论文标题
部分可观测时空混沌系统的无模型预测
Hexagonal lattice diagrams for complex curves in $\mathbb{CP}^2$
论文作者
论文摘要
We demonstrate that the geometric, topological, and combinatorial complexities of certain surfaces in $\mathbb{CP}^2$ are closely related: We prove that a positive genus surface $\mathcal{K}$ in $\mathbb{CP}^2$ that minimizes genus in its homology class is isotopic to a complex curve $\mathcal{C}_d$当且仅当$ \ Mathcal {k} $允许六角形晶格图时，它是一种特殊类型的阴影图，其中Arcs仅在桥点相交并在hexagons by $ \ mathbb {cp}^2 $的标准三角形的中央表面。这些图中有八个家族，其中两个代表有效的桥梁位置。结合了兰伯特 - 孔相关的象征表面和桥梁三触角的结果，这使我们能够在$ \ mathbb {cp}^2 $中提供对符号同位素问题的纯粹组合重新印度。最后，我们表明品种$ \ MATHCAL {V} _D = \ {[Z_1：Z__2：Z_3] \ in \ Mathbb {Cp}^2：Z_1Z_2^{D-1} {d-1} {d-1} + Z_2Z_3^ + Z_2Z_3^^{D-1} + Z_3Z_1} + Z_3Z_1^= = 0 $\mathcal{V}'_d = \{[z_1:z_2:z_3] \in \mathbb{CP}^2 : z_1^{d-1}z_2 + z_2^{d-1}z_3 + z_3^{d-1}z_1 = 0\}$ are in efficient bridge position with respect to the standard Stein trisection of $ \ mathbb {cp}^2 $，它们的阴影图与有效的六角形晶格图的两个家族一致。作为推论，我们证明了$ \ mathbb {cp}^3 $的两个无限的Hypersurfaces家族允许有效的Stein Trisextions，部分回答了Lambert-Cole and Meier的问题。
We demonstrate that the geometric, topological, and combinatorial complexities of certain surfaces in $\mathbb{CP}^2$ are closely related: We prove that a positive genus surface $\mathcal{K}$ in $\mathbb{CP}^2$ that minimizes genus in its homology class is isotopic to a complex curve $\mathcal{C}_d$ if and only if $\mathcal{K}$ admits a hexagonal lattice diagram, a special type of shadow diagram in which arcs meet only at bridge points and tile the central surface of the standard trisection of $\mathbb{CP}^2$ by hexagons. There are eight families of these diagrams, two of which represent surfaces in efficient bridge position. Combined with a result of Lambert-Cole relating symplectic surfaces and bridge trisections, this allows us to provide a purely combinatorial reformulation of the symplectic isotopy problem in $\mathbb{CP}^2$. Finally, we show that that the varieties $\mathcal{V}_d = \{[z_1:z_2:z_3] \in \mathbb{CP}^2 : z_1z_2^{d-1} + z_2z_3^{d-1} + z_3z_1^{d-1} = 0\}$ and $\mathcal{V}'_d = \{[z_1:z_2:z_3] \in \mathbb{CP}^2 : z_1^{d-1}z_2 + z_2^{d-1}z_3 + z_3^{d-1}z_1 = 0\}$ are in efficient bridge position with respect to the standard Stein trisection of $\mathbb{CP}^2$, and their shadow diagrams agree with the two families of efficient hexagonal lattice diagrams. As a corollary, we prove that two infinite families of complex hypersurfaces in $\mathbb{CP}^3$ admit efficient Stein trisections, partially answering a question of Lambert-Cole and Meier.
