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53
Towards the ample cone of ¯ Mg,n
"... To Bill Fulton on his sixtieth birthday Abstract. In this paper we study the ample cone of the moduli space Mg,n of stable npointed curves of genus g. Our motivating conjecture is that a divisor on Mg,n is ample iff it has positive intersection with all 1dimensional strata (the components of the l ..."
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To Bill Fulton on his sixtieth birthday Abstract. In this paper we study the ample cone of the moduli space Mg,n of stable npointed curves of genus g. Our motivating conjecture is that a divisor on Mg,n is ample iff it has positive intersection with all 1dimensional strata (the components of the locus of curves with at least 3g+n−2 nodes). This translates into a simple conjectural description of the cone by linear inequalities, and, as all the 1strata 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: M0,2g+n → Mg,n whose image is the locus Rg,n of curves with all components rational. Any 1strata either lies in Rg,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 Mg,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. §0 Introduction and Statement of Results. The moduli space of stable curves is among the most studied objects in algebraic geometry. Nonetheless, its birational geometry remains largely a mystery and most Mori theoretic problems in the area are entirely
The moduli space of stable quotients
"... Dedicated to William Fulton on the occasion of his 70th birthday Abstract. A moduli space of stable quotients of the rank n trivial sheaf on stable curves is introduced. Over nonsingular curves, the moduli space is Grothendieck’s Quot scheme. Over nodal curves, a relative construction is made to kee ..."
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Cited by 29 (3 self)
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Dedicated to William Fulton on the occasion of his 70th birthday Abstract. A moduli space of stable quotients of the rank n trivial sheaf on stable curves is introduced. Over nonsingular curves, the moduli space is Grothendieck’s Quot scheme. Over nodal curves, a relative construction is made to keep the torsion of the quotient away from the singularities. New compactifications of classical spaces arise naturally: a nonsingular and irreducible compactification of the moduli of maps from genus 1 curves to projective space is obtained. Localization on the moduli of stable quotients leads to new relations in the tautological ring generalizing BrillNoether constructions. The moduli space of stable quotients is proven to carry a canonical 2term obstruction theory and thus a virtual class. The resulting system of descendent invariants is proven to equal the GromovWitten theory of the Grassmannian in all genera. Stable quotients
Moduli of weighted stable maps and their gravitational descendants
, 2006
"... We study the intersection theory on the moduli spaces of maps of npointed curves f: (C,s1,... sn) → V which are stable with respect to the weight data (a1,..., an), 0 ≤ ai ≤ 1. After describing the structure of these moduli spaces, we prove a formula describing the way each descendant changes und ..."
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Cited by 19 (1 self)
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We study the intersection theory on the moduli spaces of maps of npointed curves f: (C,s1,... sn) → V which are stable with respect to the weight data (a1,..., an), 0 ≤ ai ≤ 1. After describing the structure of these moduli spaces, we prove a formula describing the way each descendant changes under a wall crossing. As a corollary, we compute the weighted descendants in terms of the usual ones, i.e. for the weight data (1,..., 1), and vice versa.
Cyclotomy and analytic geometry over F1
"... Abstract. Geometry over non–existent “field with one element ” F1 conceived by Jacques Tits [Ti] half a century ago recently found an incarnation, in at least two related but different guises. In this paper I analyze the crucial role of roots of unity in this geometry and propose a version of the no ..."
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Cited by 16 (0 self)
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Abstract. Geometry over non–existent “field with one element ” F1 conceived by Jacques Tits [Ti] half a century ago recently found an incarnation, in at least two related but different guises. In this paper I analyze the crucial role of roots of unity in this geometry and propose a version of the notion of “analytic functions” over F1. The paper combines a focused survey with some new constructions. To Alain Connes, for his sixtieth anniversary 0. Introduction: many faces of cyclotomy 0.1. Roots of unity and field with one element. The basics of algebraic geometry over an elusive “field with one element F1 ” were laid down recently in [So], [De1], [De2], [TV], fifty years after a seminal remark by J. Tits [Ti]. There are many motivations to look for F1; a hope to imitate Weil’s proof for Riemann’s
Stability conditions, wallcrossing and weighted GromovWitten invariants, Mosc
 Math. J
"... ABSTRACT. We extend B. Hassett’s theory of weighted stable pointed curves ([Has03]) to weighted stable maps. The space of stability conditions is described explicitly, and the wallcrossing phenomenon studied. This can be considered as a nonlinear analog of the theory of stability conditions in abe ..."
