This paper presents a self-contained proof of Special Termination of MMP (Minimal Model Program). By refining the assumptions and simplifying the argument, it offers a more accessible approach compared to the original...This paper presents a self-contained proof of Special Termination of MMP (Minimal Model Program). By refining the assumptions and simplifying the argument, it offers a more accessible approach compared to the original proof in BCHM (Birkar-Cascini-Hacon-McKernan).展开更多
Let E be a toric fibration arising from symplectic reduction of a direct sum of complex line bundles over (almost) Kähler base B. Then each torus-fixed point of the toric manifold fiber defines a section of t...Let E be a toric fibration arising from symplectic reduction of a direct sum of complex line bundles over (almost) Kähler base B. Then each torus-fixed point of the toric manifold fiber defines a section of the fibration. Let La be convex line bundles over B, Aa smooth divisors of B arising as the zero loci of generic sections of La , and a particular fixed-point section of E. Further assume the {Aa} to be mutually disjoint. The manifold is a new manifold with tautological line bundles over new projective spaces in the geometry, where previously there was a simpler vector bundle in the given local geometry (Section 1.5). Thus, we compute genus-0 Gromov-Witten invariants of in terms of genus-0 Gromov-Witten invariants of B and of {Aa}, the matrix used for the symplectic reduction description of the fiber of the toric fibration E→B, and the restriction maps . The proofs utilize the fixed-point localization technique describing the geometry of and its genus-0 Gromov-Witten theory, as well as the Quantum Lefschetz theorem relating the genus-0 Gromov-Witten theory of A with that of B.展开更多
In this article,we consider the problem of lifting the GW theory of a symplectic divisor to that of the ambient manifold in the context of symplectic birational geometry.In particular,we generalizeMaulik-Pandharipande...In this article,we consider the problem of lifting the GW theory of a symplectic divisor to that of the ambient manifold in the context of symplectic birational geometry.In particular,we generalizeMaulik-Pandharipande’s relative/absolute correspondence to relative-divisor/absolute correspondence.Then,we use it to lift a minimal uniruled invariant of a divisor to that of the ambient manifold.展开更多
We consider an integrable three-dimensional system of ordinary differential equations introduced by S. V. Kovalevskaya in a letter to G. Mittag- Leffler. We prove its isomorphism with the three-dimensional Euler top, ...We consider an integrable three-dimensional system of ordinary differential equations introduced by S. V. Kovalevskaya in a letter to G. Mittag- Leffler. We prove its isomorphism with the three-dimensional Euler top, and propose two integrable discretizations for it. Then we present an integrable generalization of the Kovalevskaya system, and study the problem of integrable discretization for this generalized system.展开更多
The rational ruled surface is a typical modeling surface in computer aided geometric design.A rational ruled surface may have different representations with respective advantages and disadvantages.In this paper,the au...The rational ruled surface is a typical modeling surface in computer aided geometric design.A rational ruled surface may have different representations with respective advantages and disadvantages.In this paper,the authors revisit the representations of ruled surfaces including the parametric form,algebraic form,homogenous form and Plucker form.Moreover,the transformations between these representations are proposed such as parametrization for an algebraic form,implicitization for a parametric form,proper reparametrization of an improper one and standardized reparametrization for a general parametrization.Based on these transformation algorithms,one can give a complete interchange graph for the different representations of a rational ruled surface.For rational surfaces given in algebraic form or parametric form not in the standard form of ruled surfaces,the characterization methods are recalled to identify the ruled surfaces from them.展开更多
Let Md be the moduli space of stable sheaves on P2with Hilbert polynomial dm+1.In this paper,we determine the effective and the nef cone of the space Md by natural geometric divisors.Main idea is to use the wall-cross...Let Md be the moduli space of stable sheaves on P2with Hilbert polynomial dm+1.In this paper,we determine the effective and the nef cone of the space Md by natural geometric divisors.Main idea is to use the wall-crossing on the space of Bridgeland stability conditions and to compute the intersection numbers of divisors with curves by using the Grothendieck-Riemann-Roch theorem.We also present the stable base locus decomposition of the space M6.As a byproduct,we obtain the Betti numbers of the moduli spaces,which confirm the prediction in physics.展开更多
The aim of this paper is to study 6-canonical system of a nonsingular minimal 3-fold X. If|2Kx|is not composed of pencils, it is shown that is birational with possible exceptionsfor:
文摘This paper presents a self-contained proof of Special Termination of MMP (Minimal Model Program). By refining the assumptions and simplifying the argument, it offers a more accessible approach compared to the original proof in BCHM (Birkar-Cascini-Hacon-McKernan).
