Quadratic Varieties of Small Codimension
Let ⊂ ℙⁿ⁺ᶜ be a nondegenerate smooth projective variety of dimension defined by quadratic equations. For such varieties, P. Ionescu and F. Russo proved the Hartshorne conjecture on complete intersections, which states that is a complete intersection provided that ≥ 2 + 1. As the extreme case, th...
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| Цитувати: | Quadratic Varieties of Small Codimension. Kiwamu Watanabe. SIGMA 21 (2025), 045, 14 pages |
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| author | Watanabe, Kiwamu |
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| description | Let ⊂ ℙⁿ⁺ᶜ be a nondegenerate smooth projective variety of dimension defined by quadratic equations. For such varieties, P. Ionescu and F. Russo proved the Hartshorne conjecture on complete intersections, which states that is a complete intersection provided that ≥ 2 + 1. As the extreme case, they also classified with = 2. In this paper, we classify with = 2 − 1.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 21 (2025), 045, 14 pages
Quadratic Varieties of Small Codimension
Kiwamu WATANABE
Department of Mathematics, Faculty of Science and Engineering, Chuo University,
1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan
E-mail: watanabe@math.chuo-u.ac.jp
Received January 28, 2025, in final form June 10, 2025; Published online June 15, 2025
https://doi.org/10.3842/SIGMA.2025.045
Abstract. Let X ⊂ Pn+c be a nondegenerate smooth projective variety of dimension n
defined by quadratic equations. For such varieties, P. Ionescu and F. Russo proved the
Hartshorne conjecture on complete intersections, which states that X is a complete inter-
section provided that n ≥ 2c+ 1. As the extremal case, they also classified X with n = 2c.
In this paper, we classify X with n = 2c− 1.
Key words: Hartshorne conjecture; complete intersections; Fano varieties; homogeneous
varieties
2020 Mathematics Subject Classification: 14J40; 14J45; 14M10; 14M17; 51N35
1 Introduction
Let X ⊂ Pn+c be a complex nondegenerate smooth projective variety of dimension n. Philo-
sophically, when the codimension c is small, the structure of X is subject to strong constraints.
In this direction, R. Hartshorne raised his famous conjecture.
Conjecture 1.1 ([11]). If n ≥ 2c+ 1, then X is a complete intersection.
As the extremal case, a variety X ⊂ Pn+c is called a Hartshorne variety if n = 2c and X is
not a complete intersection. The Hartshorne conjecture is still widely open. On the other hand,
P. Ionescu and F. Russo [16] proved that the Hartshorne conjecture holds for quadratic varieties,
that is, varieties scheme-theoretically defined by quadratic equations. This is a generalization
of J.M. Landsberg’s result [22, Corollary 6.29] obtained by local differential geometric methods.
Moreover, Ionescu and Russo also proved that the only quadratic Hartshorne varieties are the 6-
dimensional Grassmann variety G
(
2,C5
)
⊂ P9 and the 10-dimensional spinor variety S10 ⊂ P15.
Let us quickly review the brilliant ideas of Ionescu and Russo’s proof of [16]. If X ⊂ Pn+c is
a quadratic variety with small codimension, then X is a smooth Fano variety covered by lines;
in [16], an essentially important tool to study X is the Hilbert scheme of lines on X passing
through a general point x ∈ X. We denote it by Lx and the dual vector space of the tangent
space TxX by (TxX)∨; then Lx is naturally embedded into P
(
(TxX)∨
) ∼= Pn−1. A notable
feature of X is that Lx ⊂ Pn−1 is once again a quadratic variety, and it is scheme-theoretically
defined by at most c quadratic equations; moreover, if Lx ⊂ Pn−1 is a complete intersection,
then so is X ⊂ PN .
The purpose of this paper is to give a classification of quadratic varieties with n = 2c− 1.
Theorem 1.2. Let X ⊂ Pn+c be a nondegenerate smooth projective quadratic variety of dimen-
sion n. Assume that n = 2c − 1 and X is not a complete intersection. Then X is projectively
equivalent to one of the following:
(i) the Segre 3-fold P1 × P2 ⊂ P5;
(ii) a hyperplane section of the 6-dimensional Grassmann variety G
(
2,C5
)
⊂ P9;
mailto:watanabe@math.chuo-u.ac.jp
https://doi.org/10.3842/SIGMA.2025.045
2 K. Watanabe
(iii) a hyperplane section of the 10-dimensional spinor variety S10 ⊂ P15;
(iv) an 11-dimensional Fano variety X ⊂ P17 whose Fano index is 8. Moreover, X satisfies
the following:
(a) X is covered by lines.
(b) For a general point x ∈ X, we denote by Lx the Hilbert scheme of lines on X passing
through x ∈ X. Then
τx : Lx → P
(
(TxX)∨
)
, [ℓ] 7→ [Txℓ]
is a closed immersion, and Lx ⊂ P((TxX)∨) is projectively equivalent to a Gushel–
Mukai 6-fold (see Definition 2.12 for the definition of a Gushel–Mukai 6-fold).
