{"paper":{"title":"Fano congruences of index $3$ and alternating $3$-forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Daniele Faenzi, Emilia Mezzetti, Kristian Ranestad, Pietro De Poi","submitted_at":"2016-06-15T10:42:21Z","abstract_excerpt":"We study congruences of lines $X_\\omega$ defined by a sufficiently general choice of an alternating 3-form $\\omega$ in $n+1$ dimensions, as Fano manifolds of index $3$ and dimension $n-1$. These congruences include the $\\mathrm{G}_2$-variety for $n=6$ and the variety of reductions of projected $\\mathbb{P}^2 \\times \\mathbb{P}^2$ for $n=7$.\n  We compute the degree of $X_\\omega$ as the $n$-th Fine number and study the Hilbert scheme of these congruences proving that the choice of $\\omega$ bijectively corresponds to $X_\\omega$ except when $n=5$. The fundamental locus of the congruence is also stud"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.04715","kind":"arxiv","version":2},"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"}