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Points fattening on P1×P1 and symbolic powers of bi-homogeneous ideals
Journal of Pure and Applied Algebra 218 (2014) 1555–1562
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Journal of Pure and Applied Algebra www.elsevier.com/locate/jpaa
Points fattening on P1 × P1 and symbolic powers of bi-homogeneous ideals a ´ Magdalena Baczynska , Marcin Dumnicki b , Agata Habura a , Grzegorz Malara a , a Piotr Pokora , Tomasz Szemberg a,∗,1 , Justyna Szpond a , b ´ Halszka Tutaj-Gasinska a b
Pedagogical University of Cracow, Institute of Mathematics, Podchora¸zych ˙ 2, PL-30-084 Kraków, Poland Jagiellonian University, Institute of Mathematics, Łojasiewicza 6, PL-30-348 Kraków, Poland
a r t i c l e
i n f o
Article history: Received 21 April 2013 Received in revised form 12 November 2013 Available online 16 December 2013 Communicated by S. Kovács
a b s t r a c t We study symbolic powers of bi-homogeneous ideals of points in X = P1 × P1 and extend to this setting results on the effect of points fattening obtained in [2] and [4]. 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. © 2013 Elsevier B.V. All rights reserved.
MSC: 14C20; 14J26; 14N20; 13A15; 13F20
1. Introduction Our approach to the study of the effect of the points fattening on postulation in P2 was initiated by Bocci and Chiantini in [2]. For a non-zero homogeneous ideal I C[Pn ] its initial degree α ( I ) is defined as the least integer t such that the graded part I t 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 2 Z (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 [5, Theorem 3.14] the ideal of 2Z is the second symbolic power of I . Of course the m-fold structure m Z is defined by I (m) for all m 1. In [4] 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 there is no single, canonical way to extend the initial degree notion to this setting. 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. The main results of this note are Theorem 2.2, Theorem 2.7 and Theorem 3.1. Theorem 2.5 generalizes the proof of Chudnovsky’s theorem on polynomial interpolation in P2 [3, General Theorem 6] to the bi-homogeneous setting and could be of independent interest.
*1
Corresponding author. Szemberg research was partially supported by NCN grant UMO-2011/01/B/ST1/04875.
0022-4049/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jpaa.2013.12.007
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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 O X (a, b) for the line bundle of bi-degree (a, b), i.e.
O X (a, b) = π V∗ OP1 (a) ⊗ π H∗ OP1 (b) , where π V and π H 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 I to be the ideal I (m) = j P i j , where I m = i P i is a bi-homogeneous primary decomposition, and the intersection j P i j is taken over all components P i such that the radical
√
P i 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 dy 0 − c y 1 . If Z = { P 1 , . . . , P s } s is a finite set of points in X , then its ideal I Z is just the intersection I ( Z ) = i =1 I ( P i ). It is convenient to denote by m Z m1 ms a fat points scheme defined by the ideal I ( P 1 ) ∩ · · · ∩ I ( P s ) for an s-tuple of positive integers m = (m1 , . . . , ms ). In this notation I (m) = I (m Z ) is the ideal defining the fat points scheme m Z 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 . We denote by I the sheafification of I . 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 O X (k1 , k2 ) ⊗ I > 0 ,
α ∗ ( I ) = min k: H 0 O X (k, k) ⊗ I > 0 . Remark 1.3. Let ([x0 : x1 ], [ y 0 : y 1 ]) be bi-homogeneous coordinates on P1 × P1 . Elements in H 0 (O X (k1 , k2 )) can be considered as homogeneous polynomials of degree k1 + k2 in variables x0 , x1 , y 0 , y 1 . From this point of view α + ( I ) is exactly the initial degree α ( I ) defined in the introduction. Following Waldschmidt [11] one defines for homogeneous ideals I ⊂ C[Pn ] an asymptotic counterpart of the initial degree
α ( I (m) )
α ( I ) = lim
m→∞
m
.
This invariant is called the Waldschmidt constant of I . This notion generalizes verbatim to the multi-homogeneous setting. Definition 1.4. Let I ⊂ C[ X ] be a non-zero bi-homogeneous ideal. We define the Waldschmidt constant of I as
α • ( I ) = lim
α • ( I (m) )
m→∞
m
,
where • ∈ {∗, +}. The inclusion I () I (k) ⊆ I (+k) implies that the sequence αm := α • ( I (m) ) is sub-additive, i.e. α+k α + αk . Fekete’s Lemma [6] implies that any sub-additive sequence am of real numbers is bounded below and converges to infm amm . In particular the limit in Definition 1.4 exists, and we have in fact
α • ( I ) = inf
α • ( I (m) ) m
.
Example 1.5. Let I ( P ) be the radical ideal defining a point P in X . Then α + ( I ( P )) = 1 and α ∗ ( I ( P )) = 12 .
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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 contrast to the projective plane where one has always α ( I (m) ) < α ( I (m+1) ), it might happen on X that
α ∗ I (m+1) = α ∗ I (m) .
(1)
We illustrate this phenomenon first with a simple example. Example 2.1. Let I be the radical ideal of a single point P . The equality
I (m) = I m
for all m 1
implies then
α ∗ I (m) = α ∗ I m =
m+1 2
= α ∗ I m+1 = α ∗ I (m+1)
for all m odd. The rest of this section is devoted to the detailed characterization of geometric conditions under which the equality in (1) is possible. Theorem 2.2. Let Z = { P 1 , . . . , P s } ⊆ X be a finite set of points and let I be the radical ideal of Z . The following conditions are equivalent (a) α ∗ ( I (m+1) ) = α ∗ ( I (m) ) for some m 1; (b) there exist finite sets Z V , Z H ⊂ P1 such that Z = Z V × Z H , i.e. Z is a grid in X . Proof. We begin by showing the implication from (a) to (b). Let m be an integer such that (a) holds. Let ( p , q) be a pair of integers such that there exists a section H 0 ( X , O X ( p , q)) vanishing to order at least m + 1 in points of Z and computing the α ∗ invariant, i.e.