In this theorem, the assumption that X is quadratic is essential because infinitely many
smooth nondegenerate projective 3-folds in P5 are not complete intersections (see, for instance,
[4, Section 2.2]). On the other hand, according to D. Mumford’s famous result [28], any projec-
tive variety can be realized as a quadratic variety. Although the same method as Ionescu and
Russo [16] is used to prove Theorem 1.2, the number of subjects to be treated increases, and
the discussion becomes more complicated. The results of Ionescu and Russo and the prelimi-
nary results used in the proof are summarized in Section 2. Cases where the codimension of
Theorem 1.2 is less than or equal to 2 are handled in Section 3. Cases where the codimension is
greater than or equal to 3 are treated in Section 4. The author needs to determine whether the
fourth case that appeared in Theorem 1.2 occurs. If this variety existed, what kind of properties
it would satisfy will be discussed in Section 5.
2 Preliminaries
Notation
We employ the notation as in [12, 16, 18, 30].
� We denote by Pn the projective space of dimension n and by Qn a smooth quadric hyper-
surface of dimension n.
� We denote by G(r,Cn) the Grassmann variety parametrizing r-dimensional linear sub-
spaces of Cn. We denote by S10 the 10-dimensional spinor variety, which is defined as an
irreducible component of the orthogonal Grassmann variety OG
(
5,C10
)
⊂ G
(
5,C10
)
for
a non-degenerate quadratic form on C10 (see, for instance, [20]).
� For a vector space V , we denote by V ∨ the dual vector space of V . The projectivization
of V is defined by
P(V ) := Proj
( ∞⊕
n=0
Symk(V )
)
.
� A smooth projective variety X is called Fano if the anticanonical divisor −KX is ample.
For a smooth Fano variety X, the Fano index iX is defined as the maximal integer r such
that −KX is divisible by r in Pic(X). The coindex of X is defined by dimX + 1− iX .
� For projective varieties X, Y and F , a smooth surjective morphism f : X → Y is called
an F -bundle if any fiber of f is isomorphic to F .
� For a smooth projective variety X, we denote by ρX the Picard number of X and by TX
the tangent bundle of X.
Quadratic Varieties of Small Codimension 3
� For an embedded projective variety X ⊂ PN , we denote by Sec(X) ⊂ PN the secant variety
of X.
� For an embedded projective variety X ⊂ PN and a point o ∈ PN+1, we denote by
Cone(o,X) ⊂ PN+1 the cone over X with vertex o.
� For an embedded projective variety X ⊂ PN and a point x ∈ X, we denote by TxX the
projective tangent space of X at x.
Setup 2.1. Throughout the paper, we consider X ⊂ PN a complex nondegenerate smooth pro-
jective variety of dimension n and codimension c. Assume that X ⊂ PN is scheme-theoretically
defined by hypersurfaces of degrees d1 ≥ d2 ≥ · · · ≥ dm. We may assume that m is minimal.
Let us set d :=
∑c
i=1(di − 1). Let L be a family of lines on X. For a general point x ∈ X, we
denote by Lx the Hilbert scheme of lines on X passing through x ∈ X. Denoting by (TxX)∨
the dual vector space of TxX, the tangent map
τx : Lx → P
(
(TxX)∨
)
, [ℓ] 7→ [Txℓ]
is a closed immersion by [30, Section 2.2.1]. Throughout the paper, Lx is viewed as a closed
subscheme of P((TxX)∨) ∼= Pn−1. When X is covered by lines, put p := degNℓ/X for any line
[ℓ] ∈ Lx.
Definition 2.2. When X is scheme-theoretically an intersection of quadric hypersurfaces, i.e.,
d = c, X is called quadratic.
Example 2.3. For positive integers r < n, the Grassmann variety G(r,Cn) is scheme-theo-
retically an intersection of quadrics via the Plücker embedding. The quadrics are given by
the Plücker relations. More generally, W. Lichtenstein [24] showed that for every fundamental
representation of a semisimple linear algebraic group, the orbit of the highest vector is cut out
by a system of quadrics. Moreover, Mumford [28] proves that any projective variety can be
realized as a quadratic variety.
We start to recall some classical results of projective geometry. The following is a refinement
of Faltings’ theorem [8] due to Netsvetaev.
Theorem 2.4 (Netsvetaev’s criterion [29, Theorem 3.2]). Let X be a variety as in Setup 2.1
and assume that m ≤ n+1. If m < N − 2
3n or n ≥ 3
4N − 1
2 , then X is a complete intersection.
Theorem 2.5 (Zak’s theorem on linear normality [31, Chapter II, Corollary 2.15]). Let X be
a variety as in Setup 2.1. If N < 3
2n+ 2, then Sec(X) = PN .