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ABSTRACT. We extend B. Hassett’s theory of weighted stable pointed curves ([Has03]) to weighted stable maps. The space of stability conditions is described explicitly, and the wallcrossing phenomenon studied. This can be considered as a nonlinear analog of the theory of stability conditions in abelian and triangulated categories (cf. [GKR04], [Bri07], [Joy06, Joy07a, Joy07b, Joy08]). We introduce virtual fundamental classes and thus obtain weighted GromovWitten invariants. We show that by including gravitational descendants, one obtains an Lalgebra as introduced in [LM04] as a generalization of a cohomological field theory. §0. Introduction: Hassett’s stability conditions 0.1. Pointed curves. A nodal curve C over an algebraically closed field k is a proper nodal reduced onedimensional scheme of finite type over this field whose only singularities are nodes. The genus of C is g: = dim H1 (C, OC). Let S be a finite set. A nodal Spointed curve C is a system (C,si i ∈ S) where
The functor of toric varieties associated with Weyl chambers and LosevManin moduli spaces
 Tohoku Math. J
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Veronese quotient models of M0,n and conformal blocks
, 2012
"... The moduli space M0,n of DeligneMumford stable npointed rational curves admits morphisms to spaces recently constructed by Giansiracusa, Jensen, and Moon that we call Veronese quotients. We study divisors on M0,n associated to these maps and show that these divisors arise as first Chern classes of ..."
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Cited by 8 (4 self)
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The moduli space M0,n of DeligneMumford stable npointed rational curves admits morphisms to spaces recently constructed by Giansiracusa, Jensen, and Moon that we call Veronese quotients. We study divisors on M0,n associated to these maps and show that these divisors arise as first Chern classes of vector bundles of conformal blocks.
Closed/open string diagrammatics
 Nucl. Phys. B
"... Abstract. We introduce a combinatorial model based on measured foliations in surfaces which captures the phenomenology of open/closed string interactions. All of the predicted equations of string theory are derived in this model, and new equations can be discovered as well. In particular, several ne ..."
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Cited by 8 (2 self)
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Abstract. We introduce a combinatorial model based on measured foliations in surfaces which captures the phenomenology of open/closed string interactions. All of the predicted equations of string theory are derived in this model, and new equations can be discovered as well. In particular, several new equations together with known transformations generate the combinatorial version of open/closed duality. On the topological and chain levels, the algebraic structure discovered is new, but it specializes to a modular bioperad on the level of homology.
STABILITY CONDITIONS, WALLCROSSING AND WEIGHTED GROMOVWITTEN INVARIANTS
, 2006
"... ABSTRACT. We extend B. Hassett’s theory of weighted stable pointed curves ([Has03]) to weighted stable maps. The space of stability conditions is described explicitly, and the wallcrossing phenomenon studied. This can be considered as a nonlinear analog of the theory of stability conditions in abe ..."
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Cited by 7 (1 self)
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ABSTRACT. We extend B. Hassett’s theory of weighted stable pointed curves ([Has03]) to weighted stable maps. The space of stability conditions is described explicitly, and the wallcrossing phenomenon studied. This can be considered as a nonlinear analog of the theory of stability conditions in abelian and triangulated categories (cf. [Rud97], [Bri02b], [Joy03, Joy05, Joy04a, Joy04b]). We introduce virtual fundamental classes and thus obtain weighted GromovWitten invariants. We show that by including gravitational descendants, one obtains an Lalgebra as introduced in [LM04] as a generalization of a cohomological field theory. §0. Introduction: Hassett’s stability conditions 0.1. Pointed curves. A nodal curve C over an algebraically closed field k is a proper nodal reduced onedimensional algebraical scheme over this field whose only singularities are nodes. The genus of C is g: = dim H1 (C, OC). Let S be a finite set. A nodal Spointed curve C is a system (C,si i ∈ S) where