Towards the ample cone of $\mgn$

By A. Gibney, S. Keel, Ian A Morrison. 2000

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. In this paper we study the ample cone of the moduli space M g,n of stable n -pointed curves of genus g . Our motivating conjecture is that a divisor on M g,n is ample iﬀ it has positive intersection with all 1-dimensional strata (the components of the locus of curves with at least 3 g + n − 2 nodes). This translates into a simple conjectural description of the cone by linear inequalities, and, as all the 1-strata are rational, includes the conjecture that the Mori cone is polyhedral and generated by rational curves. Our main result is that the conjecture holds iﬀ it holds for g = 0. More precisely, there is a natural ﬁnite map r : M 0 , 2 g + n → M g,n whose image is the locus R g,n of curves with all components rational. Any 1-strata either lies in R g,n or is numerically equivalent to a family E of elliptic tails and we show that a divisor D is nef iﬀ D · E ≥ 0 and r ∗ ( D ) is nef. We also give results on contractions (i.e. morphisms with connected ﬁbers to projective varieties) of M g,n for g ≥ 1 showing that any ﬁbration factors through a tautological one (given by forgetting points) and that the exceptional locus of any birational contraction is contained in the boundary.

Subfields: Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/d5c076eccf2e284eed17f7502ab66d622c5827c4

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Citations in the corpus

Existence of minimal models for varieties of log general type (2006, unknown): “and so the pair (Yi,Γi) clearly satisfies conditions (1), (2), (4) and (7) of Lemma 6.”
Hilbert's 14th problem and Cox rings (2005, background): “n is ﬁnitely generated, then results of Keel–Hu [HK] and Keel–McKernan [KM] almost imply the “Fulton conjecture” for M0,n and therefore the description of the Mori cone of M g,n (Gibney–Keel–Morrison [GKM]). HILBERT’S 14-TH PROBLEM AND COX RINGS 3 Following Mukai [M2], we also generalize Theorem 1.1”
Log minimal model program for the moduli space of stable curves: The first flip (2008, methodology): “ Γ Mg,n(T∗(K Mps g +αδps)+(9−11α)δ 1) ≃ Proj⊕n≥0 Γ Mps g ,n(KMps g +αδps) . LOG MINIMAL MODEL PROGRAM FOR Mg: FIRST FLIP 9 We shall construct the contractions by using the powerful results of [GKM02]: Proposition 2.9. For α∈ (7/10,9/11]∩Q, there there exists a birational contraction Ψ:Mg(α)→ Mg(”