Theorem 2.6 (Zak’s classification of Severi varieties [31, Chapter IV, Theorem 4.7]). Let X be
a variety as in Setup 2.1. If N = 3
2n+ 2 and Sec(X) ̸= PN , then X is projectively equivalent to
one of the following:
(i) the Veronese surface v2
(
P2
)
⊂ P5;
(ii) the Segre 4-fold P2 × P2 ⊂ P8;
(iii) the Grassmann variety G
(
2,C6
)
⊂ P14;
(iv) the E6(ω1) ⊂ P26, which is the projectivization of the highest weight vector orbit in the
27-dimensional irreducible representation of a simple algebraic group of Dynkin type E6.
Lemma 2.7. For a variety X as in Setup 2.1, the following hold:
(i) If X is covered by lines, then p = dimLx.
(ii) If 2p ≥ n− 1, then Lx ⊂ Pn−1 is smooth, irreducible and nondegenerate.
4 K. Watanabe
Proof. See [13, Proposition 1.5, Theorems 1.4 and 2.5]. ■
Theorem 2.8 ([16, Theorems 2.4 and 3.8]). Let X be a variety as in Setup 2.1. Assume that X
is quadratic.
(i) If n ≥ c, then X is Fano.
(ii) If n ≥ c + 1, then X is covered by lines. Moreover, Lx ⊂ Pn−1 is scheme-theoretically
defined by c independent quadratic equations.
(iii) If n ≥ c+2, then X is a Fano variety with Pic(X) ∼= Z and iX = p+2. Furthermore, the
following are equivalent to each other:
(a) X ⊂ PN is a complete intersection.
(b) Lx ⊂ Pn−1 is a complete intersection of codimension c.
(c) p = n− 1− c.
As a byproduct of Theorem 2.8, Ionescu and Russo proved that the Hartshorne conjecture
on complete intersections holds for quadratic varieties. They also classified Hartshorne varieties
among quadratic varieties.
Theorem 2.9 ([16, Theorems 3.8 and 3.9]). Let X be a variety as in Setup 2.1. Assume that X
is quadratic.
(i) The Hartshorne conjecture for quadratic varieties: If n ≥ 2c + 1, then X is a complete
intersection.
(ii) Classification of Hartshorne varieties for quadratic varieties: If n = 2c and X is not
a complete intersection, then X is projectively equivalent to one of the following:
(a) the 6-dimensional Grassmann variety G
(
2,C5
)
⊂ P9;
(b) the 10-dimensional spinor variety S10 ⊂ P15.
Finally, we conclude this section by reviewing the classification of smooth Fano varieties with
large indices.
Theorem 2.10 ([9, 10]). Let X be an n-dimensional smooth Fano variety with index iX = n−1
(i.e., a smooth del Pezzo variety), whose Picard group is generated by a very ample line bundle.
Then X is isomorphic to one of the following:
(i) a hypersurface of degree 3;
(ii) a complete intersection of two quadric hypersurfaces;
(iii) a linear section of the Grassmann variety G
(
2,C5
)
⊂ P
((∧2C5
)∨)
.
Theorem 2.11 ([27]). Let X be an n-dimensional smooth Fano variety with index iX = n− 2
(i.e., a smooth Mukai variety), whose Picard group is generated by a very ample line bundle.
If n ≥ 4, then X is isomorphic to one of the following:
(i) a hypersurface of degree 4;
(ii) a complete intersection of a smooth quadric hypersurface and a cubic hypersurface;
(iii) a complete intersection of three quadric hypersurfaces;
(iv) a linear section of
∑6
10 ⊂ P10, where
∑6
10 ⊂ P10 is a smooth section of cone G̃ :=
Cone
(
o,G
(
2,C5
))
⊂ P10 over the Grassmann variety G
(
2,C5
)
⊂P
((∧2C5
)∨)
by a smooth
quadric hypersurface;
Quadratic Varieties of Small Codimension 5
(v) a linear section of the 10-dimensional spinor variety S10;
(vi) a linear section of the Grassmann variety G
(
2,C6
)
⊂ P
((∧2C6
)∨)
;
(vii) a linear section of LG
(
3,C6
)
, where LG
(
3,C6
)
is the Lagrangian Grassmann variety,
which is the variety of isotropic 3-planes for a non-degenerate skew-symmetric bilinear
form on C6;
(viii) the G2-variety which is the variety of isotropic 5-planes for a non-degenerate skew-sym-
metric 4-linear form on C7.
Definition 2.12. In Theorem 2.11,
∑6
10 ⊂ P10 is called the Gushel–Mukai 6-fold.
We refer the reader to [21] for the geometry of the Gushel–Mukai 6-fold.
3 The case c ≤ 2
For n = 2c−1 and c ≤ 2, let X be a variety as in Setup 2.1. Assume that X is quadratic and not
a complete intersection. Since we assume that X is not a complete intersection, we have c = 2.
Then n = 3. The purpose of this section is to prove the following.
Proposition 3.1. Let X be a variety as in Setup 2.1. Assume that X is quadratic and not
a complete intersection. If (n, c) = (3, 2), then X ⊂ P5 is projectively equivalent to the Segre
3-fold P1 × P2 ⊂ P5.
