Yi Hu

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• Hu, Y. (2021). Moduli of Curves of Genus One with Twisted Fields. Asian Journal of Mathematics, Volume 25(5), 683-714.
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  We construct a smooth Artin stack parameterizing the stable weighted curves of genus one with twisted fields and prove that it is isomorphic to the blowup stack of the moduli of genus one weighted curves studied by Hu and Li. This leads to a blowup-free construction of Vakil-Zinger's desingularization of the moduli of genus one stable maps to projective spaces. This construction provides the cornerstone of the theory of stacks with twisted fields, which is thoroughly studied in arXiv:2005.03384 and leads to a blowup-free resolution of the stable map moduli of genus two. [Journal_ref: ]
• Hu, Y. (2022). Moduli of Curves of Genus One with Twisted Fields. Asian Journal of Mathematics.
  More info

  We construct a smooth Artin stack parameterizing the stable weighted curves of genus one with twisted fields and prove that it is isomorphic to the blowup stack of the moduli of genus one weighted curves studied by Hu and Li. This leads to a blowup-free construction of Vakil-Zinger's desingularization of the moduli of genus one stable maps to projective spaces. This construction provides the cornerstone of the theory of stacks with twisted fields, which is thoroughly studied in arXiv:2005.03384 and leads to a blowup-free resolution of the stable map moduli of genus two. [Journal_ref: ]
• Hu, Y. (2017). Relative Resolution and Its Applications.. Proceedings of the Sixth International Congress of Chinese Mathematicians Vol. I, Adv. Lect. Math. (ALM), 36, Int. Press, Somerville, MA, 2017., Vol.I(36), 467–488.
• Hu, Y. (2014). The Space of Complete Quotients.. Pure and Applied Mathematics Quarterly, 10(1), 155-192.
• Hu, Y. (2013). Quivers, Invariants and Dualities.. Journal of Algebra,, 393(No.1), 197-216.
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  This paper studies the geometric and algebraic aspects of the moduli spaces of quivers of fence type. We first provide two quotient presentations of the quiver varieties and interpret their equivalence as a generalized Gelfand-MacPherson correspondence. Next, we introduce parabolic quivers and extend the above from the actions of reductive groups to the actions of parabolic subgroups. Interestingly, the above geometry finds its natural counterparts in the representation theory as the branching rules and transfer principle in the context of the reciprocity algebra. The last half of the paper establishes this connection.
• Yi, H. u., & Kim, S. (2013). Quivers, invariants and quotient correspondence. Journal of Algebra, 393, 197-216.
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  Abstract: This paper studies the geometric and algebraic aspects of the moduli spaces of quivers of fence type. We first provide two quotient presentations of the quiver varieties and interpret their equivalence as a generalized Gelfand-MacPherson correspondence. Next, we introduce parabolic quivers and extend the above from the actions of reductive groups to the actions of parabolic subgroups. Interestingly, the above geometry finds its natural counterparts in the representation theory as the branching rules and transfer principle in the context of the reciprocity algebra. The last half of the paper establishes this connection. © 2013 Elsevier Inc.
• Yi, H. u., & Jun, L. i. (2011). Derived resolution property for stacks, Euler classes and applications. Mathematical Research Letters, 18(4), 677-690.
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  Abstract: By resolving any perfect derived object over a Deligne-Mumford stack, we define its Euler class. We then apply it to define the Euler numbers for a smooth Calabi-Yau threefold in ℙ4. These numbers are conjectured to be the reduced Gromov-Witten invariants and to determine the usual Gromov-Witten numbers of the smooth quintic as speculated by J. Li and A. Zinger. © International Press 2011.
• Yi, H. u., Lin, J., & Shao, Y. (2011). A compactification of the space of algebraic maps from ℙ¹ to ℙⁿ. Communications in Analysis and Geometry, 19(1), 1-30.
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  Abstract: We provide a natural smooth projective compactification of the space of algebraic maps from ℙ1 to ℙn by adding a divisor with simple normal crossings.
• Yi, H. u., & Jun, L. i. (2010). Genus-one stable maps, local equations, and Vakil-Zinger's desingularization. Mathematische Annalen, 348(4), 929-963.
