{"paper":{"title":"Motivic measures of the moduli spaces of pure sheaves on $\\mathbb{P}^2$ with all degrees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Yao Yuan","submitted_at":"2015-03-21T15:28:17Z","abstract_excerpt":"Let $\\mathcal{M}(d,\\chi)$ be the moduli stack of stable sheaves of rank 0, Euler characteristic $\\chi$ and first Chern class $dH~(d>0)$, with $H$ the hyperplane class in $\\mathbb{P}^2$. We compute the $A$-valued motivic measure $\\mu_A(\\mathcal{M}(d,\\chi))$ of $\\mathcal{M}(d,\\chi)$ and get explicit formula in codimension $D:=\\rho_d-1$, where $\\rho_d$ is $d-1$ for $d=p$ or $2p$ with $p$ prime, and $7$ otherwise. As a corollary, we get the last $2(D+1)$ Betti numbers of the moduli scheme $M(d,\\chi)$ when $d$ is coprime to $\\chi$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.06309","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}