Skip to main content
SpringerLink
Log in

On homaloidal polynomial functions of degree 3 and prehomogeneous vector spaces

  • Published:
Collectanea Mathematica Aims and scope Submit manuscript

Abstract

In this paper we consider homaloidal polynomial functions f such that their multiplicative Legendre transform f *, defined as in Etingof et al. (Sel. Math. (N.S.) 8(1):27–66, 2002), Section 3.2 is again polynomial. Following Dolgachev (Michigan Math. J. 48:191–202, 2000), we call such polynomials EKP-homaloidal. We prove that every EKP-homaloidal polynomial function of degree three is a relative invariant of a symmetric prehomogeneous vector space. This provides a complete proof of Etingof et al. (Sel. Math. (N.S.) 8(1):27–66, 2002), Theorem 3.10, p. 39. Our argument may suggest a way to attack the more general problem raised in Etingof et al. (Sel. Math. (N.S.) 8(1):27–66, 2002), Section 3.4 of EKP-homaloidal polynomials of arbitrary degree.

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. Bertram, W.: The geometry of Jordan and Lie structures. In: Lecture Notes in Mathematics, vol. 1754. Springer, Berlin (2000)

  2. Chaput P.-E.: Severi varieties. Math. Z. 240(2), 451–459 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Dolgachev, I.V.: Polar Cremona transformations. Michigan Math. J. 48, 191–202 (2000) (Dedicated to William Fulton on the occasion of his 60th birthday)

  4. Etingof P., Kazhdan D., Polishchuk A.: When is the Fourier transform of an elementary function elementary?. Sel. Math. (N.S.) 8(1), 27–66 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Jacobson, N.: Structure and representations of Jordan algebras. In: American Mathematical Society Colloquium Publications, vol. 39. American Mathematical Society, Providence (1968)

  6. Kimura, T.: Introduction to prehomogeneous vector spaces. In: Translations of Mathematical Monographs, vol. 215. American Mathematical Society, Providence (2003) (Translated from the 1998 Japanese original by Makoto Nagura and Tsuyoshi Niitani and revised by the author)

  7. Zak, F.: Tangents and secants of algebraic varieties. In: Translations of Mathematical Monographs, vol. 127. American Mathematical Society, Providence (1993) (Translated from the Russian manuscript by the author)

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques, Rue de la Houssinière, BP 92208 2, 44322, Nantes Cedex 03, France

    Pierre-Emmanuel Chaput

  2. Via Delle Mimose 9 Int. 14, 00172, Rome, Italy

    Pietro Sabatino

Authors
  1. Pierre-Emmanuel Chaput
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pietro Sabatino
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pietro Sabatino.

Additional information

The authors would like to thank the anonymous referee for suggesting the short proof of item (v) in Corollary 9.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chaput, PE., Sabatino, P. On homaloidal polynomial functions of degree 3 and prehomogeneous vector spaces. Collect. Math. 64, 135–140 (2013). https://doi.org/10.1007/s13348-011-0046-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13348-011-0046-8

Keywords

Mathematics Subject Classification (2010)

Search

Navigation