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Bound on Seshadri constants on [P.sup.1] x [P.sup.1].

1 Introduction

Let X be a projective algebraic surface (over C) with an ample line bundle L. Let [P.sub.1],..., [P.sub.r] be r different points on X. Let us recall the definition introduced by Demailly in [4].

Definition 1. The Seshadri constant of L in [P.sub.1],..., [P.sub.r] is defined as the number

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Remark 2. It follows from the definition that for an ample line bundle L on X

0 < [epsilon] (L,[P.sub.1],...,[P.sub.r])[less than or equal to][square root of [L.sup.2]/r.

Notation:

For [P.sub.1],..., [P.sub.r] generic on X we will write [epsilon](L,r) instead of [epsilon](L, [P.sub.1],..., [P.sub.r]).

Finding the exact value of these constants is in general a difficult problem. For example, for [P.sup.2] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the exact values of [epsilon](L,r) are known only if r [less than or equal to] 9 or r = [k.sup.2], k [member of] N. The famous Nagata Conjecture (cf [11]) states that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This problem is still open (see [7] for more about the subject). Moreover, all known values of Seshadri constants on algebraic surfaces are rational. In general, it is hard to find the value of a Seshadri constant even in one point. The interested reader may look for example in [1], [2], [5], [10], [14] and the references therein.

In his recent paper, [6], Fuentes Garcia investigated the Seshadri constants in one point on geometrically ruled surfaces, in case of ruled surfaces with the invariant e > 0 he computes [epsilon](A, x) explicitly, whereas for surfaces with e [less than or equal to] 0 he either gives the exact value of [epsilon] or bounds for its value, depending on the position of the point on the surface.

On the other hand Syzdek in [13] studied the existence of so called Seshadri submaximal curves on [P.sup.1] x [P.sup.1] with different polarizations L.

Definition 3. 1. A Seshadri submaximal curve is a curve C on X, such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

2. A curve C in a linear system [absolute value of L] on a surface, passing through r points with multiplicities [m.sub.1],..., [m.sub.r] is Riemann-Roch expected if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Of course, when the Riemann-Roch theorem implies the existence of a submaximal curve, then the Seshadri constant is rational and less then [square root of [L.sup.2]/r]. This follows from the fact that there is a finite number of Seshadri submaximal curves on a surface, see for example [12].

Syzdek in [13] gave a list of the Riemann-Roch expected submaximal curves on [P.sup.1] x [P.sup.1]. She also proved that there exists a number Ro (depending on the type of the polarization), such that for r [greater than or equal to] [R.sub.0], there are no Riemann-Roch ex pected submaximal curves on [P.sup.1] x [P.sup.1]. In particular, she shows that for r [less than or equal to] 8, a Riemann-Roch expected submaximal curve always exists.

In this note we give a uniform lower bound for the Seshadri constant on [P.sup.1] x [P.sup.1], in case r is such, that there are no Riemann-Roch expected submaximal curves on [P.sup.1] x [P.sup.1], so [epsilon](L,r) is "expected" to be maximal (ie equal to [square root of [L.sup.2]/r). We prove the following theorem.

Theorem 4. Let L be a line bundle in [P.sup.1] x [P.sup.1], of type ([alpha],[beta]). Let r be such, that there exist no Riemann-Roch expected submaximal curves on ([P.sup.1] x [P.sup.1], L). Then

[epsilon](L, r)[greater than or equal to] [square root of 2[alpha][beta]/r + 1/2].

2 Useful facts

Lemma 5. (See [3]). Let L be a line bundle on [P.sup.1] x [P.sup.1], of type (a, b). Assume that there exists C [member of] [absolute value of L], a reduced and irreducible curve on [P.sup.1] x [P.sup.1], passing through points [P.sub.1],..., [P.sub.r] with multiplicities [m.sub.1],..., [m.sub.r], where [m.sub.j] > 0, j = 1,..., r. Then, for any chosen [m.sub.j], there exists a reduced and irreducible curve on [P.sup.2] of degree d = a + b - [m.sub.j], passing through r + 1 points on [P.sup.2] with multiplicities a - [m.sub.j], b - [m.sub.j], [m.sub.1],..., [m.sub.j-1], [m.sub.j+1],..., [m.sub.r].