Proof. As in the proof of [16, Theorem 2.4], we may find quadrics Q1, Q2 such that X is an
irreducible component of the complete intersection scheme Q1 ∩ Q2. Since X is quadratic and
not a complete intersection, this yields that degX = 3 = codimX + 1, that is, X is a variety of
minimal degree. By Bertini’s theorem [2] (see also [7]), X is projectively equivalent to the Segre
3-fold P1 × P2 ⊂ P5. ■
4 The case c ≥ 3
4.1 General properties
In this section, we work in the following setting.
Setup 4.1. Let X ⊂ PN be a variety as in Setup 2.1. Assume that X is quadratic and not
a complete intersection. Assume that n = 2c− 1 and c ≥ 3.
Lemma 4.2. Let X ⊂ PN be a variety as in Setup 4.1. Then the following hold:
(i) X ⊂ PN is a smooth Fano variety covered by lines such that Pic(X) ∼= Z and iX = p+ 2.
(ii) Lx ⊂ Pn−1 is scheme-theoretically defined by c-independent quadratic equations.
Proof. Since n = 2c− 1 ≥ c+ 2, this follows from Theorem 2.8. ■
Lemma 4.3. Let X ⊂ PN be a variety as in Setup 4.1. Then we have
2 dimLx ≥ n− 1.
In particular, Lx ⊂ Pn−1 is smooth, irreducible, and nondegenerate.
6 K. Watanabe
Proof. Since Lx ⊂ Pn−1 is defined by c quadratic equations, we have an inequality
dimLx ≥ n− 1− c =
n− 3
2
.
If the lower bound is attained, it follows from Theorem 2.8 that X is a complete intersection.
This is a contradiction. So we have an inequality dimLx > n−3
2 . Since n = 2c− 1, we have
dimLx >
(2c− 1)− 3
2
= c− 2.
As a consequence, we obtain
dimLx ≥ c− 1 =
n− 1
2
.
The latter part of our assertion follows from Lemma 2.7. ■
Proposition 4.4. Let X ⊂ PN be a variety as in Setup 4.1. Then we have
3
2
c− 3 ≤ dimLx <
3
2
c− 2.
Proof. By Theorem 2.8 (iii), Lx is not a complete intersection. As we have seen, Lx ⊂ Pn−1 is
scheme-theoretically defined by c-independent quadratic equations. Moreover, by Lemma 4.3,
we have c ≤ dimLx + 1. Thus, applying Theorem 2.4 to Lx ⊂ Pn−1, we have an inequality
3n− 9
4
≤ dimLx <
3n− 5
4
.
Since n = 2c− 1, we obtain the desired inequality. ■
4.2 The case c is even
Let us additionally assume that c is even.
Setup 4.5. Let X ⊂ PN be a variety as in Setup 4.1, and assume that c is even.
In this subsection, we shall prove the following:
Theorem 4.6. Let X be a variety as in Setup 4.1. Assume that c is even. Then c is equal to 6
and Lx ⊂ P10 is projectively equivalent to the Gushel–Mukai 6-fold
∑6
10 ⊂ P10.
Lemma 4.7. Let X ⊂ PN be a variety as in Setup 4.5. Then the following hold:
(i) dimLx = 3
2c− 3.
(ii) dimLx − 2 codimPn−1 Lx − 1 = 1
2c− 6.
Proof. The first part directly follows from Proposition 4.4. The second part follows from the
first. ■
Lemma 4.8. Let X ⊂ PN be a variety as in Setup 4.5. Then c is 4, 6, 8 or 10.
Proof. By Theorem 2.8 (iii), Lx is not a complete intersection. Then Theorem 2.9 and Lem-
ma 4.7 yield 1
2c− 6 < 0. Hence, c < 12. ■
By Lemma 4.8, the pair (c, n,dimLx, iX) satisfies one of the following.
Quadratic Varieties of Small Codimension 7
c n dimLx iX
4 7 3 5
6 11 6 8
8 15 9 11
10 19 12 14
Lemma 4.9. Let X ⊂ PN be a variety as in Setup 4.5. Then c is not 4.
Proof. Assume to the contrary that c = 4. Then X is a 7-dimensional smooth Fano vari-
ety X ⊂ P11 of coindex 3. According to Theorem 2.11, X is isomorphic to a linear section
of the 10-dimensional spinor variety S10 ⊂ P15 or a linear section of the Grassmann variety
G
(
2,C6
)
⊂ P14. Theorem 2.5 shows that these varieties cannot be isomorphically projected
into P11. ■
Proposition 4.10. Let X be a variety as in Setup 4.5. Assume that c is 6, 8 or 10. Then
Lx ⊂ P2c−2 satisfies the following:
(i) Lx ⊂ P2c−2 is a
(
3
2c− 3
)
-dimensional nondegenerate smooth Fano variety.
(ii) Lx ⊂ P2c−2 is scheme-theoretically defined by c independent quadratic equations.
(iii) Lx ⊂ P2c−2 is covered by lines.
(iv) Pic(Lx) is isomorphic to Z.
