Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

$K3$ surfaces of genus 8 and varieties of sums of powers of cubic fourfolds


Authors: Atanas Iliev and Kristian Ranestad
Journal: Trans. Amer. Math. Soc. 353 (2001), 1455-1468
MSC (2000): Primary 14J70; Secondary 14M15, 14N99
DOI: https://doi.org/10.1090/S0002-9947-00-02629-5
Published electronically: October 11, 2000
MathSciNet review: 1806733
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract:

The main result of this paper is that the variety of presentations of a general cubic form $f$ in $6$ variables as a sum of $10$cubes is isomorphic to the Fano variety of lines of a cubic $4$-fold $F'$, in general different from $F=Z(f)$.

A general $K3$ surface $S$ of genus $8$determines uniquely a pair of cubic $4$-folds: the apolar cubic $F(S)$ and the dual Pfaffian cubic $F'(S)$ (or for simplicity $F$ and $F'$). As Beauville and Donagi have shown, the Fano variety $\mathcal{F}_{F'}$ of lines on the cubic $F'$ is isomorphic to the Hilbert scheme $\operatorname{Hilb}_2S$ of length two subschemes of $S$. The first main result of this paper is that $\operatorname{Hilb}_2S$ parametrizes the variety $VSP(F,10)$ of presentations of the cubic form $f$, with $F=Z(f)$, as a sum of $10$ cubes, which yields an isomorphism between $\mathcal{F}_{F'}$ and $VSP(F,10)$. Furthermore, we show that $VSP(F,10)$ sets up a $(6,10)$ correspondence between $F'$ and $\mathcal{F}_{F'}$. The main result follows by a deformation argument.


References [Enhancements On Off] (What's this?)

  • 1. Alexander, J., Hirschowitz, A.: Polynomial interpolation in several variables, J. of Alg. Geom. 4 (1995), 201-222. MR 96f:14065
  • 2. Beauville, A., Donagi, R.: La variete des droites d'une hypersurface cubique de dimension 4. Compt. Rendu. Acad. Sc. Paris. 301 (1986) 703-706. MR 87c:14047
  • 3. Ein, L., Shepherd-Barron, N.: Some special Cremona transformations, Amer. J. Math. 111 (1989) 783-800. MR 90j:14015
  • 4. Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. GTM 150 Springer-Verlag, New York, 1995. MR 97a:13001
  • 5. Macaulay, F.S.: Algebraic theory of modular systems. Cambridge University Press, London, (1916). MR 95i:13001 (rev. reprint)
  • 6. Bayer, D., Stillman, M.: MACAULAY: A system for computation in algebraic geometry and commutative algebra, Source and object code available for Unix and Macintosh computers. Contact the authors, or download from zariski.harvard.edu via anonymous ftp.
  • 7. Mukai, S.: Curves, $K3$ surfaces and Fano 3-folds of genus $\leq 10$, in ``Algebraic Geometry and Commuatative Algebra in Honor of Masayoshi Nagata", pp. 357-377, (1988), Kinokuniya, Tokyo. MR 90b:14039
  • 8. Ranestad, K., Schreyer, F-O: Varieties of sums of powers, to appear in J. Reine Angew. Math. (2000).
  • 9. Salmon, G.: Modern Higher Algebra, 4. Edition. Hodges, Figgis, and Co., Dublin (1885)
  • 10. Zak, F. L.: Varieties of small codimension arising from group actions. Addendum to Lazarsfeld and Van de Ven: Topics in the Geometry of Projective Space, DMV Seminar 4 (1984). MR 87e:14045

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 14J70, 14M15, 14N99

Retrieve articles in all journals with MSC (2000): 14J70, 14M15, 14N99


Additional Information

Atanas Iliev
Affiliation: Institute of Mathematics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., 8, 1113 Sofia, Bulgaria
Email: ailiev@math.bas.bg

Kristian Ranestad
Affiliation: Matematisk Institutt, UiO, P.B. 1053 Blindern, N-0316 Oslo, Norway
Email: ranestad@math.uio.no

DOI: https://doi.org/10.1090/S0002-9947-00-02629-5
Received by editor(s): July 5, 1999
Published electronically: October 11, 2000
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society