Hexagonal lattice diagrams for complex curves in ℂℙ²
Alexander Zupan · Transactions of the American Mathematical Society · 2023
We demonstrate that the geometric, topological, and combinatorial complexities of certain surfaces in C P 2 \mathbb {CP}^2 are closely related: We prove that a positive genus surface K \mathcal {K} in C P 2 \mathbb {CP}^2 that minimizes genus in its homology class is isotopic to a complex curve C d \mathcal {C}_d if and only if K \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 C P 2 \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 C P 2 \mathbb {CP}^2 . Finally, we show that that the varieties V d = { [ z 1 : z 2 : z 3 ] ∈ C P 2 : z 1 z 2 d − 1 + z 2 z 3 d −