Furthermore, for a general point [ℓ] ∈ Lx ⊂ P2c−2, let M[ℓ] ⊂ P
3
2
c−4 be the Hilbert scheme of
lines passing through [ℓ] ∈ Lx. Then M[ℓ] ⊂ P
3
2
c−4 satisfies the following:
(v) M[ℓ] ⊂ P
3
2
c−4 is scheme-theoretically defined by
(
1
2c+1
)
independent quadratic equations.
(vi) 3
2c− 5 ≥ i(Lx) = dimM[ℓ] + 2 ≥ c− 2.
Proof. In Lemmas 4.2, 4.3 and 4.7, we have already seen (i) and (ii) except that Lx is Fano.
Since
dimLx − codimP2c−2 Lx − 2 = c− 6 ≥ 0,
Theorem 2.8 implies (i)–(iv). Applying Theorem 2.8 to Lx ⊂ P2c−2, M[ℓ] ⊂ P
3
2
c−4 is defined
by
(
1
2c+1
)
independent quadratic equations. Therefore, (v) holds. Moreover, it follows from (v)
dimM[ℓ] ≥
(
3
2
c− 4
)
−
(
1
2
c+ 1
)
= c− 5.
Since X is not a complete intersection, Theorem 2.8 (iii) yields that neither Lx ⊂ P2c−2 nor
M[ℓ] ⊂ P
3
2
c−4 is a complete intersection and dimM[ℓ] ≥ c− 4. By the Kobayashi–Ochiai theo-
rem [17],
3
2
c− 2 = dimLx + 1 ≥ i(Lx).
Furthermore if i(Lx) ≥ 3
2c− 3, then Lx is isomorphic to P
3
2
c−3 or Q
3
2
c−3. Then Lx ⊂ P2c−2 is a
complete intersection, because Lx ⊂ P2c−2 is covered by lines. This is a contradiction. Hence,
we have 3
2c− 4 ≥ i(Lx).
Suppose i(Lx) =
3
2c − 4. Then Lx is a del Pezzo variety. By our assumption, c equals 6, 8
or 10. Thus, according to Theorem 2.10, Lx is isomorphic to G
(
2,C5
)
⊂ P9. Since Lx ⊂ P2c−2
is covered by lines, the embedding is given by the ample generator of Pic(Lx). Then Theo-
rem 2.5 yields that Lx ⊂ P2c−2 is projectively equivalent to G
(
2,C5
)
⊂ P9. However, this is
a contradiction, because 2c− 2 ≥ 10 > 9. ■
8 K. Watanabe
Proposition 4.11. Let X be a variety as in Setup 4.5. If c is equal to 6, then Lx ⊂ P10 is
projectively equivalent to the Gushel–Mukai 6-fold
∑6
10 ⊂ P10.
Proof. Assume that c = 6. By Proposition 4.10, Lx ⊂ P10 is a nondegenerate 6-dimensional
Fano variety with coindex 3 and covered by lines; it follows from Theorem 2.11 that Lx is an
isomorphic projection of one of the varieties as follows:
(i) the Gushel–Mukai 6-fold
∑6
10 ⊂ P10;
(ii) a linear section of the 10-dimensional spinor variety S10 ⊂ P15;
(iii) a linear section of the 8-dimensional Grassmann variety G
(
2,C6
)
⊂ P14;
(iv) the 6-dimensional Lagrangian Grassmann variety LG
(
3,C6
)
⊂ P13.
By Theorems 2.5 and 2.6, the varieties (ii)–(iv) cannot be isomorphically projected into P10.
It turns out that Lx is projectively equivalent to the Gushel–Mukai 6-fold
∑6
10 ⊂ P10. ■
Proposition 4.12. Let X be a variety as in Setup 4.5. Then c is not equal to 8.
Proof. Assume that c = 8. From Theorem 4.10, it follows that dimM[ℓ] = 4 or 5, and
M[ℓ] ⊂ P8 is defined by 5 quadratic equations. By Lemma 2.7, M[ℓ] ⊂ P8 is smooth, irreducible
and nondegenerate. If dimM[ℓ] = 4, then Theorem 2.4 yields that M[ℓ] ⊂ P8 is a complete
intersection. This contradicts our assumption that X is not a complete intersection. Thus we
have dimM[ℓ] = 5. By Proposition 4.10, Lx ⊂ P14 satisfies the following:
(i) Lx ⊂ P14 is a nondegenerate smooth Fano 9-fold of index 7;
(ii) Lx ⊂ P14 is covered by lines and Pic(X) ∼= Z;
(iii) Lx ⊂ P14 is scheme-theoretically defined by 8 quadrics.
By Theorem 2.11, Lx ⊂ P14 is projectively equivalent to a hyperplane section of the 10-
dimensional spinor variety S10 ⊂ P15. This contradicts Proposition 4.13 below. ■
Proposition 4.13. Let V be a smooth hyperplane section of the 10-dimensional spinor variety
S10 ⊂ P15. Then the following hold:
(i) S10 ⊂ P15 is not scheme-theoretically defined by 8 quadrics.
(ii) V ⊂ P14 is not scheme-theoretically defined by 8 quadrics.