Definition 6. A curve D on a surface X, passing through points [P.sub.1],..., [P.sub.r] with multiplicities [m.sub.1],..., [m.sub.r], is called almost homogeneous if all but at most one [m.sub.j] are equal.

Lemma 7. (See [13], Proposition 2.10.). Let (X, L) be a polarized surface with Picard number [??]. Let [P.sub.1],..., [P.sub.r] be general points on X. If [??] = 1 or [??] = 2, then any reduced and irreducible Seshadri submaximal curve on X is almost homogeneous.

Lemma 8. (See [15], Lemma 1). Let C be a reduced and irreducible curve on a surface X, passing through a general point P [member of] X with multiplicity m [greater than or equal to] 2. Then

[C.sup.2] [greater than or equal to] [m.sup.2] - m + 1.

Lemma 9. (See [16]). Let C be a reduced and irreducible curve of degree d on [P.sup.2], passing through the general points [P.sub.1],..., [P.sub.r] with multiplicities [m.sub.1],..., [m.sub.r]. Then [d.sup.2] [greater than or equal to] [r.summation over (j=1)] [m.sup.2.sub.j] - [m.sub.q],

for any q such that [m.sub.q] > 0.

3 Proof

To prove Theorem 4 we have to exclude the existence of (reduced and irreducible) Seshadri submaximal curves C on [P.sup.1] x [P.sup.1], passing through r general points with multiplicities [m.sub.1],..., [m.sub.r] and satisfying

LC/[m.sub.1] + ... + [m.sub.r] < [square root of 2[alpha][beta]/r + 1/2. (1)

Suppose, that such a curve C exists (and is not Riemann-Roch expected). Let C be of type (a, b).

First, observe that we may exclude the situation a = 0 or b = 0. Indeed, suppose for example that b = 0. Then, as the curve is reduced and irreducible, it must be a = 1, so m = 1, and such a curve is always Riemann-Roch expected.

Then, observe that Lemma 7 implies that it is enough to exclude the existence of almost homogeneous curves, satisfying inequality (1). So, let us assume that C is of type (a, b) on [P.sup.1] x [P.sup.1], with a > 0, b > 0, passing through general points [P.sub.1],..., [P.sub.r] with multiplicity m in [P.sub.1],..., [P.sub.r-1] and with multiplicity [mu] in [P.sub.r]. We will denote such a curve by ((a, b); [m.sup.x(r-1)],[mu]) and we will write a + b =: c.

From Lemma 5 it follows that we may move our considerations to [P.sup.2], considering instead of C the curve on [P.sup.2] (in what follows also denoted by C), of degree c - m and with multiplicities a - m, b - m, [m.sup.x(r-2)], [mu] in r + 1 general points. We will denote such a curve by ((c - m); a - m, b - m, [m.sup.x(r-2)], [mu]).

Let us assume L is of type ([alpha], [beta]). We have to exclude the existence of an almost homogeneous submaximal curve on [P.sup.1] x [P.sup.1], such that

C = ((a, b); [m.sup.r-1],[mu]),

so we have to exclude the existence of a curve C, such that:

1. On [P.sup.2]: C is numerically equivalent to ((c - m); a - m, b - m, [m.sup.x(r-2)], [mu])

2. On [P.sup.1] x [P.sup.1]: LC/(r - 1)M + [mu] < [square root of 2[alpha][beta]/r + 0.5.

Then, if

LC < [square root of 2[alpha][beta]/r + 0.5 ((r - 1)m + u), (2)

we have

2 [square root of ab[alpha][beta] [less than or equal to] [alpha]b + [beta]a [less than or equal to] [square root of 2[alpha][beta/r + 0.5((r - 1)m + [mu]) = 2 [square root of [alpha][beta]/2r + 1 ((r - 1)m + [mu]), (3)

which gives

[square root of ab] < [square root of 1/2r + 1] ((r - 1)m + [mu]). (4)

Thus, our aim is to prove that there are no curves on [P.sup.2], satisfying C [equivalent to]((c-m);a - m, b - m, [m.sup.x(r-2)],[mu]) and [square root of ab] < 1/square root of 2r+1] ((r- 1)m+[mu]).

