POINTS FATTENING ON P1 × P1 AND SYMBOLIC POWERS OF
BI-HOMOGENEOUS IDEALS
MAGDALENA BACZYŃSKA, MARCIN DUMNICKI, AGATA HABURA, GRZEGORZ MALARA,
PIOTR POKORA, TOMASZ SZEMBERG, JUSTYNA SZPOND, AND HALSZKA TUTAJ-GASIŃSKA
Abstract. We study symbolic powers of bi-homogeneous ideals of points in X = P1 × P1

arXiv:1304.5775v1 [math.AG] 21 Apr 2013

and extend to this setting results on the eﬀect of points fattening obtained in [3] and [6]. We
prove a Chudnovsky-type theorem for bi-homogeneous ideals and apply it to classiﬁcation of
conﬁgurations of points with minimal or no fattening eﬀect.

1. Introduction
The study of the eﬀect of the points fattening on postulation in P2 was initiated by Bocci and
Chiantini in [3]. For a homogeneous ideal I ⊂ C[Pn ] its initial degree α(I) is deﬁned as the least
integer t such that the graded part It is non-zero. Let I be the radical ideal of a set of points
Z ⊂ P2 . Bocci and Chiantini asked how passing from Z to the double scheme structure 2Z
(this is the fattening mentioned in the title) changes the initial degrees of the associated ideals
(this is the eﬀect of fattening mentioned in the title, the bigger the diﬀerence α(2Z) − α(Z),
the bigger the eﬀect). By the classical Nagata-Zariski theorem [7, Theorem 3.14] the ideal of
2Z is the second symbolic power of I. Of course the m–fold structure mZ is deﬁned by I (m)
for all m ≥ 1. In [6] three of the authors of the present note extended Bocci-Chiantini analysis

to arbitrary symbolic powers of radical ideals of point conﬁgurations in P2 .

The purpose of this note is to study analogous questions for X := P1 × P1 . This might
appear at the ﬁrst glance as a minor modiﬁcation, yet some new phenomena appear and necessary modiﬁcations when compared with P2 indicate how similar problems could be studied on
arbitrary surfaces.
Since the ideals under consideration are now bi-homogeneous the choice of the right extension
of the initial degree notion is more facultative. Somehow intuitively, given a set of points
Z ⊂ X, this should be the least bi-degree of a curve passing through Z. But which bi-degree
is the smallest? We propose here two natural variants of answering this question, both leading
to some nice geometrical consequences, see Deﬁnition 1.2 for details. In both cases we give a
fairly complete classiﬁcation of conﬁgurations of points with relatively small eﬀect of fattening.
1991 Mathematics Subject Classification. 14C20; 14J26; 14N20; 13A15; 13F20.
Szemberg research was partially supported by NCN grant UMO-2011/01/B/ST1/04875.
Keywords: symbolic powers, fat points, conﬁgurations.
1

The main results of this note are Theorem 2.2, Theorem 2.10 and Theorem 3.1. Theorem 2.9
generalizes Chudnovsky theorem on polynomial interpolation in P2 [5, General Theorem 6] to
the bi-homogeneous setting and could be of independent interest.
The α∗ invariant introduced in Deﬁnition 1.2 is related to the anti-canonical divisor on X. As
such it can be considered on arbitrary del Pezzo surfaces. With little adjustments the questions
investigated here can be studied on arbitrary polarized surfaces (in fact even on varieties of
arbitrary dimension). We hope to come back to that in the near future.
1.1. Set-up and notation. Throughout the paper we denote by X the Cartesian product of
two projective lines P1 × P1 , We write OX (a, b) for the line bundle of bi-degree (a, b), i.e.
∗
(OP1 (b)),
OX (a, b) = πV∗ (OP1 (a)) ⊗ πH

where πV and πH denote the projections on the ﬁrst (horizontal) and the second (vertical) factor
in P1 × P1 , respectively.
It is convenient to work in the present setting with the following deﬁnition of symbolic powers.
Definition 1.1. Let I ⊆ C[X] be a bi-homogeneous ideal. We deﬁne the m-th symbolic power of
T
T
I to be the ideal I (m) = j Pij , where I m = i Pi is a bi-homogeneous primary decomposition,
√
T
and the intersection j Pij is taken over all components Pi such that the radical Pi is contained
in an associated prime of I.

