Last Updated：2026/01/01
Sentence
More
precisely,
Xⁿ⊂ℙᴺ
is
a
Severi
variety
if
and
only
if
ℙᴺ=ℙ(𝔍),
where
𝔍
is
the
Jordan
algebra
of
Hermitian
(3
×
3)-matrices
over
a
composition
algebra
𝔄,
and
X
corresponds
to
the
cone
of
Hermitian
matrices
of
rank
<1
(in
that
case
SX
corresponds
to
the
cone
of
Hermitian
matrices
with
vanishing
determinant;
cf.
Theorem
4.8).
In
other
words,
X
is
a
Severi
variety
if
and
only
if
X
is
the
“Veronese
surface”
over
one
of
the
composition
algebras
over
the
field
K
(Theorem
4.9).
Quizzes for review
More precisely, Xⁿ⊂ℙᴺ is a Severi variety if and only if ℙᴺ=ℙ(𝔍), where 𝔍 is the Jordan algebra of Hermitian (3 × 3)-matrices over a composition algebra 𝔄, and X corresponds to the cone of Hermitian matrices of rank <1 (in that case SX corresponds to the cone of Hermitian matrices with vanishing determinant; cf. Theorem 4.8). In other words, X is a Severi variety if and only if X is the “Veronese surface” over one of the composition algebras over the field K (Theorem 4.9).
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