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21
Tautological relations and the rspin Witten conjecture
"... In [23, 24], Y.P. Lee introduced a notion of universal relation for formal Gromov–Witten potentials. Universal relations are connected to tautological relations in the cohomology ring of the moduli space Mg,n of stable curves. Y.P. Lee conjectured that the two sets of relations coincide and proved ..."
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Cited by 43 (11 self)
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In [23, 24], Y.P. Lee introduced a notion of universal relation for formal Gromov–Witten potentials. Universal relations are connected to tautological relations in the cohomology ring of the moduli space Mg,n of stable curves. Y.P. Lee conjectured that the two sets of relations coincide and proved the inclusion (tautological relations) ⊂ (universal relations) modulo certain results announced by C. Teleman. He also proposed an algorithm that, conjecturally, computes all universal/tautological relations. Here we give a geometric interpretation of Y.P. Lee’s algorithm. This leads to a much simpler proof of the fact that every tautological relation gives rise to a universal relation. We also show that Y.P. Lee’s algorithm computes the tautological relations correctly if and only if the Gorenstein conjecture on the tautological cohomology ring of Mg,n is true. These results are first steps in the task of establishing an equivalence between formal and geometric Gromov–Witten theories. In particular, it implies that in any semisimple Gromov–Witten theory where arbitrary correlators can be expressed in genus 0 correlators using only tautological relations, the formal and the geometric Gromov–Witten potentials coincide.
LANDAUGINZBURG/CALABIYAU CORRESPONDENCE, GLOBAL MIRROR SYMMETRY AND ORLOV EQUIVALENCE
, 2013
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The Witten top Chern class via Ktheory
 J. Algebraic Geom
"... Abstract. The Witten top Chern class is the crucial cohomology class needed to state a conjecture by Witten relating the Gelfand–Dikiĭ hierarchies to higher spin curves. In [PV01], Polishchuk and Vaintrob provide an algebraic construction of such a class. We present a more straightforward constructi ..."
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Cited by 17 (2 self)
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Abstract. The Witten top Chern class is the crucial cohomology class needed to state a conjecture by Witten relating the Gelfand–Dikiĭ hierarchies to higher spin curves. In [PV01], Polishchuk and Vaintrob provide an algebraic construction of such a class. We present a more straightforward construction via Ktheory. In this way we shortcircuit the passage through bivariant intersection theory and the use of MacPherson’s graph construction. Furthermore, we show that the Witten top Chern class admits a natural lifting to the Ktheory ring. 1.
Towards an enumerative geometry of the moduli space of twisted curves and rth roots
, 2008
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GromovWitten theory of étale gerbes I: root gerbes, in preparation
"... Abstract. Let X be a smooth complex projective algebraic variety. Given a line bundle L over X and an integer r> 1 we study the GromovWitten theory of the stack rp L/X of rth root of L. We prove an exact formula expressing genus 0 GromovWitten invariants of rp L/X in terms of those of X. Assum ..."
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Cited by 9 (3 self)
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Abstract. Let X be a smooth complex projective algebraic variety. Given a line bundle L over X and an integer r> 1 we study the GromovWitten theory of the stack rp L/X of rth root of L. We prove an exact formula expressing genus 0 GromovWitten invariants of rp L/X in terms of those of X. Assuming that either rp L/X or X has semisimple quantum cohomology, we prove an exact formula between higher genus invariants. We also present constructions of moduli stacks of twisted stable maps to rp L/X starting from moduli stack of stable maps to X.
SYZYGIES OF TORSION BUNDLES AND THE GEOMETRY OF THE LEVEL ℓ MODULAR VARIETY OVER Mg
"... We formulate, and in some cases prove, three statements concerning the purity or, more generally, the naturality of the resolution of various modules one can attach to a generic curve of genus g and a torsion point of ℓ in its Jacobian. These statements can be viewed as analogues of Green’s Conjec ..."
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Cited by 9 (5 self)
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We formulate, and in some cases prove, three statements concerning the purity or, more generally, the naturality of the resolution of various modules one can attach to a generic curve of genus g and a torsion point of ℓ in its Jacobian. These statements can be viewed as analogues of Green’s Conjecture and we verify them computationally for bounded genus. We then compute the cohomology class of the corresponding nonvanishing locus in the moduli spaceRg,ℓ of twisted levelℓcurves of genus g and use this to derive results about the birational geometry ofRg,ℓ. For instance, we prove thatRg,3 is a variety of general type when g> 11 and the Kodaira dimension of R11,3 is greater than or equal to 19. In the last section we explain probabilistically the unexpected failure of the PrymGreen conjecture in genus 8 and level 2.
LandauGinzburg/CalabiYau correspondence for the complete intersections
 X3,3 and X2,2,2,2 , arXiv:1301.5530 [math.AG
, 2013
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SINGULARITIES OF THE MODULI SPACE OF LEVEL CURVES
"... We describe the singular locus of the compactification of the moduli space Rg,ℓ of curves of genus g paired with an ℓtorsion point in their Jacobian. Generalising previous work for ℓ ≤ 2, we also describe the sublocus of noncanonical singularities for any positive integer ℓ. For ℓ ≤ 6 and ℓ ̸ = 5 ..."
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Cited by 3 (1 self)
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We describe the singular locus of the compactification of the moduli space Rg,ℓ of curves of genus g paired with an ℓtorsion point in their Jacobian. Generalising previous work for ℓ ≤ 2, we also describe the sublocus of noncanonical singularities for any positive integer ℓ. For ℓ ≤ 6 and ℓ ̸ = 5, this allows us to provide a lifting result on pluricanonical forms playing an essential role in the computation of the Kodaira dimension of Rg,ℓ: every pluricanonical form on the smooth locus of the moduli space extends to a desingularisation of the compactified moduli space.