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6

ON THE PICARD GROUP OF LOW-CODIMENSION SUBVARIETIES

ENRIQUE ARRONDO AND JORGE CARAVANTES

Abstract. We introduce a method to determine if n-dimensional smooth subvarieties of an
ambient space of dimension at most 2n − 2 inherit the Picard group from the ambient space
(as it happens when the ambient space is a projective space, according to results of Barth and
Larsen). As an application, we give an affirmative answer (up to some mild natural numerical
conditions) when the ambient space is a Grassmannian of lines (thus improving results of Barth,
Van de Ven and Sommese) or a product of two projective spaces of the same dimension.

A (complex) smooth subvariety of small codimension is expected to be very special. The
most characteristic example of this principle is the case of subvarieties of the projective space, in
which a famous conjecture of Hartshorne (see [7]) states that a smooth n-dimensional subvariety
X ⊂ PN must be a complete intersection if 2N < 3n. The main evidence for this conjecture
is a series of results of Barth and Larsen showing that the topology of X behaves like the one
of a complete intersection. Specifically, it was first proved in [3] that H i(X, Q) ∼= H i(PN , Q) if
i ≤ 2n−N , then in [4] that π1(X) = 0 if N ≤ 2n−1 and finally in [9] that H i(X, Z) ∼= H i(PN , Z)
if i ≤ 2n − N . In particular, this implies that the Picard group of such X is generated by the
class of its hyperplane section if N ≤ 2n − 2.

More generally, one can replace the projective space with other ambient spaces to study this
kind of problems. For instance, in [5], Barth and Van de Ven prove that if X is a smooth
codimension-two subvariety of G(k,N) (the Grassmann variety of subspaces of dimension k

in PN ), then H i(X, Z) ∼= H i(G(k,N), Z) if N − k ≥ 6 and i = 0, 1, 2. A stronger result
has been obtained by Sommese in [10], in the general framework of homogeneous varieties
(with particularly strong results for Abelian varieties and products of Grassmannians). When
particularized to X ⊂ Y :=

∏

G(kl, Nl) of dimension n, his result states that πi(Y,X) = 0 if
i ≤ 2n− 2

∑

[(kl + 1)(N−kl)]+ min{Nl}+ 1. As a consequence, if Y = G(k,N), then the Picard
group of X is generated by the hyperplane section when 2(k + 1)(N − k) − N ≤ 2n − 2 (for
k = 0 this is the mentioned result for the projective space). We can also apply this to products
of projective spaces and we get that, if Y := PN × PN , then the Picard group of X is generated
by the restrictions of the generators of Pic(PN × PN ) when 3N + 2 ≤ 2n.

In this paper, we introduce a general method to study when the Picard group of an ambient
space Y of dimension N is inherited by smooth subvarieties of dimension at least N+2

2 (hence in
the range of Barth-Larsen result for projective spaces). Our starting point is the paper by the
first author [1], where it is shown that the canonical divisor of any smooth fourfold in G(1, 4)
(and in Q6 under some numerical conditions) is –numerically– a multiple of the hyperplane
section. The method there relied strongly on the fact that the codimension was two (in order
to use the so-called Hartshorne-Serre correspondence) and it seemed essential to work with
the canonical divisor and not any other; a trivial application of Porteous formula produced a
mysterious numerical relation, that eventually implied (via the Hodge index theorem) that the
surface obtained by successive hyperplane sections was (numerically) subcanonical.

Both authors were partially supported by the MCYT project BFM2003-03971.

1

http://arXiv.org/abs/math/0511267v2


2 ENRIQUE ARRONDO AND JORGE CARAVANTES

Since Sommese’s result indicates that the result of [1] should be true in higher codimension,
we tried to understand where that mysterious numerical relation actually came from, without
using such sophisticated techniques. We have thus found out that, in the general framework of
X ⊂ Y of dimension n ≥ N+2

2 , that relation could be easily obtained by just applying the self-
intersection formula for X in Y , the self-intersection formula for any smooth divisor D of X (this
shows why it is important that D has dimension at least its codimension in this ambient space)
and the relation among them. In this way, we obtain in several ambient spaces (not necessarily
with Picard number one) that any divisor (with only few restrictions) of X must be numerically
dependent on the generators of the Picard group of Y . When X is also simply connected, this
method shows that the Picard group of X (which is then torsion-free) is generated (with rational
coefficients) by the generators of the Picard group of Y .

Our first example, to which we devote Section 1, is Y = PN and in this case we reobtain (up to
divisibility) the theorem of Barth-Larsen for the Picard groups of subvarieties X ⊂ PN . In fact,
as it happened in [1], the above mentioned numerical equality becomes H2

|SD2
|S = (H|SD|S)2,

where S is the surface obtained intersecting X with n − 2 general hyperplanes. The Hodge
index theorem implies immediately that D|S is numerically a multiple of H|S. The Lefschetz
hyperplane section and the simple connectedness of X obtained in [4] completes the proof.

Following the idea of the projective case, in the second section we apply this method when
the ambient space is a Grassmannian of lines (observe that our hypothesis about the relation
between the dimension and codimension of X are much milder than the ones in the result of
Sommese). Here we prove that, when X has positive intersection with all Schubert cycles of
codimension n, then its Picard group has rank one. In this case, the key point is to write the
numerical formula as a positive linear combination of Hodge-type expressions for the surfaces
obtained by intersecting X with the different Schubert cycles of codimension n − 2 (which, by
Hodge index theorem, must be positive, and they are zero when numerical equivalence occurs),
plus some other positive terms.

In the third section we illustrate the fact that our method keep working even if the Picard
group of the ambient space Y has rank bigger than one. As a sample, we have chosen the
case in which Y is the cartesian product of two projective spaces of the same dimension. This
particular choice has the advantage that Y has even dimension (this is also one of the reasons
why we choose k = 1 in the case of Grassmannians), which simplifies the calculations. Again
we prove in this case that, in the Barth-Larsen range, the Picard group of X ⊂ Y is generated
by two elements, assuming again some nonvanishing of the coefficients of the class of X in the
Chow ring of Y . This is again an improvement of the range given in [10], where the result is
however stronger, in the sense that the Picard group of X is proved to be generated by the
restriction of the Picard group of the ambient space.

As a final section, we discuss the general validity of the method. We first briefly treat some
other examples, in order to show that the method works also for non-homogeneous ambient
spaces, and that in some cases extra assumptions are needed. We conclude with some general
remarks and conjectures.

1. Subvarieties of projective space and general method

In this section we will prove (up to divisibility of the hyperplane divisor class) Barth-Larsen
theorem for the Picard group of subvarieties of projective spaces. This will illustrate the method
that we will use later for different ambient varieties.

1.1. The Barth-Larsen result. First of all, we prove that the group of divisors up to numerical
equivalence of a low codimension variety in PN has rank 1.



ON THE PICARD GROUP OF LOW-CODIMENSION SUBVARIETIES 3

Proposition 1.1. Let X be a nonsingular projective variety of dimension n in PN with N ≤
2n − 2 and let H|X denote the restriction to X of the hyperplane class of PN . Then, for any
divisor D of X, there exists q ∈ Q so that D ≡num qH|X.

Proof. Regarding PN as a linear subspace of P2n−2, we can assume N = 2n − 2. Moreover,
changing D with D + kH for a sufficient large k ∈ Z, we can also assume that D is very ample
and hence smooth. Thus, we have the following exact sequence of vector bundles on D:

0 → ND/X → ND/P2n−2 → (NX/P2n−2)|D → 0

Hence we get an equality for the top Chern classes:

(1.1) cn−1(ND/P2n−2) = c1(ND/X)cn−2((NX/P2n−2)|D)

Let d and δ be the degrees of X and D in P2n−2 respectively. In terms of the intersection of
cycles in X we have d = Hn

|X and δ = DHn−1
|X . On the other hand, the self-intersection formula

applied to each of the inclusions D ⊂ X ⊂ P2n−2 yields

cn−1(ND/PN ) = δ2 = (DHn−1
|X )2

c1(ND/X) = D|D

cn−2(NX/PN ) = dHn−2
|D

Therefore (1.1) becomes:

(1.2) (DHn−1
|X )2 = dD2Hn−2

|X = Hn
|X(D2Hn−2

|X )

where all the intersection products are considered in X.
Taking S to be the smooth irreducible surface obtained as intersection of X with n−2 general

hyperplanes, formula (1.2) becomes

(1.3) (D|SH|S)2 = D2
|SH2

|S

Thus, Hodge index theorem implies that there exists q ∈ Q such that D|S ≡num qH|S. Finally,

by Lefschetz’s hyperplane theorem, the restriction map H2(X, Q) → H2(S, Q) is injective, hence
D ≡num qH|X .