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  Abstract: We describe an algebro-geometric approach to Vakil-Zinger's desingularization of the main component of the moduli of genus one stable maps to ℙn (Vakil and Zinger in Res Announc Am Math Soc 13:53-59, 2007; Geom Topol 12(1):1-95, 2008). Our approach is based on understanding the local structure of this moduli space; it also gives a partial desingularization of the entire moduli space. The results proved should extend to higher genera. © 2010 Springer-Verlag.
• Yi, H. u. (2008). Toric degenerations of GIT quotients, Chow quotients, and M̄ _0,N. Asian Journal of Mathematics, 12(1), 47-54.
• Yi, H. u. (2005). Topological aspects of Chow quotients. Journal of Differential Geometry, 69(3), 399-440.
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  Abstract: This paper studies the canonical Chow quotient of a smooth projective variety by a reductive algebraic group. The main purpose is to introduce the Perturbation-Translation-Specialization relation that gives a computable characterization of the Chow cycles of the Chow quotient. Also, we provide, in the languages that are familiar to topologists and differential geometers, many topological interpretations of Chow quotient that have the advantage to be more intuitive and geometric. More precisely, over the field of complex numbers, these interpretations are, symplectically, the moduli spaces of stable orbits with prescribed momentum charges; and topologically, the moduli space of stable action-manifolds.
• Yi, H. u. (2004). Factorization theorem for projective varieties with finite quotient singularities. Journal of Differential Geometry, 68(3), 545-551.
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  Abstract: In this paper, we prove that any two birational projective varieties with finite quotient singularities can be realized as two geometric GIT quotients of a non-singular projective variety by a reductive algebraic group. Then, by applying the theory of Variation of Geometric Invariant Theory Quotients ([3]), we show that they are related by a sequence of GIT wall-crossing flips.
• Yi, H. u. (2003). A compactification of open varieties. Transactions of the American Mathematical Society, 355(12), 4737-4753.
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  Abstract: In this paper we prove a general method to compactify certain open varieties by adding normal crossing divisors. This is done by showing that blowing up along an arrangement of subvarieties can be carried out. Important examples such as Ulyanov's configuration spaces and complements of arrangements of linear subspaces in projective spaces, etc., are covered. Intersection ring and (nonrecursive) Hodge polynomials are computed. Furthermore, some general structures arising from the blowup process are also described and studied.
• Hu, Y., & Yau, S. -. (2002). HyperKähler manifolds and birational transformations. Advances in Theoretical and Mathematical Physics, 6(3), 557-574.
• Yi, H. u. (2002). Combinatorics and quotients of toric varieties. Discrete and Computational Geometry, 28(2), 151-174.
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  Abstract: For linear projections of polytopes and fans of cones we introduce some new objects such as: virtual chambers, virtual cones and (locally) coherent costrings. Virtual chambers (cones) generalize real chambers (cones), while (locally) coherent costrings are linear dual to (locally) coherent strings. We establish various correspondences for these objects and their connections to toric geometry.
• Yi, H. u., Liu, C., & Yau, S. (2002). Toric morphisms and fibrations of toric Calabi-Yau hypersurfaces. Advances in Theoretical and Mathematical Physics, 6(3), 457-505.
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  Abstract: Special fibrations of toric varieties have been used by physicists, e.g. the school of Candelas, to construct dual pairs in the study of Het/F-theory duality. Motivated by this, we investigate in this paper the details of toric morphisms between toric varieties. In particular, a complete toric description of fibers - both generic and non-generic -, image, and the flattening stratification of a toric morphism are given. Two examples are provided to illustrate the discussions. We then turn to the study of the restriction of a toric morphism to a toric hypersurface. The details of this can be understood by the various restrictions of a line bundle with a section that defines the hypersurface. These general toric geometry discussions give rise to a computational scheme for the details of a toric morphism and the induced fibration of toric hypersurfaces therein. We apply this scheme to study the family of complex 4-dimensional elliptic Calabi-Yau toric hypersurfaces that appear in a recent work of Braun-Candelas-dlOssa-Grassi. Some directions for future work are listed in the end. © 2002 International Press.