{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,8,10]],"date-time":"2024-08-10T15:40:13Z","timestamp":1723304413218},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"5-6","license":[{"start":{"date-parts":[[2020,7,22]],"date-time":"2020-07-22T00:00:00Z","timestamp":1595376000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,7,22]],"date-time":"2020-07-22T00:00:00Z","timestamp":1595376000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["AAECC"],"published-print":{"date-parts":[[2020,11]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We present two algorithms determining all the complete and simplicial fans admitting a fixed non-degenerate set of vectors<jats:italic>V<\/jats:italic>as generators of their 1-skeleton. The interplay of the two algorithms allows us to discerning if the associated toric varieties admit a projective embedding, in principle for any values of dimension and Picard number. The first algorithm is slower than the second one, but it computes all complete and simplicial fans supported by<jats:italic>V<\/jats:italic>and lead us to formulate a topological-combinatoric conjecture about the definition of a fan. On the other hand, we adapt the Sturmfels\u2019 arguments on the Gr\u00f6bner fan of toric ideals to our complete case; we give a characterization of the Gr\u00f6bner region and show an explicit correspondence between Gr\u00f6bner cones and chambers of the secondary fan. A homogenization procedure of the toric ideal associated to<jats:italic>V<\/jats:italic>allows us to employing GFAN and related software in producing our second algorithm. The latter turns out to be much faster than the former, although it can compute only the projective fans supported by<jats:italic>V<\/jats:italic>. We provide examples and a list of open problems. In particular we give examples of rationally parametrized families of<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathbb {Q}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>Q<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-factorial complete toric varieties behaving in opposite way with respect to the dimensional jump of the nef cone over a special fibre.<\/jats:p>","DOI":"10.1007\/s00200-020-00452-w","type":"journal-article","created":{"date-parts":[[2020,7,22]],"date-time":"2020-07-22T08:09:31Z","timestamp":1595405371000},"page":"461-482","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Toric varieties and Gr\u00f6bner bases: the complete $$\\mathbb {Q}$$-factorial case"],"prefix":"10.1007","volume":"31","author":[{"given":"Michele","family":"Rossi","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lea","family":"Terracini","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,7,22]]},"reference":[{"issue":"3","key":"452_CR1","doi-asserted-by":"publisher","first-page":"529","DOI":"10.1016\/S0196-8858(02)00509-2","volume":"30","author":"E Babson","year":"2003","unstructured":"Babson, E., Onn, S., Thomas, R.: The Hilbert zonotope and a polynomial time algorithm for universal Gr\u00f6bner bases. Adv. Appl. Math. 30(3), 529\u2013544 (2003)","journal-title":"Adv. Appl. Math."},{"key":"452_CR2","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1155\/S1073792804130973","volume":"6","author":"F Berchtold","year":"2004","unstructured":"Berchtold, F., Hausen, J.: Bunches of cones in the divisor class group-a new combinatorial language for toric varieties. Int. Math. Res. Not. 6, 261\u2013302 (2004)","journal-title":"Int. Math. Res. Not."},{"issue":"04","key":"452_CR3","doi-asserted-by":"crossref","first-page":"103","DOI":"10.4171\/dm\/4","volume":"1","author":"JA de Loera","year":"1996","unstructured":"de Loera, J.A., Hosten, S., Santos, F., Sturmfels, B.: The polytope of all triangulations of a point configuration. Doc. Math. 1(04), 103\u2013119 (1996)","journal-title":"Doc. Math."},{"key":"452_CR4","doi-asserted-by":"crossref","unstructured":"De\u00a0Loera, J.A., Rambau, J., Santos, F.: Triangulations, vol.