σ ∈
α ∗ I (m+1) = max{ p , q}. Let D be a divisor defined by σ . Let C be an irreducible component of D of bi-degree (a, b) with mult P i C = mi for i = 1, . . . , s. The Plücker formula on P1 × P1 implies that
ab − a − b + 1 −
mi
i
2
0.
(2)
On the other hand, there is no divisor C of bi-degree (a − 1, b − 1) with multiplicities m1 − 1, . . . , ms − 1 through P 1 , . . . , P s . Indeed, otherwise the effective divisor C + ( D − C ) would have multiplicity m at all points of Z . Since its bi-degree is ( p − 1, q − 1), this clearly contradicts the equality in (a). Hence a simple conditions count and (2) yield
mi ab − a − b + 1. ab = h O X (a − 1, b − 1) 0
i
2
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. Assume to the contrary that D does consist of vertical fibers only. These fibers are then responsible for multiplicity m + 1 at every point of Z . Since the fibers are smooth, this multiplicity comes from multiple fibers through each point in Z . More precisely, for every point in Z , the divisor D contains the vertical fiber through this point with multiplicity m + 1 2. Removing the reduced union of all these fibers from D results in a divisor D vanishing in all points of Z to order m and of bi-degree strictly less than that of D . This contradicts (a). We conclude that D is the union of k > 0 vertical fibers V 1 , . . . , V k and l > 0 horizontal fibers H 1 , . . . , H l . In order to complete the proof of implication from (a) to (b) we claim that
Z = { Q i j = V i ∩ H j , i = 1, . . . , k, j = 1, . . . , l}.
(3)
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First we show the ⊇ inclusion. Assume to the contrary that some Q i j is not contained in Z . Then removing from D the union of fibers V i ∪ H j gives a divisor D = D − V i − H j of bi-degree one less than that of D and vanishing along Z with multiplicities at least m. This contradicts the assumption in (a). Now we show the ⊆ inclusion in (3). Assume to the contrary that there is a point Q ∈ Z , which is not an intersection point of one of the horizontal fibers H 1 , . . . , H l with one of the vertical fibers V 1 , . . . , V k . Without loss of generality we can assume that Q ∈ V 1 . This implies that V 1 has multiplicity at least m + 1 in D. Removing the union H 1 ∪ V 1 from D we obtain again a divisor D = D − V 1 − H 1 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 Q 11 (which is the only double point in H 1 ∪ V 1 ), since in the situation considered here we have mult Q 11 ( D ) m + 2. The assertion in (b) follows with the sets Z V = π V ( V 1 ∪ · · · ∪ V k ) and Z H = π H ( H 1 ∪ · · · ∪ H l ). Now, we turn to the implication from (b) to (a). As kindly pointed out to us by the referee, there is a paper by Guardo and Van Tuyl [9] studying in particular complete intersection zero dimensional subschemes in P1 × P1 , see also [7,8]. Everything what comes now (and in fact much more) can be read of these papers. We outline here the path of thought relevant for our purposes. To begin with assume that Z V is defined by a reduced polynomial F of degree r in [x0 : x1 ] and Z H is defined by a reduced polynomial G of degree s in [ y 0 : y 1 ]. In other words, we assume that the radical ideal I of the grid Z = Z V × Z H is generated by forms F and G (which we now treat as bi-homogeneous forms on X ). In particular I is a complete intersection and hence I (m) = I m for all m by Macaulay’s unmixedness theorem [5, Corollary 18.14]. Then we have
α ∗ I (m) = min{c: ∃i , j: i + j = m, ir c , js c }.
(4)
Note that if there exist i and j such that i + j = m and ir = js, then
α ∗ I (m) = ir = js. Otherwise i and j have to be spread possibly equally with weights r and s in order to compute that if i 0 is the least integer such that i 0 r > (m − i 0 )s, then
α ∗ ( I (m) ). By this we mean
α ∗ I (m) = min i 0 r , (m − i 0 + 1)s . Moreover, for any m the value of the difference
α ∗ I (m+1) − α ∗ I (m) = k · gcd(r , s) for some 0 k min{r , s} and any such multiple (in particular 0) occurs for infinitely many m. This shows the statement in (a) and finishes the proof of the theorem. 2 Remark 2.3. Note that under assumption (b) in the above theorem one can obtain from [9, Theorem 4.11 and Theorem 5.4] all bi-graded Betti numbers of the grid. It follows immediately from Theorem 2.2 and formula (4) that two consecutive equalities as in (1) are not possible. Corollary 2.4. 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 f ∈ I (m) of bi-degree ( p − 1, q − 1). This shows that there is always to the second set of variables, we obtain a polynomial ∗ ( m+2) ∗ ( m) ) > α ( I ). Our approach has the advantage that it does not depend on the differentiation and thus inequality α ( I could be generalized to arbitrary surfaces. 2.2. Configuration of points with the minimal positive effect of fattening Here we show that the extremely useful result of Chudnovsky [3] relating Waldschmidt constants and initial degrees generalizes to a multi-homogeneous setting. The result of Chudnovsky was inspired by slightly weaker results obtained by Waldschmidt and Skoda with analytic methods. Our approach here is modeled on the detailed algebraic proof by Harbourne and Huneke in [10, Proposition 3.1] which in turn was inspired by ideas sketched out by Chudnovsky himself. This result is of independent interest. Example 2.6 shows that the inequality in (5) cannot be improved in general.