We obtain immediately from this the following consequence of the results of Barth and Larsen
(observe that we have to use only their result about simple connectedness, which is in some sense
weaker):

Corollary 1.2. In the hypothesis of Proposition 1.1, PicX ≃ Z.

Proof. Proposition 1.1 shows that PicX has rank one, so it is enough to show that it is
torsion-free. But it is an immediate consequence of the fact that, as proved in [4], X is simply
connected.



4 ENRIQUE ARRONDO AND JORGE CARAVANTES

1.2. The general method. The above proofs suggest the following strategy to study when the
Picard group is inherited (up to divisibility) by subvarieties of small codimension. Our general
set-up will be a smooth ambient space Y of dimension N ≥ 6 and a smooth subvariety X ⊂ Y

of dimension n, and we will assume N ≤ 2n − 2. For simplicity, assume that N is even, so that
after intersecting with 2n−2−N

2 general very ample divisors (and using Lefschetz’s hyperplane
theorem) we can assume N = 2n − 2. The first step now is to take a smooth divisor D and use
the exact sequence

0 → ND/X → ND/Y → (NX/Y )|D → 0

to obtain an equality

cn−1(ND/Y ) − c1(ND/X)cn−2((NX/Y )|D) = 0

This equality becomes, using the self-intersection formulas of the three inclusions D ⊂ X ⊂ Y

the equality becomes

(1.4) P := D · D − D2 · X|X = 0,

where the first summand is a product in Y and the second one is a product in X (in this way,
when D = K, this is exactly the “mysterious” equality (5) of [1] for a subvariety of G(1, 4)).
The key ingredients to prove that the divisors on X come (up to divisibility) from divisors on
Y will be then the following, corresponding respectively to Proposition 1.1 and Corollary 1.2 in
the projective case:

Step (i): Derive from (1.4) and the Hodge index formula on the surface S (the intersection of
n− 2 general very ample divisor on X) that the restriction of D to the surface S is numerically
equivalent to a linear combination of restrictions of a set of generators of PicY . This would
imply, by Lefschetz’s hyperplane theorem, that any divisor on X is numerically equivalent (with
rational coefficients) to the restriction of some divisor on Y .

Step (ii): If X is simply connected, then from (i) we get that in fact any divisor has a multiple
that is a restriction of a divisor on Y .

Observe that step (i) was immediate in the case of the projective space. In fact, equality (1.4)
is equivalent to say (via the Hodge index theorem) that the restriction of D to S is a multiple of
the hyperplane section. In general, this will be not so easy, and we will need to decompose P as
a sum of positive summands, as it happened in [1] when Y = G(1, 4). Following this example,
one should expect some of the summands to be Hodge formulas for the intersection of X with
suitable intersections of X with cycles in Y of codimension n − 2 (one extra problem here is to
show that these surfaces to which apply the Hodge index theorem are smooth irreducible). This
is in fact the trickiest part of the method, and we do not see a way to decide a priori whether
such a decomposition is possible or not.

About step (ii), as far as we know there are no general results that lead to expect that
smooth subvarieties of small codimension are simply connected. We thus decided to concentrate
in this paper on cases for which we now a priori that the simple connectedness hold. This is
one of the reasons for having chosen Grassmannians and products of projective spaces, since
Debarre proved in [6] the kind of result we needed (also the one about the irreducibility problem
mentioned in the above paragraph).

2. Subvarieties of Grassmannians of lines

In this section we apply the method we explained in the previous section to low-codimension
subvarieties in Grassmannians. We will choose Grassmannians of lines because of two reasons.
The first one is that, as we observed, it is convenient to work on an ambient space of even
dimension. The second and main reason is that we need to use Schubert calculus, and the use of



ON THE PICARD GROUP OF LOW-CODIMENSION SUBVARIETIES 5

a compact notation can be very difficult and confusing. For instance, already for Grassmannians
of lines in Pn the rank of the Chow ring in a given codimension depends on the parity of n. We
thus decided to use integral parts in our expressions, hoping that this will not make the reading
obscure; a reasonable choice could be to read this section twice: one of them thinking of n = 2k
and another one thinking of n = 2k + 1. For Grassmannians of higher dimensional subspaces it
seems better to study in each case the concrete Grassmannian that one could need to deal with.

2.1. Statement of the result and first remarks. Since we will need it continuously, we first
recall the following:

Definition A Schubert cycle Ω(a, b) (with 0 ≤ a < b ≤ n) is the class, in the Chow ring of
G(1, n), of the (a + b − 1)-dimensional Schubert variety Ω(A,B) consisting of the lines of Pn

meeting the linear subspace A and contained in the linear subspace B, where A ⊂ B ⊂ Pn,
dim A = a and dimB = b. We refer to [8] for the needed background of Schubert cycles and
their intersections.

The main result that we will prove is the following:

Theorem 2.1. Let X ⊂ G(1, n) be a smooth subvariety of dimension n′ ≥ n and let H be denote

the hyperplane class of X (after the Plücker embedding). If all intersections of X · Hn′−n with
the Schubert cycles of codimension n are different from zero, then PicX ≃ Z.

Remark 2.2. Let X ⊂ G(1, n) be a smooth irreducible subvariety of codimension two. It will
have class

[X] = a1Ω(n − 3, n) + a2Ω(n − 2, n − 1).

It is a simple exercise in Schubert calculus to see that

[X]H = a1Ω(n − 4, n) + (a1 + a2)Ω(n − 3, n − 1).

Hence, if a1 6= 0, X is in the hypothesis of Theorem 2.1 if n ≥ 5, so that it is not necessary
to impose a2 6= 0. This shows that in general, when n′ > n, it is not necessary to impose the
sufficient condition that the intersection of X with all the Schubert cycles of codimension n′ are
different from zero.

Moreover, in these conditions, if a1 = 0 this means that the union in Pn of the (2n − 4)-
dimensional family of lines parametrized by X has dimension at most n− 1. This easily implies
that X is the Schubert variety of lines contained in a hyperplane of Pn and thus its Picard group
is generated by its hyperplane section (and hence even in this case the thesis of the theorem
remains true).

Also in the case n = 4, having in mind that the intersection of X with the Schubert cycles of
lines in a general hyperplane of P4 is smooth, it is not difficult to check that a1, a2 6= 0 unless
X is the Schubert variety of lines in a general hyperplane of P4.

Summing up, Theorem 2.1 implies that the Picard group of any smooth subvariety X ⊂
G(1, n) of codimension two is generated by one element if n ≥ 4. This improves Barth, van de
Ven and Sommese results in [5] and [10] (which require respectively n ≥ 7 and n ≥ 6), although
unfortunately for n = 4, 5 we are not able to prove that we can take the hyperplane section as
a generator of the Picard group.

Remark 2.3. In general, the numerical hypothesis in Theorem 2.1 are necessary. Consider for
example the set of all the lines of a smooth quadric in P5. This is a smooth subvariety of G(1, 5)
of codimension 3 whose intersection with the Schubert cycle of all the lines passing through a
point of P5 is clearly zero. On the other hand, identifying the smooth quadric with the Plücker
embedding of G(1, 3) in P5, it follows that X is canonically isomorphic to the incidence variety
in P3 × P3∗, whose Picard group has rank two.



6 ENRIQUE ARRONDO AND JORGE CARAVANTES

We now make some reductions to prove Theorem 2.1 in the line of subsection 1.2.

Remark 2.4. Debarre has proved in [6, Corollaire 7.4] that an X in the hypothesis of Theorem
2.1 is simply connected. Hence, as explained in subsection 1.2, we can assume n′ = n and if D

is a smooth divisor of X and S is the intersection of X with n − 2 general hyperplanes, it is
enough to prove D|S ≡num qH|S for some q ∈ Q.

We will devote the rest of the section to prove Theorem 2.1 according to Remark 2.4.

