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Transactions of the American Mathematical Society
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
Published electronically: October 11, 2000
MathSciNet review: 1806733
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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.


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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: http://dx.doi.org/10.1090/S0002-9947-00-02629-5
PII: S 0002-9947(00)02629-5
Received by editor(s): July 5, 1999
Published electronically: October 11, 2000
Article copyright: © Copyright 2000 American Mathematical Society