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When $Y$ is a projective Calabi-Yau threefold, this shows that the DT contribution of a smooth rigid curve is the signed Euler characteristic of the moduli space. This can be rephrased as a DT/PT wall-crossing type formula, which can be formulated for arbitrary smooth curves. 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Ricolfi","submitted_at":"2016-10-26T16:30:54Z","abstract_excerpt":"Let $C$ be a smooth curve embedded in a smooth quasi-projective threefold $Y$, and let $Q^n_C=\\textrm{Quot}_n(\\mathscr I_C)$ be the Quot scheme of length $n$ quotients of its ideal sheaf. We show the identity $\\tilde\\chi(Q^n_C)=(-1)^n\\chi(Q^n_C)$, where $\\tilde\\chi$ is the Behrend weighted Euler characteristic. When $Y$ is a projective Calabi-Yau threefold, this shows that the DT contribution of a smooth rigid curve is the signed Euler characteristic of the moduli space. This can be rephrased as a DT/PT wall-crossing type formula, which can be formulated for arbitrary smooth curves. 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