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Theorem 2.5. Let P 1 , . . . , P s ∈ X be mutually distinct points and let I =
α ∗ ( I (m) ) m
i
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I ( P i ) ⊂ C[ X ] be the ideal of their union. Then
α∗(I )
(5)
2
for all m 1. ∗ ∗ Proof. Let α ( I ) = a. Choose t distinct points Q 1 , . . . , Q t ∈ { P 1 , . . . , P s } with the smallest possible t such that α ( J ) = a for J = j I ( Q j ). By minimality of t, the points Z = { Q 1 , . . . , Q t } impose independent conditions in bi-degree (a − 1, a − 1) so
that t = a2 . Claim. We claim that the linear system defined by global sections of the sheaf O X (a, a) ⊗ J has no fixed components. Here J denotes the sheafification of the ideal J . Assume to the contrary that there exists a common component Γ . By the definition of Z , for any point Q ∈ Z there exists a divisor C Q in the linear system |O X (a − 1, a − 1) ⊗ I Z \{ Q } |. If Γ is not a component of C Q , then let C be a curve of bi-degree (1, 1) not containing Γ as a component. Then
C Q + C ∈ O X (a, a) ⊗ J has not Γ as a component, a contradiction. This shows that Γ must be a component of all divisors C Q with Q ∈ Z . This Γ cannot contain any point R of Z , since otherwise C R would contain R as well, a contradiction with the definition of C R . Then take any divisor C Q and let Γ be a divisor of the same bi-degree as Γ passing through Q . Then C Q − Γ + Γ is an effective divisor of bi-degree (a − 1, a − 1) vanishing along Z , a contradiction. This shows the claim. Let A be a non-zero form in ( I (m) )(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 adapted to X , we have
2ab = (a, a) · (b, b) = div( A ) · div( B ) ma2 , and hence
α ∗ ( I (m) ) m
α∗(I ) 2
.
2
Example 2.6. Continuing Example 2.1 with I the ideal of a single point, we have that the inequality in (5) cannot be improved in general.
α ∗ ( I ) = 1 and
α ∗ ( I (2m) ) 2m
= 12 . This shows
We apply now Theorem 2.5 to the case with several consecutive jumps of α ∗ ( I (m) ) equal to 1. It follows from the proof that the assumption in (6) below is equivalent to the assumption α ∗ ( I (6) ) = α ∗ ( I ) + 5. We prefer the present formulation which is quite in spirit of [4, Theorem 3.1]. Theorem 2.7. Let I be a radical bi-homogeneous ideal of a set Z = { P 1 , . . . , P s } of points in X . Assume that
α ∗ I ( 6 ) = α ∗ I ( 5 ) + 1 = · · · = α ∗ I ( 2 ) + 4 = α ∗ ( I ) + 5.
(6)
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 (6) cannot be shortened in general. Proof. The assumption α ∗ ( I (6) ) = α ∗ ( I ) + 5 together with Theorem 2.5 yields it suffices to deal with the case α ∗ ( I ) = 2. So assume α ∗ ( I ) = 2, hence
α ∗ ( I ) 2. If α ∗ ( I ) = 1, then we are done. So
There exists
There does not exist
A divisor D of bi-degree (7, 7) with multiplicities at least 6 in each P i ∈ Z
A divisor E of bi-degree (6, 6) with multiplicities at least 6 in each P i ∈ Z
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 fibers through S are disjoint from Z followed by blowing down proper transforms of these two fibers to points P and Q in P2 .
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On P2 we have now the following situation: There exists
There does not exist
A divisor D 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 )
A divisor E of degree 12 vanishing to order at least 6 at P , Q and all points in Z
with D being the proper transform of D under μ. Obviously P , Q and at least one point R = P i 0 ∈ Z are not collinear (otherwise there would be a divisor of degree 6 with vanishing orders as in the last framed box). Applying the standard Cremona transformation τ based on these points to divisor D results in There exists
There does not exist
A divisor D of degree 8 vanishing to order 6 at all points in Z = τ ( Z \ { R })
A divisor E of degree 6 vanishing to order 6 at all points in Z
This implies immediately that 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
deg( F )
1 2
(mult A F + mult B F + multC F ).
This fact applied to F := D gives a contradiction. That the result is sharp follows from Example 2.8 below.
2
The next example shows that it may happen that
α ∗ I (5 ) = α ∗ I (4 ) + 1 = · · · = α ∗ I (2 ) + 3 = α ∗ ( I ) + 4 and
α ∗ ( I ) = 1.
Example 2.8. For the set Z of points
P 1 = [1 : 0], [1 : 0] ,
P 2 = [1 : 0], [1 : 1] ,
P 3 = [1 : 1], [1 : 0]
and
P 4 = [0 : 1], [1 : 1]
and I = I ( Z ) we have
m
1 2 3 4 5 . 2 3 4 5 6
α ∗ ( I (m) )
Proof. It is easy to draw divisors with vanishing orders claimed above and of the given bi-degree. 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. 2 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.
Example 2.9. 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.
