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In the past 20 years, compactifications of the families of curves in algebraic varieties X have been studied via stable maps, Hilbert schemes, stable pairs, unramified maps, and stable quotients.
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A. Givental, Equivariant Gromov-Witten invariants
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Y. Ruan, Topological sigma model and Donaldson type invariants in Gromov theory
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SW ⇒ Gr {\rm SW}\Rightarrow{\rm Gr} : from the Seiberg-Witten equations to pseudo-holomorphic curves
C. H. Taubes, · 1996
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W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology
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E. Getzler, Intersection theory on M ¯ 1 , 4 \overline{M}_{1,4} and elliptic Gromov-Witten invariants
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B. Lian, K. Liu, and S.-T. Yau, Mirror principle I
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L. Göttsche, A conjectural generating function for numbers of curves on surfaces
1998
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L. Göttsche and R. Pandharipande, The quantum cohomology of blow-ups of ℙ 2 {\mathbb{P}}^{2} and enumerative geometry
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Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties,
J. Li and G. Tian, · 1998
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A. Beauville, Counting rational curves on K 3 K3 surfaces
1999
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X. Chen, Rational curves on K 3 K3 surfaces
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D. Cox and S. Katz, Mirror symmetry and algebraic geometry
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Stability conditions on triangulated categories
T. Bridgeland · 2007
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I. Itenberg, G. Mikhalkin, and E. Shustin, Tropical algebraic geometry
2007
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Symmetric obstruction theories and Hilbert schemes of points on threefolds
K. Behrend and B. Fantechi · 2008
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Genus zero Gopakumar-Vafa invariants of contractible curves,
S. Katz, · 2008
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Enumerative geometry of Calabi-Yau 4-folds
A. Klemm and R. Pandharipande, · 2008
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Birational Calabi-Yau 3-folds and BPS state counting
Y. Toda · 2008
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M-theory, topological strings and spinning black holes
S. Katz, A. Klemm, and C. Vafa, · 1999
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J. Kollár, Rational curves on algebraic varieties
1999
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R. Pandharipande, Hodge integrals and degenerate contributions
1999
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P. Belorousski and R. Pandharipande, A descendent relation in genus 2
2000
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The enumerative geometry of K3 surfaces and modular forms
J. Bryan and C. Leung, · 2000
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Hodge integrals and Gromov-Witten theory
C. Faber and R. Pandharipande · 2000
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R. Vakil and A. Zinger, A desingularization of the main component of the moduli space of genus-one stable maps into ℙ n \mathbb{P}^{n}
2008
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Polynomial Bridgeland stability conditions and the large volume limit
A. Bayer, · 2009
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Donaldson-Thomas invariants via microlocal geometry,
K. Behrend, · 2009
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Algebraic cobordism revisited
M. Levine and R. Pandharipande, · 2009
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Recent progress in GW-invariants of Calabi-Yau threefolds
J. Li · 2009
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Curve counting via stable pairs in the derived category
R. Pandharipande and R. P. Thomas, · 2009
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Limit stable objects on Calabi-Yau 3-folds,
Y. Toda, · 2009
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A. Zinger, The reduced genus-one Gromov-Witten invariants of Calabi-Yau hypersurfaces
2009
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I. Ciocan-Fontanine and B. Kim, Moduli stacks of stable toric quasimaps
2010
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M. Gross, R. Pandharipande, B. Siebert, The tropical vertex
2010
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Noether-Leftschetz theory and the Yau-Zaslow conjecture
A. Klemm, D. Maulik, R. Pandharipande and E. Scheidegger, · 2010
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Stable pairs and BPS invariants
R. Pandharipande and R. P. Thomas, · 2010
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Enumerative geometry of Calabi-Yau 5-folds,
R. Pandharipande and A. Zinger, · 2010
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Curve counting theories via stable objects I: DT/PT correspondence
Y. Toda, · 2010
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Hall algebras and curve-counting invariants
T. Bridgeland · 2011
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M. Gross, Tropical geometry and mirror symmetry
2011
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M. Gross and B. Siebert, From real affine geometry to complex geometry
2011
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A theory of generalized DonaldsonThomas invariants,
D. Joyce and Y. Song, · 2011
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A. Marian, D. Oprea and R. Pandharipande, The moduli space of stable quotients
2011
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Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds
D. Maulik, A. Oblomkov, A. Okounkov, and R. Pandharipande · 2011
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Hilbert schemes and stable pairs: GIT and derived category wall crossings
J. Stoppa and R. P. Thomas, · 2011
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Y. Toda, Moduli spaces of stable quotients and wall-crossing phenomena
2011
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