2.2. General set-up. Now we state the notation for the numerical data that we will use
throughout the section. We will fix a smooth n-dimensional subvariety X ⊂ G(1, n) with a
smooth divisor D on it. We will write the classes of X and D in the Chow ring of G(1, n) as

(2.1) [X] = a1Ω(1, n) + a2Ω(2, n − 1) + . . . + a[ n
2
]Ω(

[n

2

]

,
[n + 3

2

]

)

(observe that the hypothesis about X in the statement of Theorem 2.1 is saying that all ai are
different from zero)

(2.2) [D] = α1Ω(0, n) + α2Ω(1, n − 1) + . . . + α[ n+1
2

]Ω(
[n − 1

2

]

,
[n + 2

2

]

).

A standard Schubert calculus provides then the following intersection table of cycles of X,
which also defines the numbers λi:

Ω(0, n)X Ω(1, n − 1)X . . . Ω([ n−1
2

], [ n+2
2

])X DΩ(1, n)X DΩ(2, n − 1)X . . . DΩ([ n
2
], [ n+3

2
])X

HX a1 a1 + a2 . . . a[ n−1
2

] + a[ n+1
2

] α1 + α2 α2 + α3 . . . α[ n
2
] + α[ n+2

2
]

D α1 α2 . . . α[ n+1
2

] λ1 λ2 . . . λ[ n
2
]

(where the notation Ω(i, j)X represents the restriction to X of the Schubert cycle Ω(i, j)).

Remark 2.5. Observe that, if n is odd, the term a[ n+1
2

] is not defined, so that we will take it to be

zero. In the same way, we define α[ n+2
2

] to be zero if n is even. To have a compact expression for

the matrix we can also set a0=0. As we remarked at the beginning of the section, this compact
expression, although convenient for avoiding unnecessary repetitions, can be confusing. Hence
for the convenience of the reader we write the precise form of the above matrix of intersections
depending on whether n is even or odd:

(2.3) M :=























(

a1 a1 + a2 . . . ak−1 + ak α1 + α2 α2 + α3 . . . αk

α1 α2 . . . αk λ1 λ2 . . . λk

)

if n = 2k

(

a1 a1 + a2 . . . ak α1 + α2 α2 + α3 . . . αk + αk+1

α1 α2 . . . αk+1 λ1 λ2 . . . λk

)

if n = 2k + 1

If D is numerically a multiple of HX , it is clear that the matrix M must have rank one.
Reciprocally, we claim that if M has rank one, then Theorem 2.1 follows. Indeed if there is
q ∈ Q such that the second row of M is q times the first one, this means that the product of

D′ := D − qHX

with any Ω(i, n− i)X and any DΩ(j, n+1−j)X is zero. Since Hn−1
X and Hn−2

X are respectively a
linear combination of Schubert cycles of the type Ω(i, n−i)X and Ω(j, n+1−j)X , it follows that
D′Hn−1

X = D′DHn−2
X = 0. Hence, if S is the intersection of X with n − 2 general hyperplanes,

it follows:
D′

|SH|S = D′Hn−1
X = 0



ON THE PICARD GROUP OF LOW-CODIMENSION SUBVARIETIES 7

D′
|S

2
= D′(D − qHX)Hn−2

X = 0

By Hodge index theorem, we conclude that D′
|S is numerically equivalent to zero. By Remark

2.4, Theorem 2.1 follows.
We will thus devote the rest of the section to prove that M has rank one (Corollary 2.9).

2.3. End of the proof.

Lemma 2.6. The following equality holds.

(2.4) P := α2
1 + . . . + α2

[ n+1
2

]
− a1λ1 − . . . − a[ n

2
]λ[ n

2
] = 0

Proof. This is (1.4) using (2.1), (2.2) and the intersection products given by Schubert calculus
and the matrix M .

Remark 2.7. For all i = 1, . . . , [n2 ], let Si be the surface obtained intersecting X with a general
Ω(i, n − i + 1). It is easy to see that Si is nonsingular, and by [6, Théorème 8.1] it is also
irreducible.

Applying Hodge index theorem to Si and using the fact that HΩ(i, n − i + 1) = Ω(i − 1, n −
i + 1) + Ω(i, n − i), we get

(2.5) 0 ≥ Hodgei :=

∣

∣

∣

∣

∣

H2
|Si

D|Si
H|Si

D|Si
H|Si

D2
|Si

∣

∣

∣

∣

∣

=

∣

∣

∣

∣

ai−1 + ai αi + αi+1

αi λi

∣

∣

∣

∣

+

∣

∣

∣

∣

ai + ai+1 αi + αi+1

αi+1 λi

∣

∣

∣

∣

(as usual, one should be careful with the correct expression of Hodgei in the cases i = 1, [n2 ]; in
particular, notice that, if n is even and i = [n2 ], then the correct definition of Hodgei consists
only of the first summand of (2.5)).

Lemma 2.8. With P and Hodgei defined by (2.4) and (2.5), the following equality holds:

(2.6) P = −

[ n
2
]

∑

i=1

ai

ai−1 + 2ai + ai+1
Hodgei +

[ n+1
2

]−1
∑

i=1

ai

∣

∣

∣

∣

ai−1 + ai ai + ai+1

αi αi+1

∣

∣

∣

∣

2

(ai−1 + 2ai + ai+1)(ai−1 + ai)(ai + ai+1)

Proof. We regard the expressions as polynomials in the indeterminates α1, . . . , α[ n+1
2

], λ1, . . . ,

λ[ n
2
] with coefficients in the quotient field of C[a1, . . . , ak]. The only non-zero coefficients are those

of the monomials of the type α2
i , αiαi+1, λi. Hence it suffices to check that the corresponding

coefficients in the left and right hand sides coincide. This is trivial for the coefficient of λi, which
is ai in both sides.

For αiαi+1, its coefficient is zero in the left-hand side of (2.6), while in the right-hand side it
is:

2
ai

ai−1 + 2ai + ai+1
− 2

ai

(ai−1 + 2ai + ai+1)(ai−1 + ai)(ai + ai+1)
(ai + ai+1)(ai−1 + ai)

which is also zero.
For α2

i , its coefficient in the left-hand side is 1, while in the right-hand side is:

ai−1

ai−2 + 2ai−1 + ai
+

ai

ai−1 + 2ai + ai+1
+

ai−1(ai−2 + ai−1)
2

(ai−2 + 2ai−1 + ai)(ai−2 + ai−1)(ai−1 + ai)

+
ai(ai + ai+1)

2

(ai−1 + 2ai + ai+1)(ai−1 + ai)(ai + ai+1)



8 ENRIQUE ARRONDO AND JORGE CARAVANTES

and a simple calculation shows that this is also 1.
We leave to the reader to check that all these equalities remain valid when i takes the minimum

and maximum values; a special care should be taken when n is even, because (as observed at
the end of Remark 2.7), the expression ai−1 + 2ai + ai+1 appearing in both sums of (2.6) must
be regarded as (ai−1 + ai) + (ai + ai+1), thus just ai−1 + ai if i = n

2 .

Corollary 2.9. The matrix M of (2.3) has rank one.

Proof. The left-hand side of (2.6) is zero by Lemma 2.6, while the right-hand side is a sum of
nonnegative terms by (2.5). Therefore we get equalities

∣

∣

∣

∣

ai−1 + ai αi + αi+1

αi λi

∣

∣

∣

∣

+

∣

∣

∣

∣

ai + ai+1 αi + αi+1

αi+1 λi

∣

∣

∣

∣

= 0

∣

∣

∣

∣

ai−1 + ai ai + ai+1

αi αi+1

∣

∣

∣

∣

= 0

(observe that for this it was crucial the hypothesis ai 6= 0 of Theorem 2.1). The first equalities
imply that the columns of M containing a λi depend on the other columns, while the second
inequalities show that the columns not containing any λi span a one-dimensional subspace.

3. Subvarieties of products of projective spaces

In this section we will apply our method to a product of two projective spaces of the same
dimension. The main result we will prove is the following:

Theorem 3.1. Let X ⊂ Pn−1 × Pn−1 be smooth of dimension greater than or equal to n such
that the two projection maps X → Pn−1 are surjective. Then PicX is a free abelian group of
rank two, in which the pullbacks H1 and H2 of the hyperplane classes of each Pn−1 are linearly
independent.

