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This analysis describes the Picard and divisor class groups of symmetric powers, suggesting new insights on stratification.
We study several properties of the symmetric power SᵐX of a smooth variety X. We describe the Picard and divisor class groups of SᵐX when X is projective. We give a complete description of the stratification of SᵐX by iterated singular locus in terms of some combinatorial data regarding partitions of the integer $m.$ This gives a new viewpoint of a natural stratification of SᵐX by multiplicities.
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Bansal et al. (2025) studied this question.
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