A similar effect occurs on an (r , s) grid with r and s relatively prime. 3. Symbolic powers and α + invariant The behavior of the α + invariant is more similar to the initial degree in the plane, see Remark 1.3. Let f be a bihomogeneous 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 m Z . This shows that there is always the strong inequality
α + I (m) < α + I (m+1)
for all m 1. In this section we describe the situation when the effect of fattening is the minimal possible, i.e. equal to 1.
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Theorem 3.1. Let Z = { P 1 , . . . , P s } ⊆ X be a set of points and let I be the radical ideal of Z . Assume that
α + I (m+1) = α + I (m) + 1
(7)
for some m ∈ Z1 . Then all points { P 1 , . . . , P s } 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 , O X ( 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 an irreducible component of D of bi-degree (a, b) and with mult P i C = mi . The assumption in (7) implies then that
α + I ( P 1 )m1 ∩ · · · ∩ I ( P s )ms = a + b = α + I ( P 1 )m1 −1 ∩ · · · ∩ I ( P s )ms −1 + 1, where as usually a non-positive power of an ideal is understood as the whole ring. Indeed, assuming that α + ( I ( P 1 )m1 −1 ∩ · · · ∩ I ( P s )ms −1 ) = c < a + b − 1 and picking a divisor Γ defined by a form in I ( P 1 )m1 −1 ∩ · · · ∩ I ( P s )ms −1 of bi-degree (u , v ) with u + v = c, we would obtain an effective divisor D − C + Γ of bi-degree ( p − a + u , q − b + v ) vanishing along Z with multiplicity at least m. Since p − a + u + q − b + v d − 2 this contradicts (7). 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 P 1 , . . . , P s (a negative multiplicity means no vanishing condition). Hence
h0 O X (k, a + b − 2 − k) = (k + 1)(a + b − 1 − k)
mi
i
2
ab − a − b + 1,
where the last inequality is the genus bound (2). 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 (7). 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 mult Q D m + 2. Thus for D = D − ( V + H ) we have mult P D m for all points P ∈ Z . This contradicts (7) again and we are done. 2 Turning to the jumps of the
α + invariant by 2, the above proof yields immediately the following observation.
Corollary 3.2. Let Z = { P 1 , . . . , P s } ⊆ X be a set of points and let I be the radical ideal of Z . Assume that
α + I (m+1) = α + I (m) + 2
(8)
Z1 . Let D be a divisor on X of bi-degree ( p , q) such that α + ( I (m+1) )
for some m ∈ = 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
(9)
must hold for all k in the range 0, . . . , a + b − 3. We want to show that a 1 and b 1. Assuming b 2 we have a − 1 a + b − 3 hence with k = a − 1 we obtain from (9)
a(b − 1) ab − a − b + 1 which implies b 1, a contradiction. The case a 2 is excluded by symmetry and we are done with the first claim. 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 P i at most by 1 but the sum of degrees by at least 3 contradicting (8). The same argument works for horizontal fibers. 2
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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 P 1 = (0, 0), P 2 = (1, 1), P 3 = (1, 2), P 4 = (2, 2), P 5 = (3, 0), P 6 = (3, 3) and let I be the bi-homogeneous ideal of their union. Then
α + ( I ) = 4 and α + I (2) = 6.
(10)
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).
Thus we have exhibited explicitly divisors realizing inequalities in (10). We leave it to a motivated reader to check that neither for I nor for I (2) can the α + invariant be lowered. 2 Acknowledgements This note resulted from a workshop on linear series held in Lanckorona in February 2013. We thank Pedagogical University of Cracow for financial support. Furthermore we are deeply obliged to the referee for an unusually detailed report which considerably contributed to the readability and correctness of the paper! References [1] Th. Bauer, S. Di Rocco, B. Harbourne, M. Kapustka, A.L. Knutsen, W. Syzdek, T. Szemberg, A primer on Seshadri constants, in: D.J. Bates, G.-M. Besana, S. Di Rocco, C.W. Wampler (Eds.), 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, in: Contemp. Math., vol. 496, 2009, 362 pp. [2] C. Bocci, L. Chiantini, The effect of points fattening on postulation, J. Pure Appl. Algebra 215 (2011) 89–98. [3] G.V. Chudnovsky, Singular points on complex hypersurfaces and multidimensional Schwarz Lemma, in: M.-J. Bertin (Ed.), Séminaire de Théorie des Nombres, Paris, 1979–80, Séminaire Delange–Pisot–Poitou, in: Prog. Math., vol. 12, Birkhäuser, Boston, Basel, Stuttgart, 1981. ´ [4] M. Dumnicki, T. Szemberg, H. Tutaj-Gasinska, Symbolic powers of planar point configurations, J. Pure Appl. Algebra 217 (2013) 1026–1036. [5] D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Grad. Texts Math., vol. 150, Springer-Verlag, New York, 1995. [6] H. Fekete, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Z. 17 (1923) 228–249. [7] E. Guardo, Fat points schemes on a smooth quadric, J. Pure Appl. Algebra 162 (2001) 183–208. [8] E. Guardo, B. Harbourne, A. Van Tuyl, Symbolic powers versus regular powers of ideals of general points in P1 × P1 , Can. J. Math. 65 (2013) 823–842. [9] E. Guardo, A. Van Tuyl, Fat points in P1 × P1 and their Hilbert functions, Can. J. Math. 56 (2004) 716–741. [10] B. Harbourne, C. Huneke, Are symbolic powers highly evolved?, J. Ramanujan Math. Soc. 28 (2013) 311–330. [11] M. Waldschmidt, Propriétés arithmétiques de fonctions de plusieurs variables II, in: Lect. Notes Math., vol. 578, Springer-Verlag, 1977, pp. 108–135.