Remark 3.2. It is clear that the hypothesis of the surjectivity of the projections X → Pn−1 are
necessary. For instance, if the image of the first projection is contained in a subvariety Z ⊂ Pn−1

with Picard group of rank at least two, then we can find many smooth subvarieties X ⊂ Z×Pn−1

whose Picard group has rank at least three (for example taking successive hyperplane sections of
the Segre embedding of Pn−1 ×Pn−1). To give a concrete example, just consider the embedding
P1 × P1 × P1 × P1 ⊂ P3 × P3, (in which we identify the first and last P1 × P1 with a smooth
quadric in P3 after the Segre embedding).

On the other hand, without any condition about surjectivity, the results of [10] show that
PicX is generated by H1 and H2 if dim X ≥ 3n−1

2 .

Remark 3.3. After the result of Debarre showing that X is simply connected [6, Corollaire
2.4], as remarked in subsection 1.2 it is enough to assume dimX = n and prove that every
smooth divisor D on X restricted to the intersection of X with n − 2 general hyperplanes (i.e.
divisors in the class H1 +H2) is numerically equivalent to a rational combination of H1 and H2.
Observe that H1,H2 are linearly independent, since Hn

1 = Hn
2 = 0 and (H1 + H2)

n > 0.



ON THE PICARD GROUP OF LOW-CODIMENSION SUBVARIETIES 9

3.1. General set-up. We fix now the general notation that we will use throughout the section.
Let X be a smooth n-dimensional subvariety of Pn−1 × Pn−1 such that the two projections
X → Pn−1 are surjective. Let D be a smooth divisor of X and let H1,H2 be the pullback to
X of the hyperplane classes of Pn−1. We define ai, αi, and λi as the intersection products in X

according to the following table (sometimes it will be practical to set a0 = an = 0 in order to
get compact expressions):

Hn−1

1
Hn−2

1
H2 Hn−3

1
H2

2
. . . Hn−1

2
DHn−2

1
DHn−3

1
H2 DHn−4

1
H2

2
. . . DHn−2

2

H1

H2

D

0
a1

α1

a1

a2

α2

a2

a3

α3

. . .

. . .

. . .

an−1

0
αn

α1

α2

λ1

α2

α3

λ2

α3

α4

λ3

. . .

. . .

. . .

αn−1

αn

λn−1

If D depends numerically on H1 and H2, it is clear that the intersection matrix

(3.1) M :=





0 a1 a2 . . . an−1 α1 α2 α3 . . . αn−1

a1 a2 a3 . . . 0 α2 α3 α4 . . . αn

α1 α2 α3 . . . αn λ1 λ2 λ3 . . . λn−1





must have rank at least two. We claim that in fact if M has rank two then Theorem 3.1 follows.

Indeed, observe first that

∣

∣

∣

∣

0 an−1

a1 0

∣

∣

∣

∣

6= 0, since this is exactly the hypothesis that the projections

X → Pn−1 are surjective. Hence, if M has rank two, it follows that the third row is a linear
combination of the first two. This implies that there are p, q ∈ Q such that the product of

D′ := D − pH1 − qH2

with Hn−1
1 ,Hn−2

1 H1, . . . ,H
n−1
2 ,DHn−2

1 ,DHn−3
1 H2, . . . ,DHn−2

2 is zero. If S is the intersection
of X with n − 2 general hyperplanes, then it follows that

D′
|S(H1 + H2)|S = D′(H1 + H2)

n−1 = 0

D′
|S

2
= D′(D − pH1 − qH2)(H1 + H2)

n−2 = 0.

Therefore, by Hodge index theorem D′
|S is numerically equivalent to zero. By Remark 3.3, this

implies Theorem 3.1.
We will thus devote the rest of the section to prove that M has rank two (Corollary 3.16).

3.2. First numerical properties. In this subsection we prove the first relations between the
numerical invariants of a smooth subvariety X ⊂ Pn−1 × Pn−1 of dimension n.

Observe that the intersection matrix (3.1) shows that the classes of X and D in the Chow
ring of Pn−1 × Pn−1 are

(3.2) [X] = an−1H
n−2
1 + an−2H

n−3
1 H2 + . . . + a1H

n−2
2

(3.3) [D] = αnHn−1
1 + αn−1H

n−2
1 H2 + . . . + α1H

n−1
2

(we abuse momentarily the notation by indicating with H1,H2 the hyperplane classes in each
Pn−1, instead of denoting their pullback to X). This implies the following:

Lemma 3.4. The following equality holds

(3.4) P := α1αn + α2αn−1 + . . . + αnα1 − a1λn−1 − a2λn−2 − . . . − an−1λ1 = 0



10 ENRIQUE ARRONDO AND JORGE CARAVANTES

Proof. This is (1.4) using (3.2), (3.3) and the intersection products given by the matrix M of
(3.1).

A first important observation about our numerical invariants is that, in our situation, all the
numbers a1, . . . , an−1 are different from zero:

Lemma 3.5. Let X ⊂ Pn−1 × Pn−1 be irreducible of dimension n. Let a1, . . . , an−1 be defined
as above. Suppose that a1, an−1 are different from zero. Then all ai are strictly positive.

Proof. See [6, Théorème 1.3].

Definition For i = 1, . . . , n − 1, we will denote Si to the surface obtained intersecting X with
a general Pi × Pn−i (hence the class of Si in X is Hn−i−1

1 H i−1
2 ). Since both natural projections

restricted to X are surjective, it follows from [6, Théorème 1.3] that Si is a smooth irreducible
surface.

Remark 3.6. The Hodge index theorem applied to the divisors H1,H2 restricted to Si implies
the inequality:

∣

∣

∣

∣

ai−1 ai

ai ai+1

∣

∣

∣

∣

=

∣

∣

∣

∣

∣

H1
2
|Si

H1|Si
H2|Si

H1|Si
H2|Si

H2
2
|Si

∣

∣

∣

∣

∣

≤ 0

with equality if and only if H1 and H2 are numerically dependent on Si. Lemma 3.5 allows us
to write all these inequalities as:

(3.5)
a1

a2
≤

a2

a3
≤ . . . ≤

an−2

an−1
.

Thus it follows that, for all i < j, we have
∣

∣

∣

∣

ai−1 aj

ai aj+1

∣

∣

∣

∣

≤ 0

with equality if and only if
∣

∣

∣

∣

ai−1 ai

ai ai+1

∣

∣

∣

∣

=

∣

∣

∣

∣

ai ai+1

ai+1 ai+2

∣

∣

∣

∣

= . . . =

∣

∣

∣

∣

aj−1 aj

aj aj+1

∣

∣

∣

∣

= 0

i.e.
ai−1

ai
=

ai

ai+1
= . . . =

aj

aj+1
.

We suspect that the inequalities in (3.5) are never equalities. However we are not able to prove
it, so that our proofs will become sometimes more complicated when considering the cases in
which some equalities hold.

Remark 3.7. We apply now the Hodge index theorem to the restriction to Si of the divisors
H1,H2,D and get

(3.6) ri,i+1,n+i :=

∣

∣

∣

∣

∣

∣

ai−1 ai αi

ai ai+1 αi+1

αi αi+1 λi

∣

∣

∣

∣

∣

∣

=

∣

∣

∣

∣

∣

∣

∣

H2
1|Si

H1|Si
H2|Si

H1|Si
D|Si

H1|Si
H2|Si

H2
2|Si

H2|Si
D|Si

H1|Si
D|Si

H2|Si
D|Si

D2
|Si

∣

∣

∣

∣

∣

∣

∣

≥ 0

(in general ri,j,k will denote the minor of the columns i, j, k of the matrix M) with equality if and
only if H1, H2 and D are numerically dependent when restricted to Si. Hence the vanishing of
ri,i+1,n+i would immediately prove that the restriction of D to Si is numerically a combination
of H1 and H2, unless the restrictions to Si of H1 and H2 are numerically dependent.



ON THE PICARD GROUP OF LOW-CODIMENSION SUBVARIETIES 11

In this latter case (which occurs if and only if ai−1

ai
= ai

ai+1
, by Remark 3.6) the restriction of

D to Si is numerically a combination of H1 and H2 if and only if it is just a multiple of H1. By
the Hodge index theorem, this is equivalent to the inequality:

(3.7) r̃i :=

∣

∣

∣

∣

ai−1 αi

αi λi

∣

∣

∣

∣

=

∣

∣

∣

∣

∣

H2
1|Si

H1|Si
D|Si

H1|Si
D|Si

D2
|Si

∣

∣

∣

∣

∣

≤ 0

to be an equality. Moreover, it easily follows that

rk

(

ai−1 ai αi

ai ai+1 αi+1

)

= 1

since this is the multiplication matrix in Si of the divisors H1,H2 (which are numerically de-
pendent) with H1,H2,D. In particular

(3.8) αi+1 =
ai+1

ai
αi

Remark 3.8. The above remark shows that we should check how many equalities we have in
the chain (3.5). A practical way to control this is to define the function:

(3.9)

σ : {2, . . . , n} −→ {2, . . . , n}

i 7→
unique l such that

al−2

al−1
<

al−1

al
= . . . =

ai−2

ai−1
=

ai−1

ai

In this way, the equality ai−1

ai
=

aj−1

aj
is equivalent to the equality σ(i) = σ(j). In other words,

the equalities in (3.5) coincide with the equalities in

σ(2) ≤ σ(3) ≤ . . . ≤ σ(n).

