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

The main result of this paper is that the variety of presentations of a general cubic form in variables as a sum of cubes is isomorphic to the Fano variety of lines of a cubic -fold , in general different from .

A general surface of genus determines uniquely a pair of cubic -folds: the apolar cubic and the dual Pfaffian cubic (or for simplicity and ). As Beauville and Donagi have shown, the Fano variety of lines on the cubic is isomorphic to the Hilbert scheme of length two subschemes of . The first main result of this paper is that parametrizes the variety of presentations of the cubic form , with , as a sum of cubes, which yields an isomorphism between and . Furthermore, we show that sets up a correspondence between and . 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:
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