A point P in X has coordinates ([a : b], [c : d]). The ideal deﬁning P as a subscheme of X
is then a prime ideal generated by two forms of bi-degree (1, 0) and (0, 1) respectively, namely
by the forms bx0 − ax1 and dy0 − cy1 . If Z = {P1 , . . . , Ps } is a ﬁnite set of points in X, then
s
T
its ideal IZ is just the intersection I(Z) =
I(Pi ). It is convenient to denote by mZ a fat
i=1

points scheme deﬁned by the ideal I(P1 )m1 ∩ · · · ∩ I(Ps )ms for an s–tuple of positive integers

m = (m1 , . . . , ms ). In this notation I (m) = I(mZ) is the ideal deﬁning the fat points scheme

mZ for m = (m, . . . , m).
1.2. Initial degree(s) in bi-homogeneous setting. The notion of the initial degree for
surfaces with Picard number larger than 1 requires some modiﬁcations. The following two
variants seem the most natural for X.
Definition 1.2. Let I ⊂ C[X] be a bi-homogeneous ideal. We associate to I the following
integers
α+ (I) = min{k = k1 + k2 : H 0 (OX (k1 , k2 ) ⊗ I) > 0},
α∗ (I) = min{k : H 0 (OX (k, k) ⊗ I) > 0}.
2

Following Waldschmidt [11] one deﬁnes for homogeneous ideals I ⊂ C[Pn ] an asymptotic
counterpart of the initial degree
α(I (m) )
.
m−→∞
m

γ(I) = lim

This invariant is called the Waldschmidt constant of I. This notion generalizes verbatim to the
multi-homogeneous setting.
Definition 1.3. Let I ⊂ C[X] be a non-zero bi-homogeneous ideal. We deﬁne the Waldschmidt
constant of I as

α• (I (m) )
,
m−→∞
m

γ • (I) = lim
where • ∈ {∗, +}.

The inclusion I (m) I (k) ⊆ I (m+k) and the Fekete Lemma [8] imply in the standard way that
the limit in Deﬁnition 1.3 exists, and that in fact we have
γ • (I) = inf

α• (I (m) )
.
m

Example 1.4. Let I(P ) be the radical ideal deﬁning a point P in X. Then γ + (I(P )) = 1 and
γ ∗ (I(P )) =

1
2.

2. Symbolic powers and α∗ invariant
2.1. Configurations of points with no effect of fattening. In this section we consider how
α∗ (I) jumps when passing from I (m) to I (m+1) . In contrary to the projective plane where one
has always α(I (m) ) < α(I (m+1) ), it might happen on X that
α∗ (I (m) ) = α∗ (I (m+1) ).

(1)

However the equality in (1) is possible only under strong geometric constrains.
We begin with the following extremely useful Lemma which is [6, Lemma 2.1] adopted to the
present setting.
Lemma 2.1. Let Z be a set of points P1 , . . . , Ps ∈ X and let m > n be positive integers. Let

I = I(Z) and let β = α∗ (I(mZ)), γ = α∗ (I(nZ)) and ε = β − γ. Let C be an effective divisor

of bi-degree (β1 , β2 ) computing α∗ (I(mZ)), i.e. β = β1 + β2 . Furthermore let
C = C1 + C2
(j)

(j)

be a sum of two integral effective non-zero divisors. Let (β1 , β2 ) be the bi-degree of Cj for
(j)

j = 1, 2 and let mi
(j)

= ordPi Cj be the multiplicity of Cj at the point Pi . We set m(j) =

(j)

(m1 , . . . , ms ) and with
(j)

ni

n
o
(j)
= max mi − (m − n), 0
3

(j)

(j)

n(j) = (n1 , . . . , ns ). Then
(j)

(j)

i) β (j) := β1 + β2 = α(I(m(j) Z)) and
ii) α∗ (I(m(j) Z)) − α∗ (I(n(j) Z)) ≤ ε.
for j = 1, 2.
Proof. The proof is basically the same as for [6, Lemma 2.1] and we omit it here.



Theorem 2.2. Let Z = {P1 , ..., Ps } ⊆ X be a set of points and let I be the radical ideal of Z.

Assume that the condition (1) holds for some m ∈ Z≥1 . Then there are finite sets ZV , ZH ⊂ P1
such that Z = ZV × ZH , i.e. Z is a grid in X.

