{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T14:38:11Z","timestamp":1787236691311,"version":"build-2736575974"},"reference-count":38,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS--0757212"],"award-info":[{"award-number":["DMS--0757212"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006049","name":"Comisi\u00f3n Sectorial de Investigaci\u00f3n Cient\u00edfica","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100006049","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Algebra Geometry"],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>For a real projective variety $X$, the cone $\\Sigma_X$ of sums of squares of linear forms plays a fundamental role in real algebraic geometry. The dual cone $\\Sigma_X^*$ is a spectrahedron, and we show that its convexity properties are closely related to homological properties of $X$. For instance, we show that all extreme rays of $\\Sigma_X^*$ have rank 1 if and only if $X$ has Castelnuovo--Mumford regularity two. More generally, if $\\Sigma_X^*$ has an extreme ray of rank $p&gt;1$, then $X$ does not satisfy the property $N_{2,p}$. We show that the converse also holds in a wide variety of situations: the smallest $p$ for which property $N_{2,p}$ does not hold is equal to the smallest rank of an extreme ray of $\\Sigma_X^*$ greater than one. We generalize the work of Blekherman, Smith, and Velasco on equality of nonnegative polynomials and sums of squares from irreducible varieties to reduced schemes and classify all spectrahedral cones with only rank 1 extreme rays. Our results have applications to the positive semidefinite matrix completion problem and to the truncated moment problem on projective varieties.<\/jats:p>","DOI":"10.1137\/16m1084560","type":"journal-article","created":{"date-parts":[[2017,2,1]],"date-time":"2017-02-01T07:58:01Z","timestamp":1485935881000},"page":"175-199","source":"Crossref","is-referenced-by-count":14,"title":["Do Sums of Squares Dream of Free Resolutions?"],"prefix":"10.1137","volume":"1","author":[{"given":"Grigoriy","family":"Blekherman","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rainer","family":"Sinn","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mauricio","family":"Velasco","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,2,1]]},"reference":[{"key":"atypb1","first-page":"107","author":"Agler J.","year":"1987","journal-title":"BC"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-2012-00733-4"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-2014-12253-2"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2015.07.015"},{"key":"atypb5","doi-asserted-by":"crossref","unstructured":"G. Blekherman, P. A. Parrilo, and R. R. Thomas, eds.\n                      Semidefinite Optimization and Convex Algebraic Geometry\n                      , MOS-SIAM Ser. Optim. 13, SIAM, Philadelphia; Mathematical Optimization Society, Philadelphia, 2013.","DOI":"10.1137\/1.9781611972290"},{"key":"atypb6","doi-asserted-by":"crossref","unstructured":"G. Blekherman, D. Plaumann, R. Sinn, and C. Vinzant,\n                      Low-Rank Sum-of-Squares Representations on Varieties of Minimal Degree\n                      , preprint,https:\/\/arxiv.org\/abs\/1606.04387, 2016.","DOI":"10.1093\/imrn\/rnx113"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1090\/jams\/847"},{"key":"atypb8","doi-asserted-by":"crossref","unstructured":"J. Bochnak, M. Coste, and M.F. Roy,\n                      Real Algebraic Geometry\n                      , Ergeb. Math. Grenzgeb. (3) [Results in Mathematics and Related Areas (3)] 36, translated from the 1987 French original, revised by the authors, Springer-Verlag, Berlin, 1998.","DOI":"10.1007\/978-3-662-03718-8"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1007\/s00208-010-0616-1"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.2307\/2528966"},{"key":"atypb11","doi-asserted-by":"crossref","unstructured":"D. Eisenbud,\n                      The Geometry of Syzygies. A Second Course in Commutative Algebra and Algebraic Geometry\n                      , Grad. Texts in Math. 229, Springer-Verlag, New York, 2005.","DOI":"10.1017\/CBO9780511756382.005"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1112\/S0010437X05001776"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1353\/ajm.2006.0043"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1007\/BF01394316"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1353\/ajm.2006.0022"},{"key":"atypb16","doi-asserted-by":"crossref","unstructured":"R. Fro\u0308berg,\n                      On Stanley-Reisner rings\n                      , in Topics in Algebra, Part 2 (Warsaw, 1988), Banach Center Publ. 26, PWN, Warsaw, 1990, pp. 57-70.","DOI":"10.4064\/-26-2-57-70"},{"key":"atypb17","first-page":"25","volume":"538","author":"Gallego F. J.","year":"2001","journal-title":"J. Reine Angew. Math."},{"key":"atypb18","first-page":"301","volume":"67","author":"Green M.","year":"1988","journal-title":"Compositio Math."},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(84)90207-6"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-2012-05755-2"},{"key":"atypb21","doi-asserted-by":"crossref","unstructured":"R. Hartshorne,\n                      Algebraic Geometry\n                      , Grad. Texts in Math. 52, Springer-Verlag, New York, Heidelberg, 1977.","DOI":"10.1007\/978-1-4757-3849-0"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1137\/0610028"},{"key":"atypb23","unstructured":"M. Hering,\n                      Syzygies of Toric Varieties\n                      , thesis,http:\/\/www.maths.ed.ac.uk\/ mhering\/, 2006."},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1007\/s10898-015-0313-4"},{"key":"atypb25","unstructured":"J. B. Lasserre,\n                      Moments, Positive Polynomials and Their Applications\n                      , Imp. Coll. Press Optim. Ser. 1, Imperial College Press, London, 2010."},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(95)00741-5"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1007\/s004930100012"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-013-0648-x"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.4153\/CMB-2003-041-9"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2009.03.022"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1016\/0022-1236(89)90050-5"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1007\/BF01100204"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-04-07523-9"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2016.07.026"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-02-11121-1"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1214\/11-AOS957"},{"key":"atypb37","doi-asserted-by":"crossref","unstructured":"C. Voisin,\n                      Green's generic syzygy conjecture for curves of even genus lying on a K3 surface\n                      , J. Eur. Math. Soc. (JEMS), 4 (2002), pp. 363-404.","DOI":"10.1007\/s100970200042"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1112\/S0010437X05001387"}],"container-title":["SIAM Journal on Applied Algebra and Geometry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/16M1084560","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T14:00:57Z","timestamp":1787234457000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/16M1084560"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1]]},"references-count":38,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2017,1]]}},"alternative-id":["10.1137\/16M1084560"],"URL":"https:\/\/doi.org\/10.1137\/16m1084560","relation":{},"ISSN":["2470-6566"],"issn-type":[{"value":"2470-6566","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,1]]}}}