\u00a025 of Algorithms and Computation in Mathematics. Springer, Berlin, (2010). Structures for algorithms and applications","DOI":"10.1007\/978-3-642-12971-1"},{"key":"452_CR5","doi-asserted-by":"publisher","first-page":"507","DOI":"10.24033\/asens.1201","volume":"3","author":"M Demazure","year":"1970","unstructured":"Demazure, M.: Sous-groupes alg\u00e9briques de rang maximum du groupe de Cremona. Ann. Sci. \u00c9cole Norm. Sup. 3, 507\u2013588 (1970)","journal-title":"Ann. Sci. \u00c9cole Norm. Sup."},{"key":"452_CR6","unstructured":"Franz, M.: Convex: a Maple package for convex geometry (version 1.2) (2016). Available at http:\/\/www.math.uwo.ca\/faculty\/franz\/convex\/"},{"key":"452_CR7","doi-asserted-by":"publisher","first-page":"174","DOI":"10.3792\/pjaa.81.174","volume":"81","author":"O Fujino","year":"2005","unstructured":"Fujino, O., Payne, S.: Smooth complete toric varieties with no nontrivial nef line bundles. Proc. Jpn. Acad. 81, 174\u2013179 (2005)","journal-title":"Proc. Jpn. Acad."},{"issue":"1","key":"452_CR8","doi-asserted-by":"publisher","first-page":"647","DOI":"10.1112\/S1461157015000212","volume":"18","author":"J Hausen","year":"2015","unstructured":"Hausen, J., Keicher, S.: A software package for Mori dream spaces. LMS J. Comput. Math. 18(1), 647\u2013659 (2015)","journal-title":"LMS J. Comput. Math."},{"key":"452_CR9","doi-asserted-by":"crossref","unstructured":"Herzog, J., Hibi, T.: Monomial Ideals. Springer, New York (2010). Graduate Texts in Mathematic, No. 260","DOI":"10.1007\/978-0-85729-106-6"},{"key":"452_CR10","unstructured":"Huber, B.: TiGERS. Available at https:\/\/sites.math.washington.edu\/~thomas\/programs\/tigers.html"},{"issue":"3","key":"452_CR11","doi-asserted-by":"publisher","first-page":"321","DOI":"10.1080\/10586458.2000.10504409","volume":"9","author":"B Huber","year":"2000","unstructured":"Huber, B., Thomas, R.R.: Computing Gr\u00f6bner fans of toric ideals. Exp. Math. 9(3), 321\u2013331 (2000)","journal-title":"Exp. Math."},{"issue":"6","key":"452_CR12","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1142\/S0218195902000980","volume":"12","author":"H Imai","year":"2002","unstructured":"Imai, H., Masada, T., Takeuchi, F., Imai, K.: Enumerating triangulations in general dimensions. Int. J. Comput. Geom. Appl. 12(6), 455\u2013480 (2002)","journal-title":"Int. J. Comput. Geom. Appl."},{"key":"452_CR13","unstructured":"Jensen, A.N.: Gfan, a software system for Gr\u00f6bner fans and tropical varieties. Available at http:\/\/home.imf.au.dk\/jensen\/software\/gfan\/gfan.html"},{"key":"452_CR14","unstructured":"Jensen, A.N.: CaTS, a software package for computing state polytopes of toric ideals. Available at http:\/\/www.soopadoopa.dk\/anders\/cats\/cats.html"},{"issue":"2","key":"452_CR15","doi-asserted-by":"publisher","first-page":"289","DOI":"10.1016\/0040-9383(91)90015-V","volume":"30","author":"P Kleinschmidt","year":"1991","unstructured":"Kleinschmidt, P., Sturmfels, B.: Smooth toric varieties with small Picard number are projective. Topology 30(2), 289\u2013299 (1991)","journal-title":"Topology"},{"key":"452_CR16","unstructured":"Miller, E.: Alexander duality for monomial ideals and their resolutions. arXiv:math\/9812095 (1998)"},{"issue":"1","key":"452_CR17","doi-asserted-by":"publisher","first-page":"180","DOI":"10.1006\/jabr.2000.8359","volume":"231","author":"E Miller","year":"2000","unstructured":"Miller, E.: The Alexander duality functors and local duality with monomial support. J. Algebra 231(1), 180\u2013234 (2000)","journal-title":"J. Algebra"},{"issue":"2","key":"452_CR18","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1016\/S0747-7171(88)80042-7","volume":"6","author":"T Mora","year":"1988","unstructured":"Mora, T., Robbiano, L.: The Gr\u00f6bner fan of an ideal. J. Symb. Comput. 6(2), 183\u2013208 (1988)","journal-title":"J. Symb. Comput."