Proof. (i) By [6, 4.4], the linear system of quadrics containing S10 ⊂ P15 provides a rational
map P15 π
99K Q8, that can be resolved via the blow-up of P15 along S10, which is a P7-bundle
q : BlS10
(
P15
)
→ Q8 over Q8 as follows:
BlS10(P15)
p
xx
q
&&
S10 ⊂ P15 π // Q8 ⊂ P9.
Denoting the exceptional divisor of p by E, we have
p∗OP15(2)⊗OX(−E) ∼= q∗OQ8(1). (4.1)
For a quadric hypersurface D ⊂ P15 containing S10, we denote by D̃ the strict transform of D
with respect to p and by HD the closure of π
(
D \ S10
)
. By (4.1), we have
D̃ = p∗(D)− E = q∗(HD).
Quadratic Varieties of Small Codimension 9
Assume that there exist 8 quadrics D1, D2, . . . , D8 ∈ |IS10/P15(2)| such that S is a scheme-
theoretic intersection of Di’s, that is, S =
⋂8
i=1Di. This means that the base locus of the linear
system ⟨D1, D2, . . . , D8⟩ coincides with S10:
Bs(⟨D1, D2, . . . , D8⟩) = S10.
Then we see that
8⋂
i=1
D̃i = ∅ in BlS10
(
P15
)
.
However, we have
8⋂
i=1
Hi ̸= ∅ in Q8 ⊂ P9.
This yields
8⋂
i=1
D̃i =
8⋂
i=1
q−1(Hi) = q−1
(
8⋂
i=1
Hi
)
̸= ∅.
This is a contradiction.
(ii) Let H ⊂ P15 be a hyperplane such that V = S10 ∩ H as a scheme. Then we have
a standard exact sequence of ideal sheaves
0 → IS10/P15(1) → IS10/P15(2) → IV/H(2) → 0. (4.2)
Since the embedding S10 ↪→ P15 is given by the complete linear system OS10(1), we obtain
H1
(
IS10/P15(1)
)
= 0. The above exact sequence (4.2) yields a surjection
H0
(
S10, IS10/P15(2)
)
↠ H0
(
S10, IV/H(2)
)
. (4.3)
To prove our assertion, assume the contrary, that is, V ⊂ P14 is scheme-theoretically de-
fined by 8 quadrics D′
1, . . . , D
′
8. From the surjectivity of the above map (4.3), these quadrics
D′
1, . . . , D
′
8 can be extended to quadrics D̃′
1, . . . , D̃
′
8 containing S10. This means that S10 is
contained in the scheme-theoretic intersection D̃′
1 ∩ · · · ∩ D̃′
8. Since S10 does not coincide with
D̃′
1 ∩ · · · ∩ D̃′
8 by (i) and D̃′
1 ∩ · · · ∩ D̃′
8 ∩H = V , we obtain a contradiction. ■
Proposition 4.14. Let X be a variety as in Setup 4.5. Then c is not equal to 10.
Proof. Assume that c = 10. From Theorem 4.10, it follows that dimM[ℓ] = 6, 7 or 8, and
M[ℓ] ⊂ P11 is defined by 6 quadratic equations. By Lemma 2.7, M[ℓ] ⊂ P11 is smooth, irreducible
and nondegenerate. Then Theorem 2.4 implies that M[ℓ] ⊂ P11 is a complete intersection. This
contradicts our assumption that X is not a complete intersection. ■
4.3 The case c is odd
Let us consider the case where c is odd. In this subsection, we shall prove the following.
Theorem 4.15. Let X be a variety as in Setup 4.1. Assume that c is odd. Then X is projectively
equivalent to
(i) a hyperplane section of the 10-dimensional spinor variety S10 ⊂ P15 or
(ii) a hyperplane section of the 6-dimensional Grassmann variety G
(
2,C5
)
⊂ P9.
10 K. Watanabe
The following lemmas can be proved like Lemmas 4.7 and 4.8. So, the proofs are left to the
readers.
Lemma 4.16. Let X ⊂ PN be a variety as in Setup 4.1 and assume that c is odd. Then the
following hold:
(i) dimLx = 3c−5
2 .
(ii) dimLx − 2 codimPn−1 Lx − 1 = c−9
2 .
Lemma 4.17. Let X ⊂ PN be a variety as in Setup 4.1 and assume that c is odd. Then c is 3,
5 or 7.
By Lemma 4.17, the pair (c, n,dimLx, iX) satisfies one of the following.
c n dimLx iX
3 5 2 4
5 9 5 7
7 13 8 10
Proposition 4.18. Let X be a variety as in Setup 4.1. Then the following hold:
(i) If c = 3, then X ⊂ P8 is projectively equivalent to a hyperplane section of the Grassmann
variety G
(
2,C5
)
⊂ P9.
(ii) If c = 5, then X ⊂ P14 is projectively equivalent to a hyperplane section of the spinor
variety S10 ⊂ P15.
