For orientable surfaces of genus at least one, adding n points continuously to an m-point configuration is possible only if n is a multiple of m plus twice the genus minus two when m is at least two; for m equal to one it is always possible.
Eynard.Counting Surfaces
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On sections of configurations of points on orientable surfaces
For orientable surfaces of genus at least one, adding n points continuously to an m-point configuration is possible only if n is a multiple of m plus twice the genus minus two when m is at least two; for m equal to one it is always possible.