Observe that saying that there are no equalities in (3.5) (as we said, this is what we suspect) is
equivalent to saying that σ is the identity map.

We recall from Remark 3.6 that
aσ(i)−2

aσ(i)−1
<

aσ(i)−1

aσ(i)
= . . . = ai−2

ai−1
= ai−1

ai
is equivalent to

∣

∣

∣

∣

aσ(i)−1 ai−1

aσ(i) ai

∣

∣

∣

∣

= 0,

∣

∣

∣

∣

aσ(i)−2 ai−1

aσ(i)−1 ai

∣

∣

∣

∣

< 0. On the other hand, iterating (3.8) we get

(3.10) αi =
ai

aσ(i)
ασ(i)

The idea of the above function σ is that it allows to determine “virtual” equalities in the chain
(3.5), and then perform the substitutions given in (3.10). This will be very useful in the next
section, when a1, . . . , an−1 will be indeterminates rather than concrete values. This motivates
the following definition, which indicates when a function σ is obtained as in the above remark:

Definition We call partition map to a function σ : {2, . . . , n} → {2, . . . , n} satisfying:

(1) σ(i) ≤ i for all i

(2) for all j ∈ {σ(i), . . . , i}, σ(j) = σ(i).

3.3. Polynomial equalities. Throughout this subsection, we will consider ai, αi, λi as inde-
terminates, and we will consider the polynomial ring Q(a1, . . . , an−1)[α1, . . . , αn, λ1, . . . , λn−1].
In this context, the elements of Q(a1, . . . , an−1) will be regarded as coefficients (our convention
a0 = an = 0 will be useful again for writing the expressions in a compact way). We define the
polynomials (compare with (3.4), (3.6) and (3.7)):

P := α1αn + α2αn−1 + . . . + αnα1 − a1λn−1 − a2λn−2 − . . . − an−1λ1



12 ENRIQUE ARRONDO AND JORGE CARAVANTES

ri,i+1,n+i :=

∣

∣

∣

∣

∣

∣

ai−1 ai αi

ai ai+1 αi+1

αi αi+1 λi

∣

∣

∣

∣

∣

∣

r̃i :=

∣

∣

∣

∣

ai−1 αi

αi λi

∣

∣

∣

∣

ri,i+1,n−i+1 :=

∣

∣

∣

∣

∣

∣

ai−1 ai an−i

ai ai+1 an−i+1

αi αi+1 αn−i+1

∣

∣

∣

∣

∣

∣

and the coefficients

(3.11) ci :=
an−i

∣

∣

∣

∣

ai−1 ai

ai ai+1

∣

∣

∣

∣

(3.12) di :=
aian−i

∣

∣

∣

∣

ai−1 ai

ai ai+1

∣

∣

∣

∣

∣

∣

∣

∣

ai−1 an−i

ai an−i+1

∣

∣

∣

∣

∣

∣

∣

∣

ai an−i

ai+1 an−i+1

∣

∣

∣

∣

Observe that, if i = ⌈n
2 ⌉ then ri,i+1,n−i+1 = 0 and di is undefined. We will take both of them

equal to zero.

Lemma 3.9. The following equality holds in Q(a1, . . . , an−1)[α1, . . . , αn, λ1, . . . , λn−1]:

(3.13) P = −
n−1
∑

i=1

ciri,i+1,n+i −
n−1
∑

i=1

dir
2
i,i+1,n−i+1

Proof. We have to compare the coefficients of the different monomials. The only monomials
appearing in (3.13) are of the type λi, α

2
i , αiαi+1, αiαn−i+1 and αiαn−i+2. It is immediate that

the coefficient of λi is −an−i in both the left and right hand sides of (3.13).
For α2

i (we do not consider the case i = 1, ⌈n
2 ⌉, ⌈

n
2 ⌉ + 1, n, which overlaps with other cases),

we have zero on the left-hand side, while the coefficient on the right-hand side comes from
ri−1,i,n+i−1, ri,i+1,n+i, r2

i−1,i,n−i+2, r2
n−i+1,n−i+2,i and r2

i,i+1,n−i+1 and it is

ci−1ai−2 + ciai+1 − di−1

∣

∣

∣

∣

ai−2 an−i+1

ai−1 an−i+2

∣

∣

∣

∣

2

− di

∣

∣

∣

∣

ai an−i

ai+1 an−i+1

∣

∣

∣

∣

2

− dn−i+1

∣

∣

∣

∣

an−i an−i+1

an−i+1 an−i+2

∣

∣

∣

∣

2

which is also zero.
For αiαi+1, (we assume now i 6= ⌈n

2 ⌉, ⌈
n
2 ⌉ + 1 in order to avoid overlaps with other cases) we

have again zero on the left-hand side, while the coefficient on the right-hand side comes from
ri,i+1,n+i and r2

i,i+1,n−i+1, and it is

−2ciai + 2di

∣

∣

∣

∣

ai−1 an−i

ai an−i+1

∣

∣

∣

∣

∣

∣

∣

∣

ai an−i

ai+1 an−i+1

∣

∣

∣

∣

which is obviously zero.
For αiαn−i+1 (assuming now i 6= 1, ⌈n

2 ⌉, ⌈
n
2 ⌉ + 1), the coefficient on the left-hand side is 2,

while on the right-hand side, it comes from r2
i,i+1,n−i+1 and r2

n−i+1,n−i+2,i, and it is

−2di

∣

∣

∣

∣

ai an−i

ai+1 an−i+1

∣

∣

∣

∣

∣

∣

∣

∣

ai−1 ai

ai ai+1

∣

∣

∣

∣

+ 2dn−i+1

∣

∣

∣

∣

an−i an−i+1

an−i+1 an−i+2

∣

∣

∣

∣

∣

∣

∣

∣

ai−1 an−i+1

ai an−i+2

∣

∣

∣

∣

which is also 2.



ON THE PICARD GROUP OF LOW-CODIMENSION SUBVARIETIES 13

For αiαn−i+2 (again with i 6= ⌈n
2 ⌉, ⌈

n
2 ⌉ + 1), its coefficient is zero in the left-hand side, while

it appears only in the right-hand side in r2
i−1,i,n−i+2 and r2

n−i+1,n−i+2,i, with coefficient:

2di−1

∣

∣

∣

∣

ai−2 an−i+1

ai−1 an−i+2

∣

∣

∣

∣

∣

∣

∣

∣

ai−2 ai−1

ai−1 ai

∣

∣

∣

∣

+ 2dn−i+1

∣

∣

∣

∣

an−i an−i+1

an−i+1 an−i+2

∣

∣

∣

∣

∣

∣

∣

∣

ai−1 an−i

ai an−i+1

∣

∣

∣

∣

which is zero.
We are left with the coefficients of the cases we have excluded, i.e. the ones of the monomials

α2
1, α1αn, α2

k, αkαk+1, αkαk+2, α
2
k+1, αk+1αk+2 and α2

n, with k = ⌈n
2 ⌉. We omit the proof, which

is as before a simple computation, but depending now on whether n = 2k or n = 2k − 1 for the
cases involving k in an index.

Thinking of our application to subvarieties of Pn−1 ×Pn−1, Lemma 3.9 could be the analogue
of Lemma 2.8, but a priori the denominators of some coefficient ci, di could vanish, depending
on whether we have equalities in the chain (3.5). If this happens, the idea would be to use, when
necessary, the inequalities (3.7) instead of (3.6) and perform the substitutions (3.10). In order
to keep all these substitutions in the framework of polynomials, we fix a partition map σ for the
rest of the subsection.