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Journal of Pure and Applied Algebra www.elsevier.com/locate/jpaa
Points fattening on P1 × P1 and symbolic powers of bi-homogeneous ideals a ´ Magdalena Baczynska , Marcin Dumnicki b , Agata Habura a , Grzegorz Malara a , a Piotr Pokora , Tomasz Szemberg a,∗,1 , Justyna Szpond a , b ´ Halszka Tutaj-Gasinska a b
Pedagogical University of Cracow, Institute of Mathematics, Podchora¸zych ˙ 2, PL-30-084 Kraków, Poland Jagiellonian University, Institute of Mathematics, Łojasiewicza 6, PL-30-348 Kraków, Poland
a r t i c l e
i n f o
Article history: Received 21 April 2013 Received in revised form 12 November 2013 Available online 16 December 2013 Communicated by S. Kovács
a b s t r a c t We study symbolic powers of bi-homogeneous ideals of points in X = P1 × P1 and extend to this setting results on the effect of points fattening obtained in [2] and [4]. 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. © 2013 Elsevier B.V. All rights reserved.
MSC: 14C20; 14J26; 14N20; 13A15; 13F20
1. Introduction Our approach to the study of the effect of the points fattening on postulation in P2 was initiated by Bocci and Chiantini in [2]. For a non-zero homogeneous ideal I C[Pn ] its initial degree α ( I ) is defined as the least integer t such that the graded part I t 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 2 Z (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 [5, Theorem 3.14] the ideal of 2Z is the second symbolic power of I . Of course the m-fold structure m Z is defined by I (m) for all m 1. In [4] 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 there is no single, canonical way to extend the initial degree notion to this setting. 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. The main results of this note are Theorem 2.2, Theorem 2.7 and Theorem 3.1. Theorem 2.5 generalizes the proof of Chudnovsky’s theorem on polynomial interpolation in P2 [3, General Theorem 6] to the bi-homogeneous setting and could be of independent interest.
*1
Corresponding author. Szemberg research was partially supported by NCN grant UMO-2011/01/B/ST1/04875.
0022-4049/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jpaa.2013.12.007
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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 O X (a, b) for the line bundle of bi-degree (a, b), i.e.
O X (a, b) = π V∗ OP1 (a) ⊗ π H∗ OP1 (b) , where π V and π H 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 I to be the ideal I (m) = j P i j , where I m = i P i is a bi-homogeneous primary decomposition, and the intersection j P i j is taken over all components P i such that the radical
√
P i 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 dy 0 − c y 1 . If Z = { P 1 , . . . , P s } s is a finite set of points in X , then its ideal I Z is just the intersection I ( Z ) = i =1 I ( P i ). It is convenient to denote by m Z m1 ms a fat points scheme defined by the ideal I ( P 1 ) ∩ · · · ∩ I ( P s ) for an s-tuple of positive integers m = (m1 , . . . , ms ). In this notation I (m) = I (m Z ) is the ideal defining the fat points scheme m Z 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 . We denote by I the sheafification of I . 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 O X (k1 , k2 ) ⊗ I > 0 ,
α ∗ ( I ) = min k: H 0 O X (k, k) ⊗ I > 0 . Remark 1.3. Let ([x0 : x1 ], [ y 0 : y 1 ]) be bi-homogeneous coordinates on P1 × P1 . Elements in H 0 (O X (k1 , k2 )) can be considered as homogeneous polynomials of degree k1 + k2 in variables x0 , x1 , y 0 , y 1 . From this point of view α + ( I ) is exactly the initial degree α ( I ) defined in the introduction. Following Waldschmidt [11] one defines for homogeneous ideals I ⊂ C[Pn ] an asymptotic counterpart of the initial degree
α ( I (m) )
α ( I ) = lim
m→∞
m
.
This invariant is called the Waldschmidt constant of I . This notion generalizes verbatim to the multi-homogeneous setting. Definition 1.4. Let I ⊂ C[ X ] be a non-zero bi-homogeneous ideal. We define the Waldschmidt constant of I as
α • ( I ) = lim
α • ( I (m) )
m→∞
m
,
where • ∈ {∗, +}. The inclusion I () I (k) ⊆ I (+k) implies that the sequence αm := α • ( I (m) ) is sub-additive, i.e. α+k α + αk . Fekete’s Lemma [6] implies that any sub-additive sequence am of real numbers is bounded below and converges to infm amm . In particular the limit in Definition 1.4 exists, and we have in fact
α • ( I ) = inf
α • ( I (m) ) m
.
Example 1.5. Let I ( P ) be the radical ideal defining a point P in X . Then α + ( I ( P )) = 1 and α ∗ ( I ( P )) = 12 .
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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 contrast to the projective plane where one has always α ( I (m) ) < α ( I (m+1) ), it might happen on X that
α ∗ I (m+1) = α ∗ I (m) .
(1)
We illustrate this phenomenon first with a simple example. Example 2.1. Let I be the radical ideal of a single point P . The equality
I (m) = I m
for all m 1
implies then
α ∗ I (m) = α ∗ I m =
m+1 2
= α ∗ I m+1 = α ∗ I (m+1)
for all m odd. The rest of this section is devoted to the detailed characterization of geometric conditions under which the equality in (1) is possible. Theorem 2.2. Let Z = { P 1 , . . . , P s } ⊆ X be a finite set of points and let I be the radical ideal of Z . The following conditions are equivalent (a) α ∗ ( I (m+1) ) = α ∗ ( I (m) ) for some m 1; (b) there exist finite sets Z V , Z H ⊂ P1 such that Z = Z V × Z H , i.e. Z is a grid in X . Proof. We begin by showing the implication from (a) to (b). Let m be an integer such that (a) holds. Let ( p , q) be a pair of integers such that there exists a section H 0 ( X , O X ( p , q)) vanishing to order at least m + 1 in points of Z and computing the α ∗ invariant, i.e.