Let us now assume that m [greater than or equal to] 2. Consider the curve

[??] = (c - m)H - (a - m)[E.sub.1] - (b - m)[E.sub.2] - [mE.sub.3] - ... - [mE.sub.r-1] - [mu][E.sub.r+1]

on the blow up of [P.sup.2] in [P.sub.1],..., [P.sub.r-1], [P.sub.r+1], passing with multiplicity m through [P.sub.r]. To this curve we may apply Lemma 8. We get

[([??]).sup.2] = [(a+b-m).sup.2] - [(a-m).sup.2] - [(b-m).sup.2] - (r-3) [m.sup.2] - [[mu].sup.2] [greater than or equal to] [m.sup.2]-m+1,

which gives

2ab - (r - 1)[m.sup.2] - [[mu].sup.2] + m - 1 [greater than or equal to] 0. (5)

Together with the inequality (4), we get

2/2r+1([(r-1)m+[mu]).sup.2] - (r-1) [m.sup.2] - [[mu].sup.2] + m-1 [greater than or equal to] 0. (6)

Assuming first, that m = 2, from the above inequality we obtain

-(2r-1)[[mu].sup.2] + 8(r-1)[mu] - 10r + 13 [greater than or equal to] 0. (7)

This is a quadratic inequality with respect to [mu], with a negative leading term coefficient. It is easy to check that the discriminant of the quadratic function is negative for all r > 2, so there are no solutions for the above inequality.

Assume now, that m[greater than or equal to]3, so m [less than or equal to][m.sup.2]/3. From inequality (6) after multiplying by (2r + 1) we get

[2(r - 1).sup.2] [m.sup.2] + 4(r - 1)m[mu] + 2[[mu].sup.2] - (2r + 1) (r - 1)[m.sup.2]- (2r + 1)[[mu].sup.2] + (2r + 1)(m - 1) [greater than or equal to] 0, (8)

so, writing [m.sup.2]/3 instead of m - 1 we get

(-7/3 r + 10/3)[m.sup.2] + 4(r - 1)m[mu] - (2r - 1)[[mu].sup.2] [greater than or equal to] 0. (9)

Treating this as a quadratic inequality with m as a variable and r, [mu] as parameters, we see that the [m.sup.2]-coefficient is negative and the discriminant equals

4/3 [[mu].sup.2] (-2[r.sup.2] + 3r + 2), (10)

so the inequality has no solutions for r [greater than or equal to] 3.

Now assume that m = 1 and [mu] [greater than or equal to] 2. Then inequality (4) becomes

[square root of ab] < 1/[square root of 2r + 1 (r - 1 + [mu]). (11)

Apply Lemma 8 to the curve ((c - 1); a - 1, b - 1,[1.sup.x(r-2)]) passing through the last point with multiplicity [mu]. We get

[(a+b-1.sup.)2] - [(a-1).sup.2] + [(b-1).sup.2] - (r-2) - [[mu].sup.2] + [mu] - 1 [greater than or equal to]0, (12)

equivalent to

2ab - r - [[mu].sup.2] + [mu] [greater than or equal to] 0. (13)

Using inequality (11) we get

[[mu].sup.2] (1 - 2r) + [mu](4r - 1) - 5r + 2 [greater than or equal to] 0, (14)

this gives a contradiction.

Assume now that m = [mu] = 1. Consider the linear system of curves of degree c - 1 passing through two points with multiplicities a - 1, b - 1. In [11], Nagata proved that the dimension of such a system is either expected or every curve in the system contains a line. However, our curve C is reduced and irreducible and passes through at least seven more points in general position. Thus, the dimension of the system ( (c - 1); a - 1, b - 1) is expected. Moreover, the conditions on the system to pass through points in general position with multiplicity one are independent, so each such point causes the dimension of the system to become one less. So, in our case we have a curve ((c - 1); a - 1, b - 1) passing through r - 1 points with multiplicity one. This means that

(a+b-1)(a+b+2)/2 - (a-1)a/2 - (b-1)b/2 - (r-1) [greater than or equal to] 0. (15)

This implies that

ab + a + b - r [greater than or equal to] 0, (16)

and this would mean that our curve C on [P.sup.1] x [P.sup.1] is Riemann-Rock expected, contrary to our assumptions.