The first case in which we would be in trouble is when the denominator of ci vanishes (i.e.
when σ(i) = σ(i + 1), in terms of the partition map). As we observed in Remark 3.7, in this
case we should replace ri,i+1,n+i with r̃i (with an appropriate coefficient to fit in (3.13)). On the
other hand, the denominator of di would also create problems, so it looks convenient to remove
the term dir

2
i,i+1,n−i+1 from (3.13). The precise way of performing these changes is given by the

following:

Lemma 3.10. Suppose σ(i) = σ(i + 1) for some i = 1, . . . , n − 1 and set:

li :=

an−i

∣

∣

∣

∣

ai an−i+1

αi αn−i+1

∣

∣

∣

∣

2 ∣

∣

∣

∣

ai−1 ai

ai ai+1

∣

∣

∣

∣

ai

∣

∣

∣

∣

ai−1 an−i

ai an−i+1

∣

∣

∣

∣

∣

∣

∣

∣

ai an−i

ai+1 an−i+1

∣

∣

∣

∣

mi :=

an−i

∣

∣

∣

∣

ai ai+1

αi αi+1

∣

∣

∣

∣

(aian−i+1αi+1 − 2aiai+1αn−i+1 + ai+1an−i+1αi)

aiai+1

∣

∣

∣

∣

ai an−i

ai+1 an−i+1

∣

∣

∣

∣

Then,

−ciri,i+1,n+i − dir
2
i,i+1,n−i+1 = −

an−i

ai+1
r̃i − li − mi

As a consequence,

(3.14) P = −
∑

σ(i)<σ(i+1)

ciri,i+1,n+i −
∑

σ(i)=σ(i+1)

an−i

ai+1
r̃i−

−
∑

σ(i)<σ(i+1)

dir
2
i,i+1,n−i+1 −

∑

σ(i)=σ(i+1)

li −
∑

σ(i)=σ(i+1)

mi

Proof. The first part is a straightforward computation, while the second one follows immedi-
ately from the first one and (3.13).

Observe that we can still have problems with the denominator of li and mi if σ(n − i + 1)
equals σ(i) = σ(i + 1). In the same way, even if σ(i) < σ(i + 1) we can have problems with the



14 ENRIQUE ARRONDO AND JORGE CARAVANTES

denominator of di if σ(n − i + 1) equals σ(i) or σ(i + 1). Let us see that we can avoid these
problems by performing the substitutions suggested by (3.10). We thus make the following:

Definition Given any polynomial Q ∈ Q(a1, . . . , an−1)[α1, . . . , αn, λ1, . . . , λn−1], we define a
new polynomial (depending on few variables):

Qσ := Q(a1, . . . , an−1, α1,
a2

aσ(2)
ασ(2), . . . ,

an−1

aσ(n−1)
ασ(n−1), αn, λ1, . . . , λn−1)

We see now how these substitutions affect the problematic terms in (3.14).

Lemma 3.11. The following identities hold:
(i) If σ(i) = σ(i + 1), then mσ

i = 0.
(ii) If σ(i) = σ(i + 1) = σ(n − i + 1), then lσi = 0.

(iii) If σ(i) < σ(i + 1) = σ(n − i + 1), then rσ
i,i+1,n−i+1 = −ai

aσ(i)ai+1

∣

∣

∣

∣

ai an−i

ai+1 an−i+1

∣

∣

∣

∣

∣

∣

∣

∣

aσ(i) ai+1

ασ(i) αi+1

∣

∣

∣

∣

and hence

(3.15) gi := di(r
σ
i,i+1,n−i+1)

2 =

a3
i an−i

∣

∣

∣

∣

ai an−i

ai+1 an−i+1

∣

∣

∣

∣

∣

∣

∣

∣

aσ(i) ai+1

ασ(i) αi+1

∣

∣

∣

∣

2

a2
σ(i)a

2
i+1

∣

∣

∣

∣

ai−1 an−i

ai an−i+1

∣

∣

∣

∣

∣

∣

∣

∣

ai−1 ai

ai ai+1

∣

∣

∣

∣

(iv) If σ(n − i + 1) = σ(i) < σ(i + 1), then rσ
i,i+1,n−i+1 = −1

aσ(i)

∣

∣

∣

∣

ai−1 an−i

ai an−i+1

∣

∣

∣

∣

∣

∣

∣

∣

aσ(i) ai+1

ασ(i) αi+1

∣

∣

∣

∣

and

hence

(3.16) hi := di(r
σ
i,i+1,n−i+1)

2 =

aian−i

∣

∣

∣

∣

ai−1 an−i

ai an−i+1

∣

∣

∣

∣

∣

∣

∣

∣

aσ(i) ai+1

ασ(i) αi+1

∣

∣

∣

∣

2

a2
σ(i)

∣

∣

∣

∣

ai an−i

ai+1 an−i+1

∣

∣

∣

∣

∣

∣

∣

∣

ai−1 ai

ai ai+1

∣

∣

∣

∣

Proof. This is just a straightforward computation.

Corollary 3.12. With the above notations,

(3.17) P σ = −
∑

σ(i)<σ(i+1)

cir
σ
i,i+1,n+i −

∑

σ(i)=σ(i+1)

an−i

ai+1
r̃σ
i −

−
∑

σ(n−i+1)6=σ(i)<σ(i+1)6=σ(n−i+1)

di(r
σ
i,i+1,n−i+1)

2 −
∑

σ(i)<σ(i+1)=σ(n−i+1)

gi−

−
∑

σ(n−i+1)=σ(i)<σ(i+1)

hi −
∑

σ(i)=σ(i+1)6=σ(n−i+1)

lσi

Proof. Just apply Lemma 3.11 to the equality (3.14) of Lemma 3.10.

3.4. End of the proof. After the previous technical subsection we come back to the geometrical
situation of a subvariety X ⊂ Pn−1 × Pn−1 with the usual invariants. We can now state and
prove the analogue of Lemma 2.8.



ON THE PICARD GROUP OF LOW-CODIMENSION SUBVARIETIES 15

Lemma 3.13. The following equality holds:

(3.18) 0 = −
∑

ai−1
ai

<
ai

ai+1

an−i
∣

∣

∣

∣

ai−1 ai

ai ai+1

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

ai−1 ai αi

ai ai+1 αi+1

αi αi+1 λi

∣

∣

∣

∣

∣

∣

−
∑

ai−1
ai

=
ai

ai+1

an−i

ai+1

∣

∣

∣

∣

ai+1 αi+1

αi+1 λi

∣

∣

∣

∣

−

−
∑

an−i
an−i+1

6=
ai−1

ai
<

ai
ai+1

6=
an−i

an−i+1

aian−i
∣

∣

∣

∣

ai−1 ai

ai ai+1

∣

∣

∣

∣

∣

∣

∣

∣

ai an−i

ai+1 an−i+1

∣

∣

∣

∣

∣

∣

∣

∣

ai−1 an−i

ai an−i+1

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

ai−1 ai an−i

ai ai+1 an−i+1

αi αi+1 αn−i+1

∣

∣

∣

∣

∣

∣

2

and therefore

(3.19)

∣

∣

∣

∣

∣

∣

ai−1 ai αi

ai ai+1 αi+1

αi αi+1 λi

∣

∣

∣

∣

∣

∣

= 0 if ai−1

ai
< ai

ai+1

(3.20)

∣

∣

∣

∣

ai+1 αi+1

αi+1 λi

∣

∣

∣

∣

= 0 if ai−1

ai
= ai

ai+1

(3.21)

∣

∣

∣

∣

∣

∣

ai−1 ai an−i

ai ai+1 an−i+1

αi αi+1 αn−i+1

∣

∣

∣

∣

∣

∣

= 0 if an−i

an−i+1
6= ai−1

ai
< ai

ai+1
6= an−i

an−i+1

Proof. Let σ be the partition map defined by (3.9). Then (3.18) is an easy consequence
of the equality (3.17) in Corollary 3.12. Indeed, observe first that, by (3.10), any polynomial
Q ∈ Q(a1, . . . , an−1)[α1, . . . , αn, λ1, . . . , λn−1], takes the same value as Qσ when applied to the
invariants of X, provided that all denominators are different from zero. In particular, P σ vanishes
thanks to Lemma 3.4. We thus apply Corollary 3.12 and analyze each of the summands of (3.17):

It is clear that the first three summands of (3.17) coincide with the summands of (3.18) and
that their denominators do not vanish (recall that this is equivalent to say σ(i) < σ(i + 1)).
For the other summands of (3.17), if ai−1

ai
< ai

ai+1
= an−i

an−i+1
or an−i

an−i+1
= ai−1

ai
< ai

ai+1
, then

di(r
σ
i,i+1,n−i+1)

2 takes the value gi or hi respectively of (3.15) and (3.16), whose denomina-

tors do not vanish, but their numerators contain respectively the factor

∣

∣

∣

∣

ai−1 an−i

ai an−i+1

∣

∣

∣

∣

and
∣

∣

∣

∣

ai an−i

ai+1 an−i+1

∣

∣

∣

∣

, so that they vanish; and finally, if ai−1

ai
= ai

ai+1
6= an−i

an−i+1
, then the denomina-

tor of li, as defined in Lemma 3.10 does not vanish, while the numerator contains the factor
∣

∣

∣

∣

ai−1 ai

ai ai+1

∣

∣

∣

∣

, hence it vanishes.

