{"paper":{"title":"Toward an algebraic theory of Welschinger invariants","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Marc Levine","submitted_at":"2018-08-07T07:27:54Z","abstract_excerpt":"Let $S$ be a smooth del Pezzo surface over a field $k$ of characteristic $\\neq 2, 3$. We define an invariant in the Grothendieck-Witt ring $GW(k)$ for \"counting\" rational curves in a curve class $D$ of fixed positive degree (with respect to the anti-canonical bundle $-K_S$) and containing a collection of distinct closed points $\\mathfrak{p}=\\sum_ip_i$ of total degree $r:=-D\\cdot K_S-1$ on $S$. This recovers Welschinger's invariant in case $k=\\mathbb{R}$ by applying the signature map. The main result is that this quadratic invariant depends only on the $\\mathbb{A}^1$-connected component contain"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.02238","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"}