σ ∈
α ∗ I (m+1) = max{ p , q}. Let D be a divisor defined by σ . Let C be an irreducible component of D of bi-degree (a, b) with mult P i C = mi for i = 1, . . . , s. The Plücker formula on P1 × P1 implies that
ab − a − b + 1 −
mi
i
2
0.
(2)
On the other hand, there is no divisor C of bi-degree (a − 1, b − 1) with multiplicities m1 − 1, . . . , ms − 1 through P 1 , . . . , P s . Indeed, otherwise the effective divisor C + ( D − C ) would have multiplicity m at all points of Z . Since its bi-degree is ( p − 1, q − 1), this clearly contradicts the equality in (a). Hence a simple conditions count and (2) yield
mi ab − a − b + 1. ab = h O X (a − 1, b − 1) 0
i
2
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. Assume to the contrary that D does consist of vertical fibers only. These fibers are then responsible for multiplicity m + 1 at every point of Z . Since the fibers are smooth, this multiplicity comes from multiple fibers through each point in Z . More precisely, for every point in Z , the divisor D contains the vertical fiber through this point with multiplicity m + 1 2. Removing the reduced union of all these fibers from D results in a divisor D vanishing in all points of Z to order m and of bi-degree strictly less than that of D . This contradicts (a). We conclude that D is the union of k > 0 vertical fibers V 1 , . . . , V k and l > 0 horizontal fibers H 1 , . . . , H l . In order to complete the proof of implication from (a) to (b) we claim that
Z = { Q i j = V i ∩ H j , i = 1, . . . , k, j = 1, . . . , l}.
(3)
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First we show the ⊇ inclusion. Assume to the contrary that some Q i j is not contained in Z . Then removing from D the union of fibers V i ∪ H j gives a divisor D = D − V i − H j of bi-degree one less than that of D and vanishing along Z with multiplicities at least m. This contradicts the assumption in (a). Now we show the ⊆ inclusion in (3). Assume to the contrary that there is a point Q ∈ Z , which is not an intersection point of one of the horizontal fibers H 1 , . . . , H l with one of the vertical fibers V 1 , . . . , V k . Without loss of generality we can assume that Q ∈ V 1 . This implies that V 1 has multiplicity at least m + 1 in D. Removing the union H 1 ∪ V 1 from D we obtain again a divisor D = D − V 1 − H 1 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 Q 11 (which is the only double point in H 1 ∪ V 1 ), since in the situation considered here we have mult Q 11 ( D ) m + 2. The assertion in (b) follows with the sets Z V = π V ( V 1 ∪ · · · ∪ V k ) and Z H = π H ( H 1 ∪ · · · ∪ H l ). Now, we turn to the implication from (b) to (a). As kindly pointed out to us by the referee, there is a paper by Guardo and Van Tuyl [9] studying in particular complete intersection zero dimensional subschemes in P1 × P1 , see also [7,8]. Everything what comes now (and in fact much more) can be read of these papers. We outline here the path of thought relevant for our purposes. To begin with assume that Z V is defined by a reduced polynomial F of degree r in [x0 : x1 ] and Z H is defined by a reduced polynomial G of degree s in [ y 0 : y 1 ]. In other words, we assume that the radical ideal I of the grid Z = Z V × Z H is generated by forms F and G (which we now treat as bi-homogeneous forms on X ). In particular I is a complete intersection and hence I (m) = I m for all m by Macaulay’s unmixedness theorem [5, Corollary 18.14]. Then we have
α ∗ I (m) = min{c: ∃i , j: i + j = m, ir c , js c }.
(4)
Note that if there exist i and j such that i + j = m and ir = js, then
α ∗ I (m) = ir = js. Otherwise i and j have to be spread possibly equally with weights r and s in order to compute that if i 0 is the least integer such that i 0 r > (m − i 0 )s, then
α ∗ ( I (m) ). By this we mean
α ∗ I (m) = min i 0 r , (m − i 0 + 1)s . Moreover, for any m the value of the difference
α ∗ I (m+1) − α ∗ I (m) = k · gcd(r , s) for some 0 k min{r , s} and any such multiple (in particular 0) occurs for infinitely many m. This shows the statement in (a) and finishes the proof of the theorem. 2 Remark 2.3. Note that under assumption (b) in the above theorem one can obtain from [9, Theorem 4.11 and Theorem 5.4] all bi-graded Betti numbers of the grid. It follows immediately from Theorem 2.2 and formula (4) that two consecutive equalities as in (1) are not possible. Corollary 2.4. 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 f ∈ I (m) of bi-degree ( p − 1, q − 1). This shows that there is always to the second set of variables, we obtain a polynomial ∗ ( m+2) ∗ ( m) ) > α ( I ). Our approach has the advantage that it does not depend on the differentiation and thus inequality α ( I could be generalized to arbitrary surfaces. 2.2. Configuration of points with the minimal positive effect of fattening Here we show that the extremely useful result of Chudnovsky [3] relating Waldschmidt constants and initial degrees generalizes to a multi-homogeneous setting. The result of Chudnovsky was inspired by slightly weaker results obtained by Waldschmidt and Skoda with analytic methods. Our approach here is modeled on the detailed algebraic proof by Harbourne and Huneke in [10, Proposition 3.1] which in turn was inspired by ideas sketched out by Chudnovsky himself. This result is of independent interest. Example 2.6 shows that the inequality in (5) cannot be improved in general.
