164
Igor V. Dolgachev
Vol. 72 (2004)
deﬁnition, VSP(
F
d
;
N
)
o
is the subset of the projective space
P
S
N
V
∗
of ho-
mogeneous polynomials of degree
N
on
V
equal to the product of linear
forms
l
1
·
...l
N
such that
F
d
=
∑
l
d
i
. The zero sets
V
(
l
i
)o
ftheforms
l
i
’s
are hyperplanes in
P
V
,andtheset
V
(
l
1
)
,...,V
(
l
N
) was classically known
as the polar polyhedron of
F
d
. Its faces
V
(
l
i
) can be viewed as points in
the dual space
P
V
∗
,andthevar
ietyVSP(
F
d
;
N
)
o
is the subvariety of the
symmetric power
P
V
∗
(
N
)
of
P
V
∗
parametrizing the polar polyhedra of
F
d
.
The varietes VSP(
F
d
;
N
)
o
were intensively studied in the classical algebraic
geometry and the invariant theory in the works of A. Dixon, F. Palatini, T.
Reye, H. Richmond, J. Rosanes, G. Scorza, A. Terracini, and others. How-
ever, the lack of techniques of higher dimensional algebraic geometry did not
allow them to give any explicit construction of the varieties VSP(
F
d
;
N
)
o
or
to study a possible compactiﬁcation VSP(
F
d
;
N
)ofVSP(
F
d
;
N
)
o
(except in
the case
n
= 1 and a few cases where VSP(
F
2
,N
)
o
is a ﬁnite set of points).
The ﬁrst explicit construction of VSP(
F
d
;
N
)
o
was given by S. Mukai in
the cases (
n, d, N
)=(2
,
2
,
3)
,
(2
,
4
,
6)
,
(2
,
6
,
10) for a general polynomial
F
d
.