Observe now that all the summands of (3.18) are nonnegative by Remark 3.7, which implies
the equalities (3.19), (3.20) and (3.21) (again, it was crucial to know that all ai are not zero).

We are now ready to prove that the matrix M of (3.1) has rank two. We will do it in several
steps.



16 ENRIQUE ARRONDO AND JORGE CARAVANTES

Corollary 3.14. For each i = 1, . . . , n, i 6= ⌈n
2 ⌉, define the following submatrix of M :

M ′
i :=











































ai−1 ai an−i

ai ai+1 an−i+1

αi αi+1 αn−i+1



 if i < n − i





an−i ai−1 ai

an−i+1 ai ai+1

αn−i+1 αi αi+1



 if n − i < i − 1

Then the middle column of M ′
i is a linear combination of the two others.

Proof. This is immediate if an−i

an−i+1
6= ai−1

ai
< ai

ai+1
6= an−i

an−i+1
, since in this case any two columns

of M ′
i are linearly independent, and (3.21) implies that any of the columns of M ′

i is a linear

combination of the two others. It is also immediate if
ai−1

ai
= ai

ai+1
, since in this case





ai−1

ai

αi



 and





ai

ai+1

αi+1



 are multiple of each other (by the second part of Remark 3.7) and one of them is the

middle column. We are thus left with the cases ai−1

ai
< ai

ai+1
= an−i

an−i+1
and an−i

an−i+1
= ai−1

ai
< ai

ai+1
,

which are proved in the same way using again Remark 3.7.

Corollary 3.15. The submatrix

M ′ :=





0 a1 a2 . . . an−2 an−1

a1 a2 a3 . . . an−1 0
α1 α2 α3 . . . αn−1 αn





of M has rank two.

Proof. It is enough to prove for each i = 1, . . . , n that the i-th column of M ′ is a linear
combination of the first and last columns (which in turn are linearly independent). We will
prove it by induction on d(i) := min{i − 1, n − i} (i.e. the distance of i to the border of the
interval [1, n]). Of course, the statement is trivial for d(i) = 0 (i.e. i = 1 or i = n).

Assume first i < n+2
2 (hence d(i) = i − 1). Then Corollary 3.14 implies that the i-th column

is a linear combination of the (i − 1)-th and (n − i + 2)-th columns. By induction hypothesis
(observe that d(i− 1) = d(n− i+ 2) = i− 2), these two columns are in turn linear combinations
of the first and last columns of M , which completes the proof of this case.

If instead i > n+1
2 (hence d(i) = n − i), Corollary 3.14 implies now that the i-th column is

a linear combination of the (n − i + 1)-th and (i + 1)-th columns. The (i + 1)-th column is,
again by induction hypothesis, a linear combination of the first and last columns of M . For the
(n − i + 1)-th column we observe that d(n − i + 1) = n − i = d(i) and that n − i + 1 < n+2

2 .
Hence we are in the situation of the previous case, for which we already proved the inductive
step.

Corollary 3.16. The matrix M of (3.1) has rank two.

Proof. By Corollary 3.15, it is enough to prove that any column of M containing a λi is a
linear combination of the columns of M ′. To this purpose, for i = 1, . . . , n − 1 we consider the



ON THE PICARD GROUP OF LOW-CODIMENSION SUBVARIETIES 17

submatrix:

Mi :=





ai−1 ai αi

ai ai+1 αi+1

αi αi+1 λi





If
ai−1

ai
< ai

ai+1
, then the two first columns of Mi are linearly independent, and (3.19) proves

that the last column of Mi is a linear combination of the first two, as wanted.
If instead ai−1

ai
= ai

ai+1
, then by (3.8)

∣

∣

∣

∣

ai αi

ai+1 αi+1

∣

∣

∣

∣

= 0

which, together with (3.20) and ai+1 6= 0, implies that the last column of Mi is a multiple of the
middle one.

4. Further examples and remarks

We start with several examples that illustrate until which point the method explained in
subsection 1.2 can work. In two of them, we will not have any result stating that subvarieties
in the chosen ambient space Y are simply connected, so that step (ii) of the method cannot be
applied. Anyway, the key point in the method is step (i), in order to prove that the divisors on
subvarieties of X come numerically from divisors on Y . We will not do all the details, which
can be found in the forthcoming PhD thesis of the second author. We start with a negative
example.

Example 4.1. Let X ⊂ Q2n−2 be a smooth n-dimensional subvariety of the smooth (2n − 2)-
dimensional quadric. It is well known that the class of X in the Chow ring of Q2n−2 takes the
form dHn−2, where H is the hyperplane class. On the other hand, if D is a (smooth) divisor
on X, its class as a cycle in Q2n−2 will be [D] = α1A1 + α2A2, where A1, A2 are the classes of
the two families of (n − 1)-dimensional linear spaces contained in Q2n−2. Having in mind the
intersection of two linear spaces in Q2n−2 depending on the parity of n, equality (1.4) becomes

P :=







2α1α2 − dD2Hn−2 = 0 if n is even

α2
1 + α2

2 − dD2Hn−2 = 0 if n is odd

If S is the intersection of X with n − 2 general hyperplanes we can thus write (observe that
DSHS = DHn−1 = α1 + α2)

P =







1
2

(

(DSHS)2 − D2
SH2

S

)

− 1
2(α1 − α2)

2 if n is even

1
2

(

(DSHS)2 − D2
SH2

S

)

+ 1
2(α1 − α2)

2 if n is odd.

This proves (using the Hodge index theorem for S) that, when n is odd, any divisor on X is
numerically a multiple of the hyperplane section; moreover, since X is a smooth subvariety of
P2n−1, it follows from [4] that X is simply connected, and hence we conclude that its Picard
group is isomorphic to Z. However, when n is even, we get that D is a multiple of H only when
α1 = α2, i.e. DA1 = DA2. Since the equality DA1 = DA2 defines one linear condition on PicX

(which could be identically zero), it follows that PicX has rank at most two (we need to use
again that X is simply connected). The rank-two case can actually occur: consider the Segre



18 ENRIQUE ARRONDO AND JORGE CARAVANTES

embedding P1 × Pn−1 →֒ P2n−1, whose image can be defined as the subvariety X consisting of
those point for which the matrix

(

X0 X1 . . . Xn−1

Xn Xn+1 . . . X2n−1

)

has rank one. Since n is even, it follows that X is contained in the smooth quadric Q2n−2 of
equation

∣

∣

∣

∣

X0 X1

Xn Xn+1

∣

∣

∣

∣

+

∣

∣

∣

∣

X2 X3

Xn+2 Xn+3

∣

∣

∣

∣

+ . . . +

∣

∣

∣

∣

Xn−2 Xn−1

X2n−2 X2n−1

∣

∣

∣

∣

= 0.

Strengthening Conjecture 4.5 of [1], we suspect that the above example is the only smooth
n-dimensional subvariety X ⊂ Q2n−2 whose Picard group is not generated by H.

The next two examples are chosen to see that the method can keep working even for nonho-
mogeneous ambient spaces.