Proof. Suppose c = 3. Then X is a del Pezzo 5-fold. By Theorem 2.10, X is isomorphic
to a hyperplane section of the Grassmann variety G
(
2,C5
)
⊂ P9. Since X is covered by lines
and c = 3, our assertion holds. The case where c = 5 also follows from Theorem 2.11 and the
same argument as above. ■
Proposition 4.19. Let X be a variety as in Setup 4.1. Then c is not equal to 7.
Proof. Suppose c = 7. Then, the same proof as in Proposition 4.10 shows that Lx ⊂ P12 is
a nondegenerate smooth quadratic variety covered by lines. Moreover we also see that the family
of lines M[ℓ] ⊂ P7 passing through a general point [ℓ] ∈ Lx satisfies
� M[ℓ] ⊂ P7 is defined by 4 independent quadratic equations, and
� dimM[ℓ] = 3, 4 or 5.
Then, according to Theorem 2.4, we see that M[ℓ] ⊂ P7 is a complete intersection. This
contradicts our assumption that X ⊂ PN is not a complete intersection. ■
5 The remaining case
Summarizing the results so far, Theorem 1.2 holds true. In the following, we focus on the variety
as in Theorem 1.2 (iv). We work under the following setting.
Setup 5.1. Let X ⊂ PN be an 11-dimensional Fano variety X ⊂ P17 whose Fano index is 8.
Moreover, X satisfies the following:
(i) X is covered by lines.
Quadratic Varieties of Small Codimension 11
(ii) For a general point x ∈ X, we denote by Lx the Hilbert scheme of lines on X passing
through x ∈ X. Then Lx ⊂ P
(
(TxX)∨
)
= P10 is projectively equivalent to a Gushel–
Mukai 6-fold.
The ample generator of the Picard group Pic(Lx) is denoted by OLx(1). The embedding Lx ⊂
P((TxX)∨) = P10 is given by the complete linear system |OLx(1)|. Let πx : Ux → Lx be the
universal family and evx : Ux → X the evaluation morphism:
Ux
evx //
πx
��
X
Lx.
We denote by Locus(Lx) the image of evx: Locus(Lx) := evx(Ux).
Proposition 5.2. Under the setting of Setup 5.1, X satisfies the following:
(i) H2(X,Z) ∼= Z[H] and H4(X,Z) ∼= Z
[
H2
]
.
(ii) c1(X) = 8H ∈ H2(X,Z) and c2(X) = 31H2 ∈ H4(X,Z). In particular, the second Chern
character ch2(X) = 1
2
(
c1(X)2− 2c2(X)
)
= H2, that is, X is a 2-Fano variety in the sense
of [1].
(iii) The self-intersection number H11 is equal to 24.
(iv) Through two general points of X, there passes an irreducible conic contained in X, that
is, X is a conic-connected variety in the sense of [15].
Proof. (i) follows from the Barth–Larsen theorem [23, Corollary]. Since the Fano index of X
is 8, we have c1(X) = 8H ∈ H2(X,Z). Applying [1, Proposition 1.3 (1.2)] or [5, Proposition 4.2],
we have
πx∗ev
∗
x(ch2(X)) = c1(OLx(1)).
This yields that ch2(X) = H2; thus we obtain c2(X) = 31H2 ∈ H4(X,Z). As a consequence,
we see that (ii) holds.
Let Y ⊂ P10 be a general 4-dimensional linear section of X ⊂ P17 and HY a divisor on Y
which is the restriction of H to Y ; then one see that Y is a Fano 4-fold such that c1(Y ) = HY
and c2(Y ) = 3H2
Y . By the Riemann–Roch formula and various vanishing theorems, we obtain
the following equation (see [19, p. 48]):
h0(−KY ) = 1 +
(−KY )
2c2(Y )
12
+
(−KY )
4
6
= 1 +
5
12
H11.
On the other hand, according to [3, Corollary 2], the quadratic variety X is arithmetically
Cohen–Macaulay (aCM) (also called projectively Cohen–Macaulay), that is, the homogeneous
coordinate ring of X ⊂ P17 is Cohen–Macaulay (see also the sentence before [16, Theorem 3.8]).
By [26, Theorem 1.3.3], Y ⊂ P10 is also aCM and in particular linearly normal. Thus we obtain
11 = h0(−KY ) = 1 +
5
12
H11.
This yields that H11 = 24. Thus, (iii) holds.
Finally, (iv) follows from [14, Theorem 3.14]. ■
Proposition 5.3. Let X be a variety as in Setup 5.1. For a general point x ∈ X ⊂ P17,
Locus(Lx) ⊂ TxX ∼= P11 is projectively equivalent to the cone Cone(o′,Lx) ⊂ P11, where o′ is
a point in P11 \ P((TxX)∨).
12 K. Watanabe
Proof. According to [1, Section 2.2], there exists a rank 2 vector bundle E over Lx satisfying
the following:
� Ux = P(E).
� The sequence 0 → OLx → E → OLx(−1) → 0 is exact.