Proof. Let (p, q) be a pair of integers such that there exists a section σ ∈ H 0 (X, OX (p, q))
vanishing to order at least m + 1 in points of Z and computing the α∗ invariant, i.e.
α∗ (I (m+1) ) = max{p, q}.
Let D be a divisor deﬁned by σ. Let C be a irreducible component of D of bi-degree (a, b) with
multPi C = mi for i = 1, ..., s. Lemma 2.1 implies now that
(2)



α∗ I(P1 )m1 ∩ ... ∩ I(Ps )ms = α∗ I(P1 )m1 −1 ∩ ... ∩ I(Ps )ms −1 .

The Plücker formula on P1 × P1 implies that
(3)

ab − a − b + 1 −

X m i 
i

2

≥ 0.

On the other hand, by (2) there is no curve of bi-degree (a − 1, b − 1) with multiplicities
m1 − 1, ..., ms − 1 through P1 , ..., Ps , hence
0

ab = h (OX (a − 1, b − 1)) ≤

X m i 
i

2

≤ ab − a − b + 1.

This implies that a = 1 and b = 0 or b = 1 and a = 0, hence C is a ﬁber of a projection in the
product P1 × P1 .
Now we exclude the possibility that D consists solely of ﬁbers in one direction e.g. vertical
ﬁbers. If this were so then removing every ﬁber from D, the multiplicity in every point Pi
would drop by 1 and the bi-degree would drop by the number of removed ﬁbers contradicting
(1). We conclude that D is the union k > 0 vertical ﬁbers V1 , ..., Vk and l > 0 horizontal ﬁbres
H1 , ..., Hl .
In order to complete the proof we claim that
(4)

Z = {Qij = Vi ∩ Hj , i = 1, ..., k, j = 1, ..., l}.

First we show the ⊇ inclusion. Assume to the contrary that some Qij is not contained in

Z. Then removing from D the union of ﬁbers Vi ∪ Hj gives a divisor D ′ = D − Vi − Hj of
4

bi-degree one less than D and vanishing along Z with multiplicities at least m. This contradicts
the assumption (1).
Now we show the ⊆ inclusion in (4). Assume to contrary that there is a point Q ∈ Z, which
is not an intersection point of one of horizontal ﬁbers H1 , . . . , Hl with one of vertical ﬁbers
V1 , . . . , Vk . Without loss of generality we can assume that Q ∈ V1 . This implies that V1 has
multiplicity at least m + 1 in D. Removing the union H1 ∪ V1 from D we obtain again a divisor

D ′ = D − V1 − H1 of bi-degree one less than that of D vanishing along Z with multiplicities

≥ m. Note that this is indeed the case also in the point Q11 (which is the only double point in
H1 ∪ V1 ), since in the situation considered here we have multQ11 (D) ≥ m + 2.
The assertion of the Theorem follows with the sets ZV = πV (V1 ∪ ... ∪ Vk ) and ZH = πH (H1 ∪
... ∪ Hl ).



Working still with an (a, b)-grid ZV × ZH (i.e. #ZV = a and #ZH = b) we show that

somewhat surprisingly the whole sequence α∗ (I (m) ) can be computed explicitly. To begin with
for a, b ∈ Z≥0 we deﬁne inductively the following



(0, 0)



(5)
(am , bm ) = (am−1 + a, bm−1 ),




(a
m−1 , bm−1 + b),

sequence in Z2 :
m = 0,
am−1 + a ≤ bm−1 + b, m ≥ 1,
am−1 + a > bm−1 + b, m ≥ 1.

For this sequence we prove ﬁrst the following purely numerical lemma.
Lemma 2.3. For the sequence {(am , bm )}∞
m=0 defined in (5) we have am < bm + b + 1.
Proof. Assume to the contrary that
(6)

am ≥ bm + b + 1.

There are the following two possibilities:
Case (a): If (am−1 , bm−1 ) = (am − a, bm ) , then am − a + a > bm + b, which is a contradiction
with (5).
Case (b): If (am−1 , bm−1 ) = (am , bm − b), then am + a ≥ bm − b + b. Thus
am−1 = am > bm + b + 1 = (bm−1 + b) + b + 1 ≥ bm−1 + b + 1,
which is (6) with m replaced by m−1. Repeating the above argument we arrive to a1 ≥ b1 +b+1.
Now, for (a1 , b1 ) = (a, 0) we have a contradiction with (5). But (a1 , b1 ) = (0, b) contradicts
a1 ≥ b1 + b + 1.



