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We give a upper bound of instability of truncated symmetric powers $\\mathrm{T}^l(\\E)(0\\leq l\\leq\\rk(\\E)(p-1))$ in terms of $L_{\\max}(\\Omg^1_X)$, $\\mathrm{I}(\\Omg^1_X)$ and $\\mathrm{I}(\\E)$ (Theorem \\ref{InstabTl}). As an application, We obtain a upper bound of Frobenius direct image ${F_X}_*(\\E)$ and some sufficient conditions of slope semi-stability of ${F_X}_*(\\E)$. In additio"},"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":"1010.4228","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-10-20T15:22:05Z","cross_cats_sorted":[],"title_canon_sha256":"c6982df7698b77b95df85aedb260933a4d56ddb197f1a4cd0e2e7e9bd4edbd34","abstract_canon_sha256":"013d7ce29d8aacef05cafb12e7df61cab0c479918880378a52e37a28e50b270d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:05:27.738984Z","signature_b64":"AwywdVW1HdZXlnNYsYUGpADIY8CD6qf1AE/6NkF0F6FRMERVaPb8zhhU47N9RyYoIeVIlMEJ2k+N3H1/wzksCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"292700019585f004f6b0b52239fceb180ac3b0f9c6afc0c48f962acf19c9fca3","last_reissued_at":"2026-05-18T04:05:27.738393Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:05:27.738393Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Instability of Truncated Symmetric Powers of sheaves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Fei Yu, Lingguang Li","submitted_at":"2010-10-20T15:22:05Z","abstract_excerpt":"Let $X$ be a smooth projective variety of dimension $n$ over an algebraically closed field $k$ of characteristic $p>0$. Let $F_X:X\\rightarrow X$ be the absolute Frobenius morphism, and $\\E$ a torsion free sheaf on $X$. We give a upper bound of instability of truncated symmetric powers $\\mathrm{T}^l(\\E)(0\\leq l\\leq\\rk(\\E)(p-1))$ in terms of $L_{\\max}(\\Omg^1_X)$, $\\mathrm{I}(\\Omg^1_X)$ and $\\mathrm{I}(\\E)$ (Theorem \\ref{InstabTl}). As an application, We obtain a upper bound of Frobenius direct image ${F_X}_*(\\E)$ and some sufficient conditions of slope semi-stability of ${F_X}_*(\\E)$. 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