Computations with moduli of perverse point sheaves

Given a birational modification $X\to Y$ of complex projective varieties with fiber dimension 1 and rational singularities, consider the main component of Bridgeland’s moduli space $W\to Y$ of perverse point sheaves on $X\mathrm{/}Y$. We give criteria for the normalization of $W$ to coincide with the transform (flip/flop) $X^{+}\to Y$ of $X\to Y$. First, this holds if the exceptional loci of $X\to Y$ and $W\to Y$ have codimension $>1$. Second, if the ideal sheaf of $X{\times }_Y{X}^{+}$ is flat over $X^{+}$ then there is at least a morphism $X^+ \to W$. We illustrate the results using a number of examples, including surface modifications and standard Francia flips.