In this note we relate about the problem of evaluate the dimension of linear
systems through fat points defined on generic $K3$ surfaces.Comment: 4 pages, LaTe

De Volder, Cindy

Laface, Antonio

English

arXiv

Ghent University Academic Bibliography

arXiv:math/0409524v1  [math.AG]  27 Sep 2004RECENT RESULTS ON LINEAR SYSTEMS ON GENERIC K3SURFACESCINDY DE VOLDER AND ANTONIO LAFACEAbstract. In this note we relate about the problem of evaluate the dimensionof linear systems through fat points defined on generic K3 surfaces.1. Introduction and statement of the problemIn what follows we assume that the ground field is algebraically closed of charac-teristic 0. With S we always denote a smooth projective generic K3 surface, i.e.Pic (S) = 〈H〉 and let n = H2. Consider r points in general position on S, toeach one of them associate a natural number mi called the multiplicity of the point.We will denote by L = Ln(d,m1, . . . ,mn) the linear system |dH | through the rpoints with the given multiplicities. Define the virtual dimension of the system asv(L) = d2n/2+1−∑mi(mi+1)/2 and its expected dimension by e = max{v,−1}.Observe that e ≤ dim(L) and that the inequality may be strict if the conditionsimposed by the points are dependent. In this case we say that the system is spe-cial. By S′ we will denote the blow-up of S along the r points, given two curvesA,B on S, the intersection AB will be defined as the intersection of their stricttransforms on S′. The problem of classifying special systems has been largely stud-ied for linear systems on the plane [2, 6, 11] and more generally for systems onrational surfaces [7, 8]. The main conjecture on the structure of such systems hasbeen formulated in [8]. In this note we report about some recent results in thecase of generic K3 surfaces. In [3] the authors proved that on the projective planethis conjecture is equivalent to an older one given by Segre in [11]. The advantageof Segre conjecture is that it can be formulated in the same way on any surface.Starting from this idea we proved in [4] the equivalence of Conjecture 2.1 with Con-jecture 2.2 on a generic K3 surface. An attempt to prove Conjecture 2.2 has beendone in [5] by using a degeneration technique inspired by [1]. The main result, byusing this technique, is Theorem 3.1 which relates the speciality of some linear sys-tems through points of the same multiplicity with the speciality of systems throughjust one point.1991 Mathematics Subject Classification. 14C20, 14J28.Key words and phrases. Linear systems, fat points, generic K3 surfaces.The first author is a Postdoctoral Fellow of the Fund for Scientific Research-Flanders (Belgium)(F.W.O.-Vlaanderen).The second author would like to thank the European Research and Training Network EAGERfor the support provided at Ghent University. He also acknowledges the support of the MIUR ofthe Italian Government in the framework of the National Research Project “Geometry in AlgebraicVarieties” (Cofin 2002).12 CINDY DE VOLDER AND ANTONIO LAFACE2. The equivalence of the two conjecturesAs stated in the introduction we consider here an extension, to any surface, of Segreconjecture about special linear systems.Conjecture 2.1. If L is non-empty and reduced linear system on a surface S, thenit is non-special.By Bertini second theorem, this conjecture tell us that if L is special, then thereexists an irreducible curve C such that 2C ⊆ Bs (L). This means that, if Conjec-ture 2.1 is true, then in order to give a classification of special systems on a surfacewe should be able to classify the type of the curve C. In the case of generic K3surfaces we proved the equivalence of the preceding conjecture with the following(see [4]).Conjecture 2.2. Let L and S be as above, then(i) L is special if and only if L = L4(d, 2d) or L = L2(d, d2) with d ≥ 2;(ii) if L is non-empty then its general divisor has exactly the imposed multiplic-ities in the points pi;(iii) if L is non-special and has a fixed irreducible component C thena) L = L2(m+ 1,m+ 1,m) = mC + L2(1, 1) with C = L2(1, 12) orb) L = 2C with C ∈ {L4(1, 13),L6(1, 2, 1),L10(1, 3)} orc) L = C.