Theorem 2.4. Let I be the bi-homogeneous ideal associated with an (a, b)-grid of points Z ⊂ X.
Then for all m ≥ 1

α∗ (I (m) ) = max{am , bm },
5

where am , bm are defined by (5).
Proof. Note that for αm =

am
a

bm
∗
b the divisor αm πV Zv
α∗ (I (m) ) ≤ max{am , bm }.

and βm =

grid points to order at least m, hence

∗ Z
+ βm πH
H vanishes at all

We claim that in fact the

equality holds. Assume without loss of generality that am ≥ bm and assume to the contrary

that α∗ (I (m) ) ≤ am − 1.

Let C be a divisor of the bi-degree (p, q) computing α∗ (I (m) ), i.e. α∗ (I (m) ) = max {p, q}. We

claim that
(7)

am − 1 < (m − (αm − 1))b.

Taking (7) for granted, let P be a point in ZV and let VP be the ﬁber over P , which is numerically
a (1, 0)-class. Intersecting C with VP we have C · VP = q. On the other hand, on VP there are b
points of C of multiplicity m. Using (7) and repeatedly Bezout’s theorem we see that VP must
be a multiplicity αm component of C. The same argument works for any point in ZV so that C
has then at least αm a = am vertical components counted with multiplicities. This contradicts
the condition p ≤ am − 1.
Turning to the proof of (7), note that it follows directly from Lemma 2.3 and the equality
m − αm = βm .



It follows immediately from Theorem 2.2 and Theorem 2.4 that two consecutive equalities as
in (1) are not possible.
Corollary 2.5. There is no set of points Z ⊂ X such that for the ideal I of Z the equality
α∗ (I (m+2) ) = α∗ (I (m+1) ) = α∗ (I (m) )
holds for any positive integer m.
Of course the same result can be proved along the following lines. Let f be an element in
I (m+2) of bi-degree (p, q) such that α∗ (I (m+2) ) = max {p, q}. Taking partial derivatives of f
with respect to the ﬁrst set of variables and then with respect to the second set of variables,
we obtain a polynomial fe ∈ I (m) of bi-degree (p − 1, q − 1). This shows that there is always

inequality α∗ (I (m+2) ) > α∗ (I (m) ). Our approach has the advantage that it does not call back

to diﬀerentiation and thus could be generalized to arbitrary surfaces.
It is natural to introduce the following function.
Definition 2.6. Let I be a bi-homogenous radical ideal of a set of points Z. We deﬁne the
jump function
f (Z; m) = α∗ (I (m) ) − α∗ (I (m−1) )
for all m ∈ Z≥1 with α∗ (I (0) ) = 0.

6

The following result is a straightforward consequence of Theorem 2.4.
Corollary 2.7. Assume that a, b ∈ Z≥1 . Working under assumptions of Theorem 2.4, the
jump function f (Z; m) has infinitely many jumps equal to 0 and infinitely many jumps equal to
min{a, b}.
Proof. The idea of the proof is to show that there exists m such that (am , bm ) = (ab, ab).



The following remark shows that a grid can be recovered from the jump function.
Remark 2.8. Let f (Z; m) be a jump function for a certain (a, b)-grid in X with a ≤ b. Then
P
a = f (Z; 1) and for r = min{j : f (Z; j) < a} the number b = ri f (Z; i).
2.2. Configuration of points with the minimal positive effect of fattening. Here we
show that the extremely useful result of Chudnovsky [5] relating Waldschmidt constants and
initial degrees generalizes to a multi-homogeneous setting. The original result of Chudnovsky
is proved with analytic methods. Our approach is modeled on the algebraic proof in [10,
Proposition 3.1]. This result is of independent interest.
Theorem 2.9. Let P1 , . . . , Ps ∈ X be mutually distinct points and let I =
the ideal of their union. Then

T

i I(Pi )

