Skip to main content
SpringerLink
Log in

Generalized varieties of sums of powers

  • Published:
Bulletin of the Brazilian Mathematical Society, New Series Aims and scope Submit manuscript

Abstract

Let X ⊂ PN be an irreducible, non-degenerate variety. The generalized variety of sums of powers V S P H X(h) of X is the closure in the Hilbert scheme Hilbh (X) of the locus parametrizing collections of points {x 1,..., x h} such that the (h -1)-plane >x 1,..., x h> passes through a fixed general point p ∈ PN. When X = V d n is a Veronese variety we recover the classical variety of sums of powers V S P(F, h) parametrizing additive decompositions of a homogeneous polynomial as powers of linear forms. In this paper we study the birational behavior of V S P H X(h). In particular, we show how some birational properties, such as rationality, unirationalityand rational connectedness, of V S P H X(h) are inherited from the birational geometry of variety X itself.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Bolognesi and A. Massarenti. Moduli of abelian surfaces, symmetric theta structures and theta characteristics, arXiv:1409.5033v1.

  2. F. Campana. Connexité rationnelle des variétés de Fano. Ann. Sci. École Norm. Sup., 25 (1992), 539–545.

    Article  MathSciNet  MATH  Google Scholar 

  3. L. Chiantini and C. Ciliberto. Weakly defective varieties. Trans. Amer.Math. Soc., 354(1) (2002), 151–178.

    Article  MathSciNet  MATH  Google Scholar 

  4. H. Clemens and P. Griffiths. The intermediate Jacobian of the cubic threefold. Ann. of Math, 95 (1972), 281–356.

    Article  MathSciNet  MATH  Google Scholar 

  5. I.V. Dolgachev. Dual homogeneous forms and varieties of power sums. Milan J. Math., 72 (2004), 163–187.

    Article  MathSciNet  MATH  Google Scholar 

  6. I.V. Dolgachev and V. Kanev. Polar covariants of plane cubics and quartics. Adv. Math., 98 (1993), 216–301.

    Article  MathSciNet  MATH  Google Scholar 

  7. D. Eisenbud and J. Harris. On varieties of minimal degree (a centennial account). Proc. Sympos. Pure Math, 46, Part 1, Amer. Math. Soc, Providence, RI, (1987).

  8. A. Franchetta. Sulla curva doppia della proiezione di una superficie generale dell’S4, da un punto generico su un S3. Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8), 2 (1947), 276–279.

    MathSciNet  MATH  Google Scholar 

  9. T. Graber, J. Harris and J. Starr. Families of rationally connected varieties. J. Amer. Math. Soc., 16(1) (2003), 57–67.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Harris. Lectures on rationally connected varieties. EAGER Advanced School in Algebraic Geometry in Levico Terme, Trento, September 2001. Notes by J. Kock.

    Google Scholar 

  11. D. Hilbert. Letter adresseé à M. Hermite. Gesam. Abh, Vol. II, pp. 148–153.

  12. A. Iliev and K. Ranestad. K3 surfaces of genus 8 and varieties of sums of powers of cubic fourfolds.Trans. Amer.Math. Soc., 4 (2001), 1455–1468.

    Article  MATH  Google Scholar 

  13. J. Kollár, Y. Miyaoka and S. Mori. Rational connectedness and boundedness of Fano manifolds. J. Diff. Geom., 36 (1992), 765–769.

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Kollár. Rational Curves on Algebraic Varieties. Ergebnisse derMathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin (1996).

    Google Scholar 

  15. J.M. Landsberg and L. Manivel. On the Ideals of Secant Varieties of Segre Varieties. Foundations of Computational Mathematics November, 2004, Vol. 4, Issue 4, pp. 397–422.

    Article  MathSciNet  MATH  Google Scholar 

  16. J.M. Landsberg and G. Ottaviani. New lower bounds for the border rank of matrix multiplication, to appear in Theory of Computing, arXiv:1112.6007v3.

  17. P.Y. Le Barz. Formules multisécantes pour les courbes gauches quelconques. Enumerative geometry and classical algebraic geometry, Progress in math, 24, Birkhaüser, Boston, Mass (1982).

    Chapter  Google Scholar 

  18. A.F. Lopez and Z. Ran. On the irreducibility of secant cones, and an application to linear normality. Duke Math. J., 117(3) (2003), 389–401.

    Article  MathSciNet  MATH  Google Scholar 

  19. S. Marchesi and A. Massarenti. Varieties connected by chains of lines. Geometriae Dedicata, 2012, doi: 10.1007/s10711-012-9817-y.

    Google Scholar 

  20. A. Massarenti and M. Mella. Birational aspects of the geometry of Varieties of Sums of Powers. Advances in Mathematics, 2013, no. 243, pp. 187–202.

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Massarenti and E. Raviolo. On the rank of n × n matrix multiplication. Linear Algebra and its Applications, 438 (2013), 4500–4509.

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Mella. Base loci of linear systems and the Waring problem. Proc. Am. Math. Soc., 137 (2009), 91–98.

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Mella. Singularities of linear systems and the Waring problem. Trans. Amer. Math. Soc., 358(12) (2006), 5523–5538.

    Article  MathSciNet  MATH  Google Scholar 

  24. S. Mukai. Fano 3-folds. Lond. Math. Soc. Lect. Note Ser., 179 (1992), 255–263.

    MathSciNet  MATH  Google Scholar 

  25. S. Mukai. Polarized K3 surfaces of genus 18 and 20. Complex Projective Geometry, Lond. Math. Soc. Lect. Note Ser, Cambridge University Press, 1992, pp. 264–276.

    Chapter  Google Scholar 

  26. F. Palatini. Sulla rappresentazione delle forme ternarie mediante la somma di potenze di forme lineari. Rom. Acc. L. Rend., 12 (1903), 378–384.

    MATH  Google Scholar 

  27. K. Ranestad and F.O. Schreyer. Varieties of sums of powers. J. ReineAngew. Math., 525 (2000), 147–181.

    MathSciNet  MATH  Google Scholar 

  28. K. Ranestad and C. Voisin. Variety of power sums and divisors in the moduli space of cubic fourfolds, arXiv:1309.1899v2.

  29. H.W. Richmond. On canonical forms. Q. J. Pure Appl.Math., 33 (1904), 967–984.

    Google Scholar 

  30. J.J. Sylvester. The collected mathematical papers of James Joseph Sylvester. Cambridge University Press (1904).

    Google Scholar 

  31. I. Takagi and F. Zucconi. Scorza quartics of trigonal spin curves and their varieties of power sums. Math. Ann., 349(3) (2011), 623–645.

    Article  MathSciNet  MATH  Google Scholar 

  32. A. Terracini. Sulle Vk per cui la varietá degli Sh (h + 1)-seganti ha dimensione minore dell’ordinario. Rend. Circ. Mat. Palermo, 31 (1911), 392–396.

    Article  MATH  Google Scholar 

  33. F.L. Zak. Tangents and Secants of Algebraic Varieties. Translations of Mathematical Monographs, 127 (1993).

Download references

Author information

Authors and Affiliations

  1. IMPA Estrada Dona Castorina, 110, 22460-320, Rio de Janeiro, RJ, Brazil

    Alex Massarenti

Authors
  1. Alex Massarenti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alex Massarenti.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Massarenti, A. Generalized varieties of sums of powers. Bull Braz Math Soc, New Series 47, 911–934 (2016). https://doi.org/10.1007/s00574-016-0196-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00574-016-0196-0

Keywords

Mathematical subject classification

Search

Navigation