文摘Let E be a toric fibration arising from symplectic reduction of a direct sum of complex line bundles over (almost) Kähler base B. Then each torus-fixed point of the toric manifold fiber defines a section of the fibration. Let La be convex line bundles over B, Aa smooth divisors of B arising as the zero loci of generic sections of La , and a particular fixed-point section of E. Further assume the {Aa} to be mutually disjoint. The manifold is a new manifold with tautological line bundles over new projective spaces in the geometry, where previously there was a simpler vector bundle in the given local geometry (Section 1.5). Thus, we compute genus-0 Gromov-Witten invariants of in terms of genus-0 Gromov-Witten invariants of B and of {Aa}, the matrix used for the symplectic reduction description of the fiber of the toric fibration E→B, and the restriction maps . The proofs utilize the fixed-point localization technique describing the geometry of and its genus-0 Gromov-Witten theory, as well as the Quantum Lefschetz theorem relating the genus-0 Gromov-Witten theory of A with that of B.
文摘In this article,we consider the problem of lifting the GW theory of a symplectic divisor to that of the ambient manifold in the context of symplectic birational geometry.In particular,we generalizeMaulik-Pandharipande’s relative/absolute correspondence to relative-divisor/absolute correspondence.Then,we use it to lift a minimal uniruled invariant of a divisor to that of the ambient manifold.
文摘We consider an integrable three-dimensional system of ordinary differential equations introduced by S. V. Kovalevskaya in a letter to G. Mittag- Leffler. We prove its isomorphism with the three-dimensional Euler top, and propose two integrable discretizations for it. Then we present an integrable generalization of the Kovalevskaya system, and study the problem of integrable discretization for this generalized system.
基金supported by Beijing Natural Science Foundation under Grant No.Z190004the National Natural Science Foundation of China under Grant No.61872332+2 种基金the University of Chinese Academy of Sciences and by FEDER/Ministerio de CienciaInnovación y Universidades Agencia Estatal de Investigación/MTM2017-88796-P(Symbolic Computation:New challenges in Algebra and Geometry together with its applications)the Research Group ASYNACS(Ref.CCEE2011/R34)。
文摘The rational ruled surface is a typical modeling surface in computer aided geometric design.A rational ruled surface may have different representations with respective advantages and disadvantages.In this paper,the authors revisit the representations of ruled surfaces including the parametric form,algebraic form,homogenous form and Plucker form.Moreover,the transformations between these representations are proposed such as parametrization for an algebraic form,implicitization for a parametric form,proper reparametrization of an improper one and standardized reparametrization for a general parametrization.Based on these transformation algorithms,one can give a complete interchange graph for the different representations of a rational ruled surface.For rational surfaces given in algebraic form or parametric form not in the standard form of ruled surfaces,the characterization methods are recalled to identify the ruled surfaces from them.
基金supported by TJ Park Science Fellowship of POSCO TJ Park Foundation and National Research Foundation of Korea(Grant No.2013R1A1A2006037)
文摘Let Md be the moduli space of stable sheaves on P2with Hilbert polynomial dm+1.In this paper,we determine the effective and the nef cone of the space Md by natural geometric divisors.Main idea is to use the wall-crossing on the space of Bridgeland stability conditions and to compute the intersection numbers of divisors with curves by using the Grothendieck-Riemann-Roch theorem.We also present the stable base locus decomposition of the space M6.As a byproduct,we obtain the Betti numbers of the moduli spaces,which confirm the prediction in physics.
文摘The aim of this paper is to study 6-canonical system of a nonsingular minimal 3-fold X. If|2Kx|is not composed of pencils, it is shown that is birational with possible exceptionsfor:
