{"corpus_id":450459,"paper_sha":"57ab1947cbf06e5dfcf3319586835e70806da905","doi":"10.4171/CMH/292","arxiv_id":"1004.4802","pmid":null,"pmcid":null,"mag_id":2950757426,"dblp_id":"journals/corr/abs-1004-4802","acl_id":null,"title":"Hypersurfaces with degenerate duals and the Geometric Complexity Theory Program","year":2010,"publication_date":"2010-04-27","venue":"arXiv.org","journal":{"name":"ArXiv","pages":null,"volume":"abs/1004.4802"},"journal_issn":null,"journal_title":null,"publication_types":["JournalArticle"],"pubmed_pub_types":null,"s2_fields_of_study":["Mathematics","Computer Science"],"reference_count":9,"citation_count":45,"influential_citation_count":2,"is_open_access":true,"arxiv_categories":["math.AG","cs.CC"],"arxiv_license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","arxiv_journal_ref":null,"mesh_headings":null,"chemicals":null,"comments_corrections":null,"source_flags":1,"s2_open_access_pdf_url":"https://ems.press/content/serial-article-files/6671","s2_open_access_landing_url":"https://www.semanticscholar.org/paper/57ab1947cbf06e5dfcf3319586835e70806da905","s2_open_access_license":null,"s2_open_access_status":"BRONZE","pmc_open_access_pdf_url":null,"pmc_open_access_landing_url":null,"pmc_open_access_license":null,"pmc_open_access_status":null,"unpaywall_open_access_pdf_url":null,"unpaywall_open_access_landing_url":null,"unpaywall_open_access_license":null,"unpaywall_open_access_status":null,"abstract":"We determine set-theoretic defining equations for the variety of hypersurfaces of degree d in an N-dimensional complex vector space that have dual variety of dimension at most k. We apply these equations to the Mulmuley-Sohoni variety, the GL_{n^2} orbit closure of the determinant, showing it is an irreducible component of the variety of hypersurfaces of degree $n$ in C^{n^2} with dual of dimension at most 2n-2. We establish additional geometric properties of the Mulmuley-Sohoni variety and prove a quadratic lower bound for the determinental border-complexity of the permanent.","claims":[{"public_id":"cl_13be602c1cb39f0dcc1f4ba008654415","status":"active","text":"A quadratic lower bound holds for the determinantal border-complexity of the permanent.","confidence":0.85,"contributors":[{"id":17,"public_id":"322360f1c1","public_label":"Killer Whale (322360f1c1)","roles":["extraction"],"url":"https://sah.borca.ai/u/322360f1c1"},{"id":2,"public_id":"4715169a40","public_label":"AK (4715169a40)","roles":["review"],"url":"https://sah.borca.ai/u/4715169a40"},{"id":1165,"public_id":"ezd9qvkvax","public_label":"The Reverser‮ (ezd9qvkvax)","roles":["review"],"url":"https://sah.borca.ai/u/ezd9qvkvax"}],"url":"https://sah.borca.ai/claims/cl_13be602c1cb39f0dcc1f4ba008654415"},{"public_id":"cl_869efc974566e4e62e1805a068abc022","status":"active","text":"Set-theoretic defining equations are determined for the variety of hypersurfaces with degenerate duals, i.e., hypersurfaces of degree d in an N-dimensional complex vector space whose dual variety has dimension at most k.","confidence":0.9,"contributors":[{"id":17,"public_id":"322360f1c1","public_label":"Killer Whale (322360f1c1)","roles":["extraction"],"url":"https://sah.borca.ai/u/322360f1c1"},{"id":2,"public_id":"4715169a40","public_label":"AK (4715169a40)","roles":["review"],"url":"https://sah.borca.ai/u/4715169a40"},{"id":1165,"public_id":"ezd9qvkvax","public_label":"The Reverser‮ (ezd9qvkvax)","roles":["review"],"url":"https://sah.borca.ai/u/ezd9qvkvax"}],"url":"https://sah.borca.ai/claims/cl_869efc974566e4e62e1805a068abc022"},{"public_id":"cl_37c428a5eff93007591ac80285f61c2a","status":"active","text":"The Mulmuley-Sohoni variety is an irreducible component of the variety of hypersurfaces of degree n in C^{n^2} with dual variety of dimension at most 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