Since we have
dimExt1(OLx(−1),OLx) = h1(OLx(1)) = 0,
E is isomorphic to OLx ⊕ OLx(−1). By tensoring OLx(1) to E , we see that Ux is isomorphic
to P(OLx(1)⊕OLx). Denoting by OUx(1) the tautological line bundle of Ux = P(OLx(1)⊕OLx),
the evaluation morphism
evx : Ux = P(OLx(1)⊕OLx) → Locus(Lx)
is determined by a linear system Λ ⊂ |OUx(1)|. The complete linear system |OUx(1)| determines
the morphism Ux = P(OLx(1) ⊕ OLx) → Cone(o′,Lx), which is nothing but the blow-up of
Cone(o′,Lx) at the vertex o′. Hence, the evaluation morphism evx factors through the morphism
Ux = P(OLx(1)⊕OLx) → Cone(o′,Lx), and the morphism p : Cone(o′,Lx) → Locus(Lx) is given
by a projection:
Ux = P(OLx(1)⊕OLx) //
evx
))
Cone(o′,Lx)
p
��
Locus(Lx).
Since there exists one-to-one correspondence between the set of lines in Cone(o′,Lx) passing
through the vertex o′ and the set of lines in Locus(Lx) passing through x via p, the map p is
bijective. By the Zak’s linear normality [31, Chapter II, Corollary 2.11] (see also [30, Theo-
rem 5.1.6]), the secant variety Sec(Lx) coincides with the whole space P10. This implies that
Sec(Cone(o′,Lx)) = P11. Thus we conclude that Λ = |OUx(1)| and Locus(Lx) ∼= Cone(o′,Lx).
As a consequence, our assertion holds. ■
Lemma 5.4. Let X be a variety as in Setup 5.1. Then a double cover does not exist from X to
a smooth quadratic variety Y ⊂ P16.
Proof. Assume the contrary, that is, there exists a double cover f : X → Y to a smooth
quadratic variety Y ⊂ P16. By Theorem 2.9, Y should be a complete intersection; thus we have
deg Y = 32. This yields that degX = 2deg Y = 64. This contradicts Proposition 5.2 (iii). ■
Remark 5.5. Let X be a variety as in Setup 5.1. By Lemma 5.4, if we can show that X has
a double cover as in Lemma 5.4, we can see that case (iv) of Theorem 1.2 does not occur.
Acknowledgments
The author would like to express his sincere gratitude to the anonymous referees for their metic-
ulous reading of his manuscript. Their insightful comments and suggestions have significantly
improved various parts of this work. The author knew the proof of Proposition 4.13 in [25] and
would like to thank them for teaching how to prove it. Additionally, the author would like to
thank Professor Wahei Hara for explaining the parts of the proof that were not understood and
for providing a detailed explanation. The author is partially supported by JSPS KAKENHI
Grant Number 21K03170, 25K06940.
Quadratic Varieties of Small Codimension 13
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1 Introduction
2 Preliminaries
3 The case c leq 2
4 The case c geq 3
4.1 General properties
4.2 The case c is even
4.3 The case c is odd
5 The remaining case
References
|
| id | nasplib_isofts_kiev_ua-123456789-213531 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-17T11:03:57Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Watanabe, Kiwamu 2026-02-18T11:26:17Z 2025 Quadratic Varieties of Small Codimension. Kiwamu Watanabe. SIGMA 21 (2025), 045, 14 pages 1815-0659 2020 Mathematics Subject Classification: 14J40; 14J45; 14M10; 14M17; 51N35 arXiv:2405.04002 https://nasplib.isofts.kiev.ua/handle/123456789/213531 https://doi.org/10.3842/SIGMA.2025.045 Let ⊂ ℙⁿ⁺ᶜ be a nondegenerate smooth projective variety of dimension defined by quadratic equations. For such varieties, P. Ionescu and F. Russo proved the Hartshorne conjecture on complete intersections, which states that is a complete intersection provided that ≥ 2 + 1. As the extreme case, they also classified with = 2. In this paper, we classify with = 2 − 1. The author would like to express his sincere gratitude to the anonymous referees for their meticulous reading of his manuscript. Their insightful comments and suggestions have significantly improved various parts of this work. The author knew the proof of Proposition 4.13 in [25] and would like to thank them for teaching how to prove it. Additionally, the author would like to thank Professor Wahei Hara for explaining the parts of the proof that were not understood and for providing a detailed explanation. The author is partially supported by JSPS KAKENHI Grant Numbers 21K03170 and 25K06940. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Quadratic Varieties of Small Codimension Article published earlier |
| spellingShingle | Quadratic Varieties of Small Codimension Watanabe, Kiwamu |
| title | Quadratic Varieties of Small Codimension |
| title_full | Quadratic Varieties of Small Codimension |
| title_fullStr | Quadratic Varieties of Small Codimension |
| title_full_unstemmed | Quadratic Varieties of Small Codimension |
| title_short | Quadratic Varieties of Small Codimension |
| title_sort | quadratic varieties of small codimension |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/213531 |
| work_keys_str_mv | AT watanabekiwamu quadraticvarietiesofsmallcodimension |