Example 4.2. Let Y be the blowing up of P6 at a point. Then the Picard group (and in fact
the whole Chow ring) of Y is generated by H (the pullback of the hyperplane section of P6) and
E (the exceptional divisor). Let X ⊂ Y be a smooth codimension-two subvariety and let D be
a smooth divisor on X. We write their classes in the Chow ring of Y as:

[X] = a1H
2 + a2E

2

[D] = α1H
3 + α2E

3

From the intersection table in X:

(4.1)

H3 E3 DH2 DE2

H

E

D

a1

0
α1

0
−a2

−α2

α1

0
λ1

0
−α2

λ2

we get that (1.4) takes the form:

P := α2
1 − α2

2 − a1λ1 − a2λ2 = 0

Then we see that we can get a decomposition

P = −
a1 − a2

a1

∣

∣

∣

∣

a1 α1

α1 λ1

∣

∣

∣

∣

+
1

a1

∣

∣

∣

∣

∣

∣

a1 0 α1

0 −a2 −α2

α1 −α2 λ1 + λ2

∣

∣

∣

∣

∣

∣

The first determinant in this decomposition corresponds to the intersection matrix of the divisors
H and D in the surface obtained as the intersection of X with two general divisors of class H

(this is clearly a smooth irreducible surface and H is very ample on it), hence by Hodge index
theorem this determinant is nonpositive. Similarly, the second determinant corresponds to the
intersection matrix of the divisors H, E and D in the surface obtained as the intersection of
X with two general divisors of class H − E. If we assume for instance that H − E is ample
on X, this surface is smooth irreducible, and hence the determinant is nonnegative by Hodge
index theorem (observe that those three divisors generate the very ample divisor 2H−E). Since
a1 − a2 = (H − E)4 (hence strictly positive if we assume H − E is ample) both determinants
vanish and a simple argument shows that the rank of the intersection matrix (4.1) is two. As
we have seen in the previous sections, this implies (under the assumption that H − E is ample
on X) that any divisor on X depends numerically on H and E.



ON THE PICARD GROUP OF LOW-CODIMENSION SUBVARIETIES 19

The interest of the next example is that shows that the method works even if the Picard group
of the ambient variety is not a finitely generated group (and in particular it has torsion). It also
shows (if C = P1) that the result of section 3 should remain valid for products of projective
spaces of different dimension.

Example 4.3. Set Y = C×P5, where C is a smooth curve. The Picard group of Y is generated
by H (the pull-back of the hyperplane section of P5) and the pull-back of the Picard group of
C by the projection p1 : Y → C. From the numerical point of view, all the fibers of p1 are
equivalent, and we will denote with F the class of a fiber. Let X ⊂ Y be a smooth subvariety
of codimension two and let D ⊂ X be a smooth divisor. We can write the numerical classes of
X and D as

[X]num = a1H
2 + a2HF

[D]num = α1H
3 + α2H

2F

We consider now the intersection matrix:

(4.2)

H3
|X H2

|XF|X DH2
|X DH|XF|X

H|X

FX

D

a2

a1

α2

a1

0
α1

α2

α1

λ2

α1

0
λ1

Equality (1.4) takes now the form:

P := 2α1α2 − a1λ2 − a2λ1,

which can be decomposed as

P =
1

a1

∣

∣

∣

∣

∣

∣

a2 a1 α2

a1 0 α1

α2 α1 λ2

∣

∣

∣

∣

∣

∣

−

∣

∣

∣

∣

a1 α1

α1 λ1

∣

∣

∣

∣

The first determinant in the decomposition corresponds to the intersection matrix of the classes
of H,F,D in the surface obtained by intersecting X with two divisors of the class H, while the
second determinant corresponds to the intersection matrix of the divisors H,D in the surface
obtained as the intersection of X with a general fiber of p1 and a hyperplane of P5. Under
the appropriate assumptions for the irreducibility of those surfaces (even if C = P1, Debarre’s
result cannot provide the irreducibility of the intersection with HF as in section 3), we can
apply Hodge index theorem to conclude that both determinants are zero. In the usual way we
conclude that the rank of the matrix in (4.2) has rank two and hence D depends numerically
on H|X and F|X .

As these examples (and the rest of the paper) illustrates, our method works only numerically
(and in a mysterious way: we guess that we are missing something deep that explains when
the expression (1.4) can be decomposed as a sum of nonnegative summands). Only when there
are some topological results (like the ones by Debarre) we are able to say something about the
Picard group, but not as much as we would expect. In fact we suspect that, in the hypothesis of
Theorems 2.1 and 3.1 one could conclude that the Picard group of X is generated respectively
by H and H1,H2 (and more generally, that the cohomology coincides in the same range as
Barth-Larsen theorems).

The fact that analogues of Barth-Larsen theorems seem to work in general suggests that
also there should be a kind of Hartshorne’s conjecture when 2N < 3n. However, the possible
existence of low-rank vector bundles in general ambient spaces makes difficult to dare with a
reasonable conjecture. For instance, G(1, n) has a universal rank-two vector bundle, so that
any zero locus of a twist of it would be subcanonical of codimension two, but not a complete



20 ENRIQUE ARRONDO AND JORGE CARAVANTES

intersection. In this spirit, the first author suggested in [1, Conjecture 4.3] that the only smooth
codimension-two subvarieties in G(1, n) should be complete intersections or zero loci of twists of
the rank-two universal quotient bundle. In [2] there is some evidence for this conjecture, proving
it when n = 4 and the degree of the variety is at most 25.

In arbitrary codimension it seems very risky to state a similar conjecture for a general ambient
space Y of dimension N . One could expect that, under certain general conditions, the restriction
map H i(Y, Z) → H i(X, Z) is an isomorphism if i ≤ 2n − N . Hence, algebraic cycles on X of
codimension at most 2n−N

2 would lift to cycles on Y . Since the top Chern class of the normal
bundle N of X in Y always lifts to X (by the self-intersection formula) this would mean that
all the Chern classes of N would lift to classes in Y as long as N − n − 1 ≤ 2n−N

2 . Thus a
natural question is to ask under which conditions a smooth subvariety X ⊂ Y is the zero locus
of a vector bundle of rank N − n on Y if 3N − 2 ≤ 4n (the only non-trivial counterexample we
know is the Segre embedding of P1 × Pn−1 in Q2n−2 if n is even). Observe that this inequality
is stronger than the one in Hartshorne’s conjecture except for N = 6, in which our inequality
includes the case n = 4 (for which Hartshorne-Serre correspondence gives a positive answer).

Acknowledgements: We thank José Carlos Sierra and Giorgio Ottaviani for many useful
discussions. It was José Carlos Sierra who pointed us the paper [6], and helped us to pass our
results from numerical equivalence to linear equivalence.

References

[1] E. Arrondo, Subcanonicity of codimension two subvarieties, Rev. Mat. Compl. 18 (2005), 69–80.
[2] E, Arrondo, M.L. Fania, Evidence to subcanonicity of codimension two subvarieties of G(1, 4), to appear

on Int. J. Math.
[3] W. Barth, Transplanting cohomology classes in complex projective space, Amer. J. Math. 92 (1970),

951–967.
[4] W. Barth, M.E. Larsen, On the homotopy-groups of complex projective manifolds, Math Scand. 30 (1972),

88-94.
[5] W. Barth and A. Van de Ven, On the geometry in codimension 2 of Grassmann manifolds, Lecture Notes

in Math. 412, Springer Verlag (1974), 1–35.
[6] O. Debarre, Théorèmes de connexité pour les produits d’espaces projectifs et les grassmanniennes, Amer.

J. Math. 118 (1996), no. 6, 1347–1367.
[7] R. Hartshorne, Varieties of small codimension in projective space, Bull. Amer. Math. Soc. 80 (1974),

1017–1032.
[8] S.L. Kleiman, D. Laksov, Schubert calculus, Amer. Math. Monthly 79 (1972), 1061-1082.
[9] M.E. Larsen, On the topology of projective manifolds, Invent. Math. 19 (1973), 251–260.

[10] A.J. Sommese, Complex subspaces of homogeneous complex manifolds. II. Homotopy results, Nagoya
Math. J. 86, 101–129.

Departamento de Álgebra, Facultad de Ciencias Matemáticas, Universidad Complutense de

Madrid, 28040 Madrid, Spain

E-mail address: arrondo@mat.ucm.es, jcaravan@mat.ucm.es


	1. Subvarieties of projective space and general method
	1.1. The Barth-Larsen result
	1.2. The general method

	2. Subvarieties of Grassmannians of lines
	2.1. Statement of the result and first remarks
	2.2. General set-up
	2.3. End of the proof

	3. Subvarieties of products of projective spaces
	3.1. General set-up
	3.2. First numerical properties
	3.3. Polynomial equalities
	3.4. End of the proof

	4. Further examples and remarks
	References