⊂ C[X] be

α∗ (I (m) )
α∗ (I)
≥
.
m
2
Proof. Let α∗ (I) = a. Choose distinct points Q1 , . . . , Qt ∈ {P1 , . . . , Ps } with the smallest
T
possible t such that α∗ (J) = a for J = j I(Qj ). By minimality of t, the points Qj impose

independent conditions in bi-degree (a − 1, a − 1) so that t = a2 . Thus the ideal J is generated
in bi-degree (a, a) and hence the only base points of J(a,a) are the points Qj . In particular, J(a,a)
(m)

is ﬁxed component free. Let A be a nonzero form in I(b,b) , where b = α∗ (I (m) ). Since J(a,a)
is ﬁxed component free, we can choose an element B ∈ J(a,a) with no common factor with A.
Using Bezout’s Theorem adopted to X, we have
2ab = (a, a) · (b, b) = div(A) · div(B) ≥ ma2 ,
and hence
α∗ (I (m) )
α∗ (I)
≥
.
m
2

We apply the above Theorem to the case with several consecutive jumps of α∗ (I (m) ) equal
to 1.
7

Theorem 2.10. Let I be a radical bi-homogeneous ideal of a set Z = {P1 , . . . , Ps } of points in
X. Assume that
(8)

α∗ (I (6) ) = α∗ (I (5) ) + 1 = . . . = α∗ (I (2) ) + 4 = α∗ (I) + 5.

Then α∗ (I) = 1, i.e. Z is contained in a divisor of bi-degree (1, 1).
Moreover in order to conclude that Z is contained in a divisor of bi-degree (1, 1) the sequence
of equalities in (8) cannot be shortened in general.
Proof. The assumption α∗ (I (6) ) = α∗ (I) + 5 together with Theorem 2.9 yields α∗ (I) ≤ 2. If

α∗ (I) = 1, then we are done. So it suﬃces to deal with the case α∗ (I) = 2.

Let D be a divisor of bi-degree (7, 7) with multiplicities at least 6 in each point Pi ∈ Z. By

deﬁnition of α∗ there is no curve of bi-degree (6, 6) with this property.

Now, the idea is to transplant the situation to P2 via the standard birational transformation
µ : P1 × P1 −→ P2 , i.e. µ is the composition of the blowing up of a point S such that the
horizontal and the vertical ﬁbres through S are disjoint from Z followed by blowing down
proper transforms of these two ﬁbers to points P and Q in P2 .
On P2 we have now the following situation. If D ′ is the proper transform of D under µ, then
it is a divisor of degree 14 vanishing to order at least 7 at P and at Q and to order at least 6 in
all points in Z ′ = µ(Z). We know also that there is no divisor of degree 12 vanishing to order at
least 6 at P , Q and all points in Z ′ . Obviously P , Q and at least one point R = Pi0 ∈ Z are not

collinear. Applying the standard Cremona transformation τ based on these points to divisor D ′

results in a divisor D ′′ of degree 8 vanishing to order 6 at all points in Z ′′ = τ (Z ′ \ {R}). Again

not all points in Z ′′ can be collinear. On the other hand if F is an arbitrary divisor in P2 and
points A, B, C are not collinear, then the multi-point Seshadri constant of OP2 (1) at A, B, C is
1/2 (see [1] for deﬁnitions and properties of Seshadri constants) and this yields the inequality
1
deg(F ) ≥ (multA F + multB F + multC F ).
2
This fact applied to F := D ′′ gives a contradiction.
That the result is sharp follows from Example 2.11 below.
Example 2.11. For the set Z of points
P1 = ([1 : 0], [1 : 0]), P2 = ([1 : 0], [1 : 1]), P3 = ([1 : 1], [1 : 0]) and P4 = ([0 : 1], [1 : 1])
and I = I(Z) we have
m

1 2 3 4 5

α∗ (I (m) ) 2 3 4 5 6
8

.



Proof. It is easy to draw divisors with vanishing orders claimed above and of the given bidegree. On the other hand, recursive use of Bezout’s Theorem excludes the possibilities that
the bi-degrees could be lower. We leave the details to the reader.



It might easily happen that for a radical ideal I of points in X there are inﬁnitely many
jumps by 1 in the sequence α∗ (I (m) ), m ≥ 1 and nevertheless α∗ (I) is arbitrarily large. The
following example is an easy consequence of Theorem 2.2.
Example 2.12. Let a ≥ 5 be an integer and let Z be an (a, a) grid in X minus a single point.
Then
α∗ (I (2k−1) ) = ka − 1 and α∗ (I (2k) ) = ka.
In particular α∗ (I) = a − 1.
3. Symbolic powers and α+ invariant
The behavior of the α+ invariant is more similar to the initial degree in the plane. Let f be
a bi-homogeneous polynomial of bi-degree (a, b) vanishing along a fat point scheme (m + 1)Z
for some set of points Z ⊂ X. Taking a partial derivative of f with respect to the ﬁrst or
the second set of variables, we obtain a non-zero polynomial of bi-degree (a − 1, b) or (a, b − 1)
vanishing along mZ. This shows that there is always the strong inequality
α+ (I (m) ) < α+ (I (m+1) )
for all m ≥ 1.
We describe now the situation when the eﬀect of fattening is the minimal possible, i.e. equal to
1.
Theorem 3.1. Let Z = {P1 , ..., Ps } ⊆ X be a set of points and let I be the radical ideal of Z.
Assume that
(9)