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Theorem 2.5. Let P 1 , . . . , P s ∈ X be mutually distinct points and let I =
α ∗ ( I (m) ) m
i
1559
I ( P i ) ⊂ C[ X ] be the ideal of their union. Then
α∗(I )
(5)
2
for all m 1. ∗ ∗ Proof. Let α ( I ) = a. Choose t distinct points Q 1 , . . . , Q t ∈ { P 1 , . . . , P s } with the smallest possible t such that α ( J ) = a for J = j I ( Q j ). By minimality of t, the points Z = { Q 1 , . . . , Q t } impose independent conditions in bi-degree (a − 1, a − 1) so
that t = a2 . Claim. We claim that the linear system defined by global sections of the sheaf O X (a, a) ⊗ J has no fixed components. Here J denotes the sheafification of the ideal J . Assume to the contrary that there exists a common component Γ . By the definition of Z , for any point Q ∈ Z there exists a divisor C Q in the linear system |O X (a − 1, a − 1) ⊗ I Z \{ Q } |. If Γ is not a component of C Q , then let C be a curve of bi-degree (1, 1) not containing Γ as a component. Then
C Q + C ∈ O X (a, a) ⊗ J has not Γ as a component, a contradiction. This shows that Γ must be a component of all divisors C Q with Q ∈ Z . This Γ cannot contain any point R of Z , since otherwise C R would contain R as well, a contradiction with the definition of C R . Then take any divisor C Q and let Γ be a divisor of the same bi-degree as Γ passing through Q . Then C Q − Γ + Γ is an effective divisor of bi-degree (a − 1, a − 1) vanishing along Z , a contradiction. This shows the claim. Let A be a non-zero form in ( I (m) )(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 adapted to X , we have
2ab = (a, a) · (b, b) = div( A ) · div( B ) ma2 , and hence
α ∗ ( I (m) ) m
α∗(I ) 2
.
2
Example 2.6. Continuing Example 2.1 with I the ideal of a single point, we have that the inequality in (5) cannot be improved in general.
α ∗ ( I ) = 1 and
α ∗ ( I (2m) ) 2m
= 12 . This shows
We apply now Theorem 2.5 to the case with several consecutive jumps of α ∗ ( I (m) ) equal to 1. It follows from the proof that the assumption in (6) below is equivalent to the assumption α ∗ ( I (6) ) = α ∗ ( I ) + 5. We prefer the present formulation which is quite in spirit of [4, Theorem 3.1]. Theorem 2.7. Let I be a radical bi-homogeneous ideal of a set Z = { P 1 , . . . , P s } of points in X . Assume that
α ∗ I ( 6 ) = α ∗ I ( 5 ) + 1 = · · · = α ∗ I ( 2 ) + 4 = α ∗ ( I ) + 5.
(6)
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 (6) cannot be shortened in general. Proof. The assumption α ∗ ( I (6) ) = α ∗ ( I ) + 5 together with Theorem 2.5 yields it suffices to deal with the case α ∗ ( I ) = 2. So assume α ∗ ( I ) = 2, hence
α ∗ ( I ) 2. If α ∗ ( I ) = 1, then we are done. So
There exists
There does not exist
A divisor D of bi-degree (7, 7) with multiplicities at least 6 in each P i ∈ Z
A divisor E of bi-degree (6, 6) with multiplicities at least 6 in each P i ∈ Z
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 fibers through S are disjoint from Z followed by blowing down proper transforms of these two fibers to points P and Q in P2 .
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On P2 we have now the following situation: There exists
There does not exist
A divisor D 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 )
A divisor E of degree 12 vanishing to order at least 6 at P , Q and all points in Z
with D being the proper transform of D under μ. Obviously P , Q and at least one point R = P i 0 ∈ Z are not collinear (otherwise there would be a divisor of degree 6 with vanishing orders as in the last framed box). Applying the standard Cremona transformation τ based on these points to divisor D results in There exists
There does not exist
A divisor D of degree 8 vanishing to order 6 at all points in Z = τ ( Z \ { R })
A divisor E of degree 6 vanishing to order 6 at all points in Z
This implies immediately that 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
deg( F )
1 2
(mult A F + mult B F + multC F ).
This fact applied to F := D gives a contradiction. That the result is sharp follows from Example 2.8 below.
2
The next example shows that it may happen that
α ∗ I (5 ) = α ∗ I (4 ) + 1 = · · · = α ∗ I (2 ) + 3 = α ∗ ( I ) + 4 and
α ∗ ( I ) = 1.
Example 2.8. For the set Z of points
P 1 = [1 : 0], [1 : 0] ,
P 2 = [1 : 0], [1 : 1] ,
P 3 = [1 : 1], [1 : 0]
and
P 4 = [0 : 1], [1 : 1]
and I = I ( Z ) we have
m
1 2 3 4 5 . 2 3 4 5 6
α ∗ ( I (m) )
Proof. It is easy to draw divisors with vanishing orders claimed above and of the given bi-degree. 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. 2 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.
Example 2.9. 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.
