Moduli dimensions of lattice polygons

Given a lattice polygon P with g interior points, we can associate to $$P$$ two moduli spaces: the space of algebraic curves that are non-degenerate respect and tropical genus Newton P. We completely classify possible dimensions such have in case. For non-hyperelliptic polygons, dimension must be between $$2g+1$$ take on any integer value this range, exceptions only cases 3, 4, 7. provide similar result for hyperelliptic which range is from $$2g-1$$ . In case our results also hold