{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:32:19Z","timestamp":1760059939197,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,7,20]],"date-time":"2025-07-20T00:00:00Z","timestamp":1752969600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Let X be a complex projective variety embedded in a complex projective space. The dimensions of the secant varieties of X have an expected value, and it is important to know if they are equal or at least near to this expected value. Blomenhofer and Casarotti proved important results on the embeddings of G-varieties, G being an algebraic group, embedded in the projectivations of an irreducible G-representation, proving that no proper secant variety is a cone. In this paper, we give other conditions which assure that no proper secant varieties of X are a cone, e.g., that X is G-homogeneous. We consider the Segre product of two varieties with the product action and the case of toric varieties. We present conceptual tests for it, and discuss the information we obtained from certain linear projections of X. For the Segre\u2013Veronese embeddings of Pn\u00d7Pn with respect to forms of bidegree (1,d), our results are related to the simultaneous rank of degree d forms in n+1 variables.<\/jats:p>","DOI":"10.3390\/axioms14070542","type":"journal-article","created":{"date-parts":[[2025,7,21]],"date-time":"2025-07-21T09:33:53Z","timestamp":1753090433000},"page":"542","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Tame Secant Varieties and Group Actions"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1432-7413","authenticated-orcid":false,"given":"Edoardo","family":"Ballico","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Trento, 38123 Trento, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"213","DOI":"10.7146\/math.scand.a-12200","article-title":"Joins and higher secant varieties","volume":"61","year":"1987","journal-title":"Math. Scand."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Bernardi, A., Carlini, E., Catalisano, M.V., Gimigliano, A., and Oneto, A. (2018). The Hitchhiker guide to: Secant varieties and tensor decomposition. Mathematics, 6.","DOI":"10.3390\/math6120314"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Landsberg, J.M. (2012). Tensors: Geometry and Applications, American Mathematical Society. Graduate Studies in Mathematics.","DOI":"10.1090\/gsm\/128"},{"key":"ref_4","unstructured":"Blomenhofer, A.T., and Casarotti, A. (2023). Nondefectivity of invariant secant varieties. arXiv."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"589","DOI":"10.1090\/bproc\/248","article-title":"Non-defectivity of Segre-Veronese varieties","volume":"11","author":"Abo","year":"2024","journal-title":"Proc. Am. Math. Soc. Ser. B"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1007\/s00209-024-03573-x","article-title":"On the non-defectivity of Segre-Veronese embeddings","volume":"308","author":"Ballico","year":"2024","journal-title":"Math. Z."},{"key":"ref_7","first-page":"16","article-title":"Varieties with an extremal number of degenerate higher secant varieties","volume":"392","year":"1988","journal-title":"J. Reine Angew. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1353\/ajm.1996.0012","article-title":"The possible dimension of the higher secant varieties","volume":"118","year":"1996","journal-title":"Am. J. Math."},{"key":"ref_9","first-page":"3","article-title":"On higher secany varieties of rational normal scrolls","volume":"51","year":"1996","journal-title":"Le Mat."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Ballico, E. (2024). Secant Varieties and Their Associated Grassmannians. Mathematics, 12.","DOI":"10.3390\/math12091274"},{"key":"ref_11","unstructured":"Zak, F.L. (1993). Tangents and Secants of Algebraic Varieties, American Mathematical Society. Translations of Mathematical Monographs."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1007\/BF02559525","article-title":"Zak\u2019s theorem on superadditivity","volume":"32","author":"Holme","year":"1994","journal-title":"Ark. Mat."},{"key":"ref_13","first-page":"151","article-title":"Weakly defective varieties","volume":"454","author":"Chiantini","year":"2002","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Hartshorne, R. (1977). Algebraic Geometry, Springer.","DOI":"10.1007\/978-1-4757-3849-0"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1021","DOI":"10.1007\/s00208-014-1150-3","article-title":"On maximum, typical and generic ranks","volume":"362","author":"Blekherman","year":"2015","journal-title":"Math. Ann."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1090\/S1056-3911-10-00537-0","article-title":"Secant varieties of \u21191\u00d7\u22ef \u21191 (n-times) are not defective for n\u22655","volume":"20","author":"Catalisano","year":"2011","journal-title":"J. Algebr. Geom."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1455","DOI":"10.1007\/s00208-012-0890-1","article-title":"Secant varieties of Segre-Veronese embeddings of (\u21191)r","volume":"356","author":"Laface","year":"2013","journal-title":"Math. Ann."},{"key":"ref_18","first-page":"1851","article-title":"Ranks of tensors and a generalization of secant varieties","volume":"438","author":"Landsberg","year":"2013","journal-title":"Linear Algebra Its Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1453","DOI":"10.1137\/18M1225422","article-title":"Partially symmetric variants of Comon\u2019s problem via simultaneous rank","volume":"40","author":"Gesmundo","year":"2019","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Cox, D., Little, J., and Schenck, H. (2011). Toric Varieties, American Mathematical Society. Graduate Studies in Mathematics.","DOI":"10.1090\/gsm\/124"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"651","DOI":"10.1016\/j.jpaa.2006.07.008","article-title":"Secant variety of toric varieties","volume":"209","author":"Cox","year":"2007","journal-title":"J. Pure Appl. Algebra"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"428","DOI":"10.1016\/j.laa.2019.12.008","article-title":"Secant varieties of toric varieties arising from simplicial complexes","volume":"588","author":"Khadam","year":"2020","journal-title":"Linear Algebra Its Appl."},{"key":"ref_23","first-page":"411","article-title":"Un lemme d\u2019Horace diff\u00e9rentiel: Application aux singularit\u00e9 hyperquartiques de P5","volume":"1","author":"Alexander","year":"1992","journal-title":"J. Algebr. Geom."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/7\/542\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:12:50Z","timestamp":1760033570000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/7\/542"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,20]]},"references-count":23,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2025,7]]}},"alternative-id":["axioms14070542"],"URL":"https:\/\/doi.org\/10.3390\/axioms14070542","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,7,20]]}}}