Counts and end-curves in two-parameter persistence

Given a finite dimensional, bigraded module over the polynomial ring in two variables, we define its two-parameter count, a natural number, and its end-curves, a set of plane curves. These are two-dimensional analogues of the notions of bar-count and endpoints of singly-graded modules over the polynomial ring in one variable, from persistence theory. We show that our count is the unique one satisfying certain natural conditions; as a consequence, several inclusion-exclusion formulas in two-parameter persistence yield the same positive number, which equals our count, and which in turn equals the number of end-curves, giving geometric meaning to this count. We show that the end-curves determine the classical Betti tables by showing that they interpolate between generators, relations, and syzygies. Using the band representations of a certain string algebra, we show that the set of end-curves admits a canonical partition, where each part forms a closed curve on the plane; we call this the boundary of the module. As an invariant, the boundary is neither weaker nor stronger than the rank invariant, but, in contrast to the rank invariant, it is a complete invariant on the set of spread-decomposable representations. Our results connect several lines of work in multiparameter persistence, and their extension to modules over the real-exponent polynomial ring in two variables relates to two-dimensional Morse theory.