Remark 10. Harbourne and Roe showed us that from their Theorem I.2.1.(a) in [9], it follows easily that for r [greater than or equal to] 3 [([alpha] + [beta]).sup.2]/[alpha][beta]

[epsilon](L, r)[greater than or equal to] [square root of 2[alpha][beta]/r + r/2r-5].

They suggested as well, that using their method one might be able to improve the result, to get the bound asymptotically [square root of 2[alpha][beta]/r+1/3]. This will be the aim of our future project.

Harbourne also pointed out that from his paper LS] applied to our situation, it follows, that if [L.sup.2]r = 2[alpha][beta]r is a square of a natural number and r [greater than or equal to] [L.sup.2], then the Seshadri constant [epsilon](L, r) has maximal possible value, [epsilon](L, r) = [square root of [L.sup.2]/r].

Acknowledgements: The second author would like to thank the organizers of the Workshop "Linear Systems and Subschemes", Cindy De Volder, Luca Chiantini and Marc Coppens for the invitation and for their hospitality. Both authors would like to thank Brian Harbourne and Joaquim Roe for their useful comments and suggestions.

References

[1] Bauer, Th.: Seshadri constants on algebraic surfaces. Math. Ann. 313, (1999) 547-583

[2] Bauer, Th., Szemberg, T.: Seshadri constants on abelian surfaces. (An appendix to: Bauer, Th: Seshadri constants and periods of polarized abelian varieties). Math. Ann. 312, (1998) 618-623

[3] De Volder, C., Laface, A. On linear systems on [P.sup.3] through multiple points. J. Algebra 310 (2007) no. 1, 207-217

[4] Demailly, J.-P : Singular Hermitian metrics on positive line bundles. Lect. Notes Math. 1507, Springer 1992, 87-104

[5] Ein, L., Kuchle, O., Lazarsfeld, R.: Local positivity of ample line bundles. J. Diff. Geom. 42, (1995)193-219

[6] Garcia, Fuentes L.: Seshadri constants on ruled surfaces: the rational and elliptic cases. manuscripta math. 119, (2006), 483-505

[7] Harbourne, B.: On Nagata's conjecture. J. Algebra 236 (2001) 692-702

[8] Harbourne, B.: Seshadri constants and very ample divisors on algebraic surfaces J. Reine Angew. Math. 559, (2003), 115-122

[9] Harbourne, B., Roe, J.: Discrete behaviour of Seshadri constants on surfaces. http://mat.uab.es/ roe/papers/SeshSurfaces.pdf

[10] Kiichle, O.: Multiple point Seshadri constants and the dimension of adjoint linear series. Ann. Inst. Fourier 46, (1996) 63-71

[11] Nagata, M., On the 14-th problem of Hilbert, Am. J. Math. 81(1959), 766-772

[12] Syzdek, W.: Nagata submaximal curves on [P.sup.1] x [P.sup.1], Ann. Polon. Math. 80 (2003), 223-230

[13] Syzdek, W.: Seshadri constants and geometry of surfaces, preprint, http://www.uni-essen.de/ mat903/alggeo.html

[14] Szemberg, T.: Global and local positivity of line bundles. Habilitationsshrift Essen 2001

[15] Xu, G.: Ample line bundles on smooth surfaces, J. Reine Angew. Math. 469, (1995),199209

[16] Xu, G.: Curves in [P.sup.2] and symplectic packings, Math. Ann. 299, (1994), 609-613

Cindy De Volder * Halszka Tutaj-Gasinska

* The first author is the corresponding author 2000 Mathematics Subject Classification : 14C20.

Ghent University, Department of Pure Mathematics and Computeralgebra, Ghent, Belgium

email:cindy.devolder@gmail.com

Jagiellonian University, Institute of Mathematics, Krakow, Poland

email:htutaj@im.uj.edu.pl
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Author:De Volder, Cindy; Tutaj-Gasinska, Halszka
Publication:Bulletin of the Belgian Mathematical Society - Simon Stevin
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Date:Nov 1, 2009
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