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We are interested in the splitting type $(a,b)$ of $C$, where $\\mathcal{O}_{\\mathbb{P}^1}(-a-d)\\oplus \\mathcal{O}_{\\mathbb{P}^1}(-b-d)$ gives the syzigies of the ideal $(f_0,f_1,f_2)\\subset K[s,t]$, and $(f_0,f_1,f_2)$ is a parameterization of $C$. We want to describe in which cases $(a,b)=(k,d-k)$ ($2k\\leq d)$, via a geometric description; namely we show that $(a,b)=(k,d-k)$ if and only if $C$ is the projection of a rational curve on a rational normal surface in $\\mathbb{P}^{k+1}$."},"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":"1507.02227","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-07-08T17:16:40Z","cross_cats_sorted":[],"title_canon_sha256":"96814528a3f552a47aa2090978149245103946482710fa88a59ce9ecb007b990","abstract_canon_sha256":"3762da9c7b2152ba34d4118fb645c3fce35c15151c03d75bdc704926587f2cb7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:37:08.340325Z","signature_b64":"IVV6bl+qOuCDPUtZE5OsnuguMFuCNOU5oTBRmwMXMyy2RtNjX11fstdTtxVznoL+PhcmuNUdhKAjXgGhbdSMCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a1c6b152e4b4c9a02508404f9815963125a4c0e826f2a99eae45a121e795be76","last_reissued_at":"2026-05-18T01:37:08.339658Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:37:08.339658Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Parameterizations of plane rational curves and their syzygies","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Alessandra Bernardi, Alessandro Gimigliano, Monica Id\\`a","submitted_at":"2015-07-08T17:16:40Z","abstract_excerpt":"Let $C$ be a plane rational curve of degree $d$ and $p:\\tilde C \\rightarrow C$ its normalization. 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