while Logan [Log03] and Farkas [Far09b] have extended the work of Eisenbud, Harris, and Mumford to Mg;n (see [Far09”
- Versality of algebraic group actions and rational points on twisted varieties (2011, background): “a form of P1 (a 1-dimensional Brauer-Severi variety) has no rational points. Remark 6.5. If g> 23 then Mg,0 is not unirational, and hence, neither is Mg,n for any n> 1. For each g6 22, A. Logan [Lo03] showed that there is an explicit function f(g) such that Mg,n is not unirational for any n>”
- Nonvarying sums of Lyapunov exponents of Abelian differentials in low genus (2011, background): “The class of these pointed divisors has been calculated in many special cases, in particular in [34] and later by Farkas in [18].”
- Quadratic differentials in low genus: exceptional and non-varying (2012, background): “The results are basically contained in the literature ([17], [10]), but in several cases not all boundary terms were calculated in full detail.”
- Burniat surfaces I: fundamental groups and moduli of primary Burniat surfaces (2009, background): “effective geometric classes. Examplesof such a program being carried out, can be found in [EH2], [HM]-for the case of Brill-Noether divisors on Mg consisting of curves with a gr d when ρ(g,r,d) = −1, [Log]-where pointed Brill-Noether divisors on Mg,n are studied,and [F1]-for the case of Koszul divis”
- Effective divisors on \bar{M}_g, curves on K3 surfaces and the Slope Conjecture (2003, methodology): “rr arbitrarily large, such that KM 10,n −αKn−βDn−(aλ−birrδirr+ Pn i=1 ciψi) is a positive combination of boundary classes δi:S. As the class aλ−birrδirr+ Pn i=1 ciψiis big for such a choice (cf. e.g. [Log], Theorem 2.9), this implies that M10,nis of general type. Using again the divisor K we can sho”