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Moreover, this map is biholomorphic onto the image outside the Satake boundary. It is not biholomorphic on the whole in the cases $g\\ge 6$. Igusa also proved that in the cases $g\\le 2$ this map is biholomorphic onto the image. In this paper we extend this result to the case $g=3$. So we show that the theta map $$X_3(4,8)\\to P^{35}$$ is biholomorphic onto"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1606.04468","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2016-06-14T17:33:57Z","cross_cats_sorted":[],"title_canon_sha256":"2b2d7298e96e46e0a800bd101750dcc4beaf90a0413eb50c8f01384d70976d42","abstract_canon_sha256":"a13393982d24c7389d27323dec7a8b0d6f8fea42a1cfabb5e043c097bb7f5511"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:33:23.611211Z","signature_b64":"XIsoNZ8aq/sNPgEQ7teu2bLORHge8xy0eoOAH4/e1WWw2q8zsQTV2GJJFL8njN+yKdQ8dErlSPp0sfExH0UfCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"33a2a254794be9a72c0668cedae270ec0fdd544cf867078678850ca920815351","last_reissued_at":"2026-05-18T00:33:23.610538Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:33:23.610538Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the variety associated to the ring of theta constants in genus 3","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Eberhard Freitag, Riccardo Salvati Manni","submitted_at":"2016-06-14T17:33:57Z","abstract_excerpt":"Due to fundamental results of Igusa and Mumford the $N=2^{g-1}(2^g+1)$ even theta constants define for each genus $g$ an injective holomorphic map of the Satake compactification $X_g(4,8)=H_g/\\Gamma_g[4,8]$ into the projective space $P^{N-1}$. Moreover, this map is biholomorphic onto the image outside the Satake boundary. It is not biholomorphic on the whole in the cases $g\\ge 6$. Igusa also proved that in the cases $g\\le 2$ this map is biholomorphic onto the image. In this paper we extend this result to the case $g=3$. 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