A similar effect occurs on an (r , s) grid with r and s relatively prime. 3. Symbolic powers and α + invariant The behavior of the α + invariant is more similar to the initial degree in the plane, see Remark 1.3. Let f be a bihomogeneous 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 m Z . This shows that there is always the strong inequality
α + I (m) < α + I (m+1)
for all m 1. In this section we describe the situation when the effect of fattening is the minimal possible, i.e. equal to 1.
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Theorem 3.1. Let Z = { P 1 , . . . , P s } ⊆ X be a set of points and let I be the radical ideal of Z . Assume that
α + I (m+1) = α + I (m) + 1
(7)
for some m ∈ Z1 . Then all points { P 1 , . . . , P s } 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 , O X ( 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 an irreducible component of D of bi-degree (a, b) and with mult P i C = mi . The assumption in (7) implies then that
α + I ( P 1 )m1 ∩ · · · ∩ I ( P s )ms = a + b = α + I ( P 1 )m1 −1 ∩ · · · ∩ I ( P s )ms −1 + 1, where as usually a non-positive power of an ideal is understood as the whole ring. Indeed, assuming that α + ( I ( P 1 )m1 −1 ∩ · · · ∩ I ( P s )ms −1 ) = c < a + b − 1 and picking a divisor Γ defined by a form in I ( P 1 )m1 −1 ∩ · · · ∩ I ( P s )ms −1 of bi-degree (u , v ) with u + v = c, we would obtain an effective divisor D − C + Γ of bi-degree ( p − a + u , q − b + v ) vanishing along Z with multiplicity at least m. Since p − a + u + q − b + v d − 2 this contradicts (7). 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 P 1 , . . . , P s (a negative multiplicity means no vanishing condition). Hence
h0 O X (k, a + b − 2 − k) = (k + 1)(a + b − 1 − k)
mi
i
2
ab − a − b + 1,
where the last inequality is the genus bound (2). 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 (7). 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 mult Q D m + 2. Thus for D = D − ( V + H ) we have mult P D m for all points P ∈ Z . This contradicts (7) again and we are done. 2 Turning to the jumps of the
α + invariant by 2, the above proof yields immediately the following observation.
Corollary 3.2. Let Z = { P 1 , . . . , P s } ⊆ X be a set of points and let I be the radical ideal of Z . Assume that
α + I (m+1) = α + I (m) + 2
(8)
Z1 . Let D be a divisor on X of bi-degree ( p , q) such that α + ( I (m+1) )
for some m ∈ = 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
(9)
must hold for all k in the range 0, . . . , a + b − 3. We want to show that a 1 and b 1. Assuming b 2 we have a − 1 a + b − 3 hence with k = a − 1 we obtain from (9)
a(b − 1) ab − a − b + 1 which implies b 1, a contradiction. The case a 2 is excluded by symmetry and we are done with the first claim. 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 P i at most by 1 but the sum of degrees by at least 3 contradicting (8). The same argument works for horizontal fibers. 2
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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 P 1 = (0, 0), P 2 = (1, 1), P 3 = (1, 2), P 4 = (2, 2), P 5 = (3, 0), P 6 = (3, 3) and let I be the bi-homogeneous ideal of their union. Then
α + ( I ) = 4 and α + I (2) = 6.
(10)
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).
Thus we have exhibited explicitly divisors realizing inequalities in (10). We leave it to a motivated reader to check that neither for I nor for I (2) can the α + invariant be lowered. 2 Acknowledgements This note resulted from a workshop on linear series held in Lanckorona in February 2013. We thank Pedagogical University of Cracow for financial support. Furthermore we are deeply obliged to the referee for an unusually detailed report which considerably contributed to the readability and correctness of the paper! References [1] Th. Bauer, S. Di Rocco, B. Harbourne, M. Kapustka, A.L. Knutsen, W. Syzdek, T. Szemberg, A primer on Seshadri constants, in: D.J. Bates, G.-M. Besana, S. Di Rocco, C.W. Wampler (Eds.), 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, in: Contemp. Math., vol. 496, 2009, 362 pp. [2] C. Bocci, L. Chiantini, The effect of points fattening on postulation, J. Pure Appl. Algebra 215 (2011) 89–98. [3] G.V. Chudnovsky, Singular points on complex hypersurfaces and multidimensional Schwarz Lemma, in: M.-J. Bertin (Ed.), Séminaire de Théorie des Nombres, Paris, 1979–80, Séminaire Delange–Pisot–Poitou, in: Prog. Math., vol. 12, Birkhäuser, Boston, Basel, Stuttgart, 1981. ´ [4] M. Dumnicki, T. Szemberg, H. Tutaj-Gasinska, Symbolic powers of planar point configurations, J. Pure Appl. Algebra 217 (2013) 1026–1036. [5] D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Grad. Texts Math., vol. 150, Springer-Verlag, New York, 1995. [6] H. Fekete, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Z. 17 (1923) 228–249. [7] E. Guardo, Fat points schemes on a smooth quadric, J. Pure Appl. Algebra 162 (2001) 183–208. [8] E. Guardo, B. Harbourne, A. Van Tuyl, Symbolic powers versus regular powers of ideals of general points in P1 × P1 , Can. J. Math. 65 (2013) 823–842. [9] E. Guardo, A. Van Tuyl, Fat points in P1 × P1 and their Hilbert functions, Can. J. Math. 56 (2004) 716–741. [10] B. Harbourne, C. Huneke, Are symbolic powers highly evolved?, J. Ramanujan Math. Soc. 28 (2013) 311–330. [11] M. Waldschmidt, Propriétés arithmétiques de fonctions de plusieurs variables II, in: Lect. Notes Math., vol. 578, Springer-Verlag, 1977, pp. 108–135.