Koszul divisors on moduli spaces of curves

By G. Farkas. 2006

139 citations recorded by the source corpus.

Given a moduli space, how can one construct the ``best'' (in the sense of higher dimensional algebraic geometry) effective divisor on it? We show that, at least in the case of the moduli space of curves, the answer is provided by the Koszul divisor defined in terms of the syzygies of the parameterized objects. In this paper, we find a formula for the slopes of all Koszul divisors on $\overline{{\cal M}}_g$. In particular, we obtain the first infinite series of counterexamples to the Harris-Morrison Slope Conjecture and we prove the Maximal Rank Conjecture in the case when the Brill-Noether number of the corresponding linear series equals~$0$. We also find shorter proofs for the formulas of the class of the Brill-Noether and Gieseker-Petri divisors. Finally, we improve most of Logan's results on the Kodaira dimension of the moduli spaces $\overline{{\cal M}}_{g, n}$ of pointed stable curves.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/03ea790822f618900dbd64067b283d4e23733ba4

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In Geometry and Topology

1. David Jensen
2. Ingrid Bauer
3. F. Catanese
4. Dawei Chen
5. Matthew Baker
6. Sam Payne
7. Martin Moeller
8. A. Verra

In Mathematical Physics

1. Ingrid Bauer
2. Dawei Chen
3. F. Catanese
4. Martin Moeller
5. David Jensen
6. A. Verra
7. Maksym Fedorchuk
8. D. Smyth

Citations in the corpus

• Burniat surfaces I: fundamental groups and moduli of primary Burniat surfaces (2009, methodology): “d 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 divisors on Mg, which provide counterexamples to the Slope Conjecture on”
• The geometry of the moduli space of odd spin curves (2010, background): “ll exceptional components, then µ∗(ηC yq) = η (in particular η is trivial along exceptional components), and finally ordx(σ) ≥2. Using general constructions of stacks of limit linear series cf. [EH1], [F2], it is clear that Xg is a Deligne-Mumford stack. There exists a proper morphism τ : Xg →eS − g ”
• Degeneration of Linear Series from the Tropical Point of View and Applications (2015, background): “l linear series of degree dand rank r on a general curve is complete, and for this reason, most of the work in the subject focuses on the case where V = L(D). We note that the arguments of [BE85] and [Far09] involve degenerations to unions of two curves that meet in more than one point. Since such c”
• The classification of universal Jacobians over the moduli space of curves (2010, methodology): “nd 0.3. A keyrole is played by the effective divisor Dg := {[C,x 1,...,xg] ∈ Mg,g : h 0 C,OC(x 1+··· +xg) ≥ 2}. The class of the closure of Dg inside Mg,g has been computed, see [Log] Theorem 5.4, or [F1] Theorem 4.6, for an alternative proof: Dg ≡ −λ+ Xg i=1 ψi − [Xg/2] i=0 X T⊂{1,...,g} |#(T)−i| ”
• Nonvarying sums of Lyapunov exponents of Abelian differentials in low genus (2011, background): “1;2/ D 7 C 7!1;relC 2!2;rel ı0 ıother; for g D 5: Since this divisor class was not explicitly written out in [18], below we give a proof.”
• Alternate Compactifications of Moduli Spaces of Curves (2010, background): “Fedorchuk and David Ishii Smyth 49 are positive. From this, one sees that the restricted eective cone of M0;n is spanned by the two rays indicated in Table 3. Finally, while Logan [Log03] and Farkas [Far09b] have extended the work of Eisenbud, Harris, and Mumford to Mg;n (see [Far09b, Theorem 1.10] ”
• Tropical independence II: The maximal rank conjecture for quadrics (2015, background): “Farkas proved the case where ρ( g , r , d ) is zero and dim Sym 2 L ( D X ) = dim L ( 2 D X ) , and used this to deduce an infinite sequence of counterexamples to the slope conjecture [Farkas 2009, Theorem 1.5].”
• Koszul modules and Green’s conjecture (2018, methodology): “ert’s theorem [17, Corollary III.12.9] and Proposition 6.4, we conclude that Bis locally free, and by construction, the fiber B([C]) is given as in (96). To construct A, we proceed in a way similar to [12]. We begin by observing as in the proof of Proposition 6.4 that Vi MC(2) is resolved by a Koszul”