Abstract
Let \({\mathbb{X}\subset\mathbb{P}(V)}\) be a projective variety, which is not contained in a hyperplane. Then every vector v in V may be written as a sum of vectors from the affine cone over \({\mathbb{X}}\). The minimal number of summands in such a sum is called the rank of v. In this paper, we classify all equivariantly embedded homogeneous projective varieties \({\mathbb{X}\subset\mathbb{P}(V)}\) whose rank function is lower semi-continuous. Classical examples are: the variety of rank one matrices (Segre variety with two factors) and the variety of rank one quadratic forms (quadratic Veronese variety). In the general setting, \({\mathbb{X}}\) is the orbit in \({\mathbb{P}(V)}\) of a highest weight line in an irreducible representation V of a reductive algebraic group G. Thus, our result is a list of all irreducible representations of reductive groups, for which the corresponding rank function is lower semi-continuous.
References
Alexander J., Hirschowitz A.: Polynomial interpolation in several variables. J. Alg. Geom. 4(2), 201–222 (1995)
Bala P., Carter R.W.: Classes of unipotent elements in simple algebraic groups I. Math. Proc. Camb. Philos. Soc. 79(3), 401–425 (1976)
Baur K., Draisma J.: Higher secant varieties of the minimal adjoint orbit. J. Algebra 280, 743–761 (2004)
Buczyński J., Landsberg J.M.: Rank of tensors and a generalization of secant varieties. Linear Algebra Appl. 438(2), 668–689 (2013)
Carlini E., Enrico M., Catalisano V., Geramita A.V.: The solution to the Waring problem for monomials and the sum of coprime monomials. J. Algebra 370, 5–14 (2012)
Catalisano M.V., Geramita A.V., Gimigliano A.: Rank of tensors, secant varieties of Segre varieties and fat points. Linear Algebrs Appl. 355, 263–285 (2002)
Catalisano M.V., Geramita A.V., Gimigliano A.: Secant varieties of Grassmann varieties. Proc. AMS 133(3), 633–642 (2005)
Charlton P.: The geometry of pure spinors, with applications. Dissertation, University of Newcastle (1997)
Collingwood D.H., McGovern W.M.: Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold mathematics series. Van Nostrand Reinhold Co., New York (1993)
Comas G., Seiguer M.: On the rank of a binary form. Found. Comput. Math. 11, 65–78 (2011)
Dynkin E.B.: Semisimple subalgebras of semisimple Lie algebras. Math. Sbornik N.S. 30(72), 349–462 (1952) (Russian)
Fulton W., Hansen J.: A connectedness theorem for projective varieties, with applications to intersections and singularities of mappings. Ann. Math. (2) 110(1), 159–166 (1979)
Fulton, W., Lazarsfeld, R.: Connectivity and its applications in algebraic geometry, in algebraic geometry. Lecture Notes in Math. vol. 62, pp. 26–92. Springer, Berlin (1981)
Goodman, R., Wallach, N.R.: Symmetry, representations, and invariants, graduate texts in mathematics. Vol. 255, Springer, (2009)
de Graaf W.A., Vinberg É.B., Yakimova O.S.: An effective method to compute closure ordering for nilpotent orbits of \({\theta}\) -representations. J. Algebra 371(1), 38–62 (2012)
Gurevich, G.B.: Osnovy teorii algebraiceskih invariantov [Foundations of the theory of algebraic invariants]. OGIZ, Moscow-Leningrad (1948) (Russian)
Haris S.J.: Some irreducible representations of exceptional algebraic groups. Am. J. Math. 93, 75–106 (1971)
Kac V.: Some remarks on nilpotent obits. J. Algebra 64, 190–213 (1980)
Kaji H.: Secant varieties of adjoint varieties. Algebra meeting (Rio de Janeiro, 1996). Math. Contemp. 14, 75–87 (1998)
Kaji H., Yasukura O.: Secant varieties of adjoint varieties: orbit decomposition. J. Algebra 227, 26–44 (2000)
Knop, F.: Some remarks on multiplicity free spaces, Representation theories and algebraic geometry (Montreal, PQ, 1997), 301317, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 514, Kluwer, Dordrecht (1998)
Landsberg J.M.: On degenerate secant and tangential varieties and local differential geometry. Duke Math. J. 85(3), 605–634 (1996)
Landsberg, J.M.: Tensors: geometry and applications Grad. Stud. Math., vol. 128, AMS (2012)
Landsberg J.M., Manivel L.: On the projective geometry of rational homogeneous varieties. Comment. Math. Helv. 78, 65–100 (2003)
Landsberg J.M., Manivel L.: Series of Lie groups. Mich. Math. J. 52, 453–479 (2004)
Landsberg J.M., Teitler Z.: On the ranks and border ranks of symmetric tensors. Found. Comput. Math. 10, 339–366 (2010)
Onishchik, A.L., Vinberg, É.B.: Lie groups and algebraic groups, Translated from the Russian and with a preface by DA Leites. Springer Series in Soviet Mathematics. Springer, Berlin (1990)
Reichel, W.: Über triliniare alternierende formen in 6 und 7 Veränderlichen Diss., Greifwald (1907)
Roberts J.: Generic projections of algebraic varieties. Am. J. Math. 93, 191–214 (1971)
Vinberg É.B.: The Weyl group of a graded Lie algebra. Math. USSR Izv. 10(3), 463–495 (1976)
Watanabe K.: Classification of embedded projective manifolds swept out by rational homogeneous varieties of codimension one. Pac. J. Math. 252(2), 493–497 (2011)
Zak, F.L.: Tangents and secants of algebraic varieties, translations of mathematical monographs, Vol. 127, AMS (1993)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Petukhov, A.V., Tsanov, V.V. Homogeneous projective varieties with semi-continuous rank function. manuscripta math. 147, 269–303 (2015). https://doi.org/10.1007/s00229-014-0723-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-014-0723-5