α+ (I (m+1) ) = α+ (I (m) ) + 1

for some m ∈ Z≥1 . Then all points {P1 , ..., Ps } lie on a single vertical or horizontal fiber.
Proof. The assertion is trivial for s = 1, so we assume s ≥ 2. The ﬁrst part of the proof is quite

analogous to that of Theorem 2.2. Let σ ∈ H 0 (X, OX (p, q)) be a section vanishing to order at
least m + 1 in points in Z with (p, q) minimal, i.e.
α+ (I (m+1) ) = p + q = d

Let D be a divisor deﬁned by σ. Let C be a irreducible component of D of bi-degree (a, b) and
with multPi C = mi . A simple modiﬁcation of [6, Lemma 2.1] implies that


α+ I(P1 )m1 ∩ ... ∩ I(Ps )ms = a + b = α+ I(P1 )m1 −1 ∩ ... ∩ I(Ps )ms −1 + 1.
9

In particular, there is no curve of bi-degree (k, a+b−2−k), where k is an arbitrary non-negative
integer less or equal than a + b − 2, with multiplicities m1 − 1, ..., ms − 1 through P1 , ..., Ps .
Hence
0

h (OX (k, a + b − 2 − k)) = (k + 1)(a + b − 1 − k) ≤

X m i 
i

2

≤ ab − a − b + 1,

where the last inequality is the genus bound (3). Since the above inequality holds for all k’s,
we can take k = a − 1. Then
ab ≤ ab − a − b + 1.
This implies that a = 1 and b = 0 or b = 1 and a = 0, hence C is a ﬁber of a projection in the
product P1 × P1 .
In the next step we exclude the possibility that D has two or more vertical or horizontal
components. Suppose to the contrary that V and W are two vertical ﬁbers contained in the
support of D. Then D ′ = D − (V + W ) vanishes at all points in Z to order at least m (since
we remove smooth disjoint components). This contradicts however (9). The proof if there are
two horizontal components is completely analogous.
It remains to exclude the possibility that D is supported on a union V + H of a vertical ﬁber
V and a horizontal ﬁber H. Let Q = H ∩ V . Since s ≥ 2, there must be some other point
R ∈ Z contained either in V or in H. Then either (m + 1)V or (m + 1)H is contained in D, so

that in particular multQ D ≥ m + 2. Thus for D ′ = D − (V + H) we have multP D ′ ≥ m for all
points P ∈ Z. This contradicts (9) again and we are done.



Turning to the jumps of the α+ invariant by 2, the above proof yields immediately the
following observation.
Corollary 3.2. Let Z = {P1 , ..., Ps } ⊆ X be a set of points and let I be the radical ideal of Z.
Assume that
(10)

α+ (I (m+1) ) = α+ (I (m) ) + 2

for some m ∈ Z≥1 . Let D be a divisor on X of bi-degree (p, q) such that α+ (I (m+1) ) = p + q.
Then any irreducible component of D has bi-degree (1, 0), or (1, 1) or (0, 1). Moreover there are
at most two horizontal or vertical components.
Proof. Let C be a component of D of bi-degree (a, b). Arguing as in the proof of Theorem 3.1
and keeping the notation from that proof, we obtain that the inequality
(k + 1)(a + b − 2 − k) ≤ ab − a − b + 1
10

must hold for all k in the range 0, . . . , a + b − 1. For k = a − 1 we obtain
a(b − 1) ≤ ab − a − b + 1
which implies b ≤ 1. Analogously, for k = b − 1 we get a ≤ 1. This leaves only three bi-degrees
possible for C.
Turning to the second claim, assume that there were at least 3 vertical components in D.
Then removing them from D lowers the multiplicities at all points Pi at most by 1 but the sum
of degrees by at least 3 contradicting (10). The same argument works for horizontal ﬁbers. 
We conclude with the example showing that all types of components listed in the above
Corollary can occur simultaneously. For simplicity all coordinates are aﬃne, a point (a, b) is
the point ((1 : a), (1 : b)) in X.
Example 3.3. Consider the following set of points P1 = (0, 0), P2 = (1, 1), P3 = (1, 2), P4 =
(2, 2), P5 = (3, 0), P6 = (3, 3) and let I be the bi-homogeneous ideal of their union. Then
α+ (I) = 4 and α+ (I (2) ) = 6.

