Title: Counting Nodal Spectral Curves via Homological Mirror Symmetry
Abstract: For a toric surface, we study the counts of rational nodal curves in a linear system with prescribed intersection points on the boundary divisor. These counts can be interpreted as logarithmic Gromov–Witten invariants. Following a Yau–Zaslow type argument, we show that such counts coincide with the Euler number of the moduli space of a family of compactified Jacobians, which is a variation of the Beauville integrable system. Using homological mirror symmetry, we identify this integrable system with a moduli space of certain constructible sheaves. We then show that this moduli space admits a stratification called "ruling decomposition", and thereby compute its Euler number using combinatorial data. This is a joint work in progress with Tom Graber and Eric Zaslow.
Eric Zaslow
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