{"paper":{"title":"Abelian varieties analogs of two results about algebraic curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Decomposable principally polarized abelian varieties of the form E×B are identified by failures in normal generation and second-order Gaussian map surjectivity, in direct analogy to hyperelliptic curves.","cross_cats":[],"primary_cat":"math.AG","authors_text":"Giuseppe Pareschi, Nelson Alvarado","submitted_at":"2025-02-16T22:21:55Z","abstract_excerpt":"We characterize decomposable principally polarized abelian varieties of the form $E\\times B$, with $E$ an elliptic curve, in two different ways, which are, surprisingly, completely analogous to classical results of curve theory concerning hyperelliptic curves. The first one is by the failure of a normal generation property, namely the generation in degree zero of a certain graded module over the symmetric algebra over $H^0(2\\Theta)$.\n  This appears to be the first result of this type in the realm of p.p.a.v.'s. The second characterization is by the failure of surjectivity of second order gauss"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We characterize decomposable principally polarized abelian varieties of the form E×B, with E an elliptic curve, in two different ways, which are, surprisingly, completely analogous to classical results of curve theory concerning hyperelliptic curves.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The assumption that the normal generation property and the surjectivity of Gaussian maps fail precisely for these decomposable varieties, relying on the specific choice of the polarization Θ and the equivalence to Nakamaye's theorem.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Characterizes decomposable p.p.a.v.s E×B via failure of normal generation of a module over Sym H^0(2Θ) and via non-surjectivity of second-order Gaussian maps for 6Θ (equiv. 3Θ fails to separate 2-jets), plus equivalence to effective Nakamaye theorem on minimal Seshadri constant.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Decomposable principally polarized abelian varieties of the form E×B are identified by failures in normal generation and second-order Gaussian map surjectivity, in direct analogy to hyperelliptic curves.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"0874f224b3cb57942977448a14fed0fa7cdd4a1cdeb5788f96f75e2036999fae"},"source":{"id":"2502.11288","kind":"arxiv","version":4},"verdict":{"id":"9c1322c7-271d-4fd6-9ecc-c022f70cd21a","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-23T02:33:12.176255Z","strongest_claim":"We characterize decomposable principally polarized abelian varieties of the form E×B, with E an elliptic curve, in two different ways, which are, surprisingly, completely analogous to classical results of curve theory concerning hyperelliptic curves.","one_line_summary":"Characterizes decomposable p.p.a.v.s E×B via failure of normal generation of a module over Sym H^0(2Θ) and via non-surjectivity of second-order Gaussian maps for 6Θ (equiv. 3Θ fails to separate 2-jets), plus equivalence to effective Nakamaye theorem on minimal Seshadri constant.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The assumption that the normal generation property and the surjectivity of Gaussian maps fail precisely for these decomposable varieties, relying on the specific choice of the polarization Θ and the equivalence to Nakamaye's theorem.","pith_extraction_headline":"Decomposable principally polarized abelian varieties of the form E×B are identified by failures in normal generation and second-order Gaussian map surjectivity, in direct analogy to hyperelliptic curves."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2502.11288/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}