},{"key":"452_CR19","unstructured":"Oda, T.: Torus Embeddings and Applications, vol.\u00a057 of Tata Institute of Fund. Research. Springer, Berlin (1978). (Based on a joint work with K.\u00a0Miyake)"},{"key":"452_CR20","unstructured":"Oda, T.: Convex Bodies and Algebraic Geometry, vol.\u00a015 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer, Berlin (1988). An introduction to the theory of toric varieties, Translated from the Japanese"},{"key":"452_CR21","doi-asserted-by":"publisher","first-page":"135","DOI":"10.1016\/j.laa.2004.01.003","volume":"384","author":"P Pis\u00f3n-Casares","year":"2004","unstructured":"Pis\u00f3n-Casares, P., Vigneron-Tenorio, A.: N-solutions to linear systems over z. Linear Algebra Appl. 384, 135\u2013154 (2004)","journal-title":"Linear Algebra Appl."},{"key":"452_CR22","doi-asserted-by":"publisher","first-page":"256","DOI":"10.1016\/j.laa.2016.01.039","volume":"495","author":"M Rossi","year":"2016","unstructured":"Rossi, M., Terracini, L.: $$\\mathbb{Z}$$-linear gale duality and poly weighted spaces (PWS). Linear Algebra Appl. 495, 256\u2013288 (2016)","journal-title":"Linear Algebra Appl."},{"key":"452_CR23","doi-asserted-by":"publisher","first-page":"325","DOI":"10.1007\/s10231-016-0574-7","volume":"196","author":"M Rossi","year":"2017","unstructured":"Rossi, M., Terracini, L.: A $${\\mathbb{Q}}$$-factorial complete toric variety is a quotient of a poly weighted space. Ann. Mat. Pur. Appl. 196, 325\u2013347 (2017)","journal-title":"Ann. Mat. Pur. Appl."},{"issue":"9","key":"452_CR24","doi-asserted-by":"publisher","first-page":"2648","DOI":"10.1016\/j.jpaa.2017.10.012","volume":"222","author":"M Rossi","year":"2018","unstructured":"Rossi, M., Terracini, L.: A $$\\mathbb{Q}$$-factorial complete toric variety with Picard number 2 is projective. J. Pure Appl. Algebra 222(9), 2648\u20132656 (2018)","journal-title":"J. Pure Appl. Algebra"},{"key":"452_CR25","volume-title":"Theory of Linear and Integer Programming","author":"A Schrijver","year":"1986","unstructured":"Schrijver, A.: Theory of Linear and Integer Programming. Wiley-Interscience Series in Discrete Mathematics. Wiley, Chichester (1986)"},{"key":"452_CR26","doi-asserted-by":"crossref","unstructured":"Sturmfels, B.: Gr\u00f6bner Bases and Convex Polytopes, vol.\u00a08 of University Lecture Series. American Mathematical Society, Providence, RI (1996)","DOI":"10.1090\/ulect\/008"},{"key":"452_CR27","first-page":"357","volume":"77","author":"B Sturmfels","year":"1997","unstructured":"Sturmfels, B., Thomas, R.: Variation of cost functions in integer programming. Math. Program. 77, 357\u2013387 (1997)","journal-title":"Math. Program."},{"key":"452_CR28","unstructured":"Ziegler, G.: Lectures on Polytopes, vol.\u00a0152 of Graduate Texts in Mathematics"}],"container-title":["Applicable Algebra in Engineering, Communication and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00200-020-00452-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00200-020-00452-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00200-020-00452-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,10]],"date-time":"2024-08-10T14:36:25Z","timestamp":1723300585000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00200-020-00452-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,22]]},"references-count":28,"journal-issue":{"issue":"5-6","published-print":{"date-parts":[[2020,11]]}},"alternative-id":["452"],"URL":"https:\/\/doi.org\/10.1007\/s00200-020-00452-w","relation":{},"ISSN":["0938-1279","1432-0622"],"issn-type":[{"type":"print","value":"0938-1279"},{"type":"electronic","value":"1432-0622"}],"subject":[],"published":{"date-parts":[[2020,7,22]]},"assertion":[{"value":"20 September 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 March 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 April 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 July 2020","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}