(11)

Proof. The following ﬁgures show a divisor computing α+ (I) (there are many possibilities) and
the divisor computing α+ (I (2) ) (this divisor is unique). The diagonal line is a divisor of bi-degree
(1, 1).
r

r
r

r

r

r

r
r

r

r

r

r

Thus we have exhibited explicitly divisors realizing ≤ inequalities in (11). We leave it to a
motivated reader to check that neither for I nor for I (2) the α+ invariant can be lowered.



Acknowledgement. This note resulted from a workshop on linear series held in Lanckorona in
February 2013. We thank Pedagogical University of Cracow for the ﬁnancial support.
References
[1] Bauer, Th., Di Rocco, S., Harbourne, B., Kapustka, M., Knutsen, A. L., Syzdek, W., Szemberg T.: A
primer on Seshadri constants, Interactions of Classical and Numerical Algebraic Geometry, Proceedings of
a conference in honor of A. J. Sommese, held at Notre Dame, May 22–24 2008. Contemporary Mathematics
vol. 496, 2009, eds. D. J. Bates, G-M. Besana, S. Di Rocco, and C. W. Wampler, 362 pp.
[2] Bauer, Th., Bocci, C., Cooper, S., Di Rocco, S., Dumnicki, M., Harbourne, B., Jabbusch, K., Knutsen, A.
L., Kuronya, A., Miranda, R., Roe, J., Schenck, H., Szemberg, T., Teitler, Z.: Recent developments and
open problems in linear series, in: Impanga Lecture Notes ”Contributions to Algebraic Geometry” (2012).
11

[3] Bocci, Ch., Chiantini, L.: The eﬀect of points fattening on postulation, J. Pure and Applied Algebra 215,
89 – 98 (2011).
[4] Bocci, C., Harbourne, B., Comparing Powers and Symbolic Powers of Ideals, J. Algebraic Geometry 19, 399
– 417 (2010).
[5] Chudnovsky, G. V.: Singular points on complex hypersurfaces and multidimensional Schwarz Lemma,
Séminaire de Théorie des Nombres, Paris 1979–80, Séminaire Delange-Pisot-Poitou, Progress in Math vol.
12, M-J Bertin, editor, Birkhäuser, Boston-Basel-Stutgart (1981).
[6] Dumnicki, M., Szemberg, T., Tutaj-Gasińska, H.: Symbolic powers of planar point conﬁgurations, J. Pure.
Appl. Algebra 217, 1026 – 1036 (2013).
[7] Eisenbud, D.: Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995.
[8] Fekete, H., Űber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeﬃzienten, Mathematische Zeitschrift 17, 228 – 249 (1923).
[9] Guardo, E., Harbourne, B., Van Tuyl, A.: Symbolic powers versus regular powers of ideals of general points
in P1 × P1 , to appear in Canadian Journal of Mathematics, http://dx.doi.org/10.4153/CJM-2012-045-3.
[10] Harbourne, B., Huneke, C.: Are symbolic powers highly evolved ?, to appear in J. Ramanujan Math. Soc.,
arXiv:1103.5809v3.
[11] Waldschmidt, M.: Propriétés arithmétiques de fonctions de plusieurs variables II, Lecture Notes Math. 578,
Springer-Verlag 1977, 108–135

Marcin Dumnicki, Halszka Tutaj-Gasińska, Jagiellonian University, Institute of Mathematics, Łojasiewicza 6, PL-30-348 Kraków, Poland
Magdalena Baczyńska, Agata Habura, Grzegorz Malara, Piotr Pokora, Tomasz Szemberg, Justyna
Szpond, Pedagogical University of Cracow, Institute of Mathematics, Podchora̧żych 2, PL-30-084 Kraków,
Poland

12

