Hexagonal lattice diagrams for complex curves in
A Zupan�- arXiv preprint arXiv:2209.04274, 2022 - arxiv.org
arXiv preprint arXiv:2209.04274, 2022•arxiv.org
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�…
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�…
We demonstrate that the geometric, topological, and combinatorial complexities of certain surfaces in are closely related: We prove that a positive genus surface in that minimizes genus in its homology class is isotopic to a complex curve if and only if 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 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 . 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 , 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 admit efficient Stein trisections, partially answering a question of Lambert-Cole and Meier.
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