{"paper":{"title":"Asymptotic constructions and invariants of graded linear series","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Chih-Wei Chang, Shin-Yao Jow","submitted_at":"2019-03-14T13:08:35Z","abstract_excerpt":"Let $X$ be a complete variety of dimension $n$ over an algebraically closed field $\\mathbf{K}$. Let $V_\\bullet$ be a graded linear series associated to a line bundle $L$ on $X$, that is, a collection $\\{V_m\\}_{m\\in\\mathbb{N}}$ of vector subspaces $V_m\\subseteq H^0(X,L^{\\otimes m})$ such that $V_0=\\mathbf{K}$ and $V_k\\cdot V_\\ell\\subseteq V_{k+\\ell}$ for all $k,\\ell\\in\\mathbb{N}$. For each $m$ in the semigroup \\[\n  \\mathbf{N}(V_\\bullet)=\\{m\\in\\mathbb{N}\\mid V_m\\ne 0\\},\\] the linear series $V_m$ defines a rational map \\[ \\phi_m\\colon X\\dashrightarrow Y_m\\subseteq\\mathbb{P}(V_m), \\] where $Y_m$ d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.05967","kind":"arxiv","version":1},"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"}