(iv) if L has no fixed components then either its general element is irreducibleor L = L2(2, 2).The proof of this result proceeds by analyzing the base locus of the system L.Assume that there exist distinct irreducible curves Ci and Dj such thatL =a∑i=1µiCi +b∑i=1Di +M,where µi ≥ 2 and M has no fixed components. By putting A,B to be two of theirreducible curves into the fixed part of L and assuming conjecture 2.1 to be true,we have that v(A) = v(B) = v(A+B) = 0. Since v(A+B) = v(A)+v(B)+AB−1,this implies that AB = 1. Hence this gives that CiCj = CiDj = DiDj = 1 andC2i ≤ 1. Now, it is possible to prove (see [4]) that given two distinct irreduciblecurves A and B on S then either AB 6= 1 or A = L2(1, 12) and B is an irreducibleelement of L2(1, 1).3. A degeneration of K3 surfacesIn this section we consider an attempt to prove conjecture 2.2 by using a degener-ation of K3 surfaces to a union of planes and the blow-up of a K3 along points.Let ∆ be an open disk and let X be the blow-up of S ×∆ along b general pointsof S × {0}. The threefold X is equipped with two projections p1, p2 on ∆ and Srespectively and the general fiber Xt of p1 is isomorphic to S, while X0 is a re-ducible surface given by the union of b planes with a surface S. The last surface isthe blow-up of S along the b points. Each one of the b planes Pi cuts a curve Rion S which is a line in Pi and a (−1)-curve in S. Now given a line bundle L onRECENT RESULTS ON LINEAR SYSTEMS ON GENERIC K3 SURFACES 3S it is possible to construct infinitely many line bundles (depending on the integerk) OX(L, k) := p∗2(L)⊗OX(kS) on X such that each one restricted to Xt gives L.Defining X (L, k) as the restriction to X0 we have thatX (L, k)|Pi = OP2(k)X (L, k)|S = b∗(L)⊗OS(−∑bi=1 kEi),where b : S → S is the blow-up map. This construction allows us to degeneratea system on S to a union of systems on the Pi’s and S in the following way. LetZ := m1q1+ · · ·+mrqr be a subscheme of S with points in general position. Chosena1, . . . , ab positive integers such that a1+ · · ·+ab ≤ r, let Z′i be the specialization ofai points of Z to points of Pi (with the same multiplicities). Let Z′Sbe the residualsubscheme, made of r −∑ai general points of S. Given Z′ := Z ′1 + . . .+ Z′b + Z′S,one has that X (L, k) ⊗ IZ′ , is a degeneration of L ⊗ IZ . In this way, the startingsystem L through r degenerate to the system L0 on X0 made by the Li on the Piand by the LS on S. Observe that the last system corresponds to a system on Sthrough less than r points. In this way, by using the fact that the homogeneousplanar systems L2(d,m4), L2(d,m9) are never special, it is possible to use thepreceding degeneration in an inductive way. So, for example consider the systemLn(d,m4h), take b = 4h−1 and put four general points on each of the Pi. In thisway the speciality of the starting system is related to that of Ln(d,m4h−1) and soon. More generally we have the following (see [5]).Theorem 3.1. If Ln(d,m) is non-special for all non-negative integers (d,m) thenLn(d′,m′4h9k) is non-special for all non-negative integers (d′,m′, h, k).Unfortunately it is an open problem to evaluate if a system through just one pointis special or not. The only known example is L4(d, 2d) as stated in Conjecture 2.2.4 CINDY DE VOLDER AND ANTONIO LAFACEReferences[1] Anita Buckley and Marina Zompatori. Linear systems of plane curves with a compositenumber of base points of equal multiplicity. Trans. Amer. Math. Soc., 355(2):539–549(electronic), 2003.[2] Ciro Ciliberto and Rick Miranda. Degenerations of planar linear systems. J. ReineAngew. Math., 501:191–220, 1998.[3] C. Ciliberto and R. Miranda. The Segre and Harbourne-Hirschowitz conjectures. In Ap-plications of algebraic geometry to coding theory, physics and computation (Eilat, 2001),volume 36 of NATO Sci. Ser. II Math. Phys. Chem., pages 37–51. Kluwer Acad. Publ.,Dordrecht, 2001.[4] Cindy De Volder and Antonio Laface. Linear systems on generic K3 surfaces. Preprint,math.AG/0309073, 2003.[5] Cindy De Volder and Antonio Laface. Degeneration of linear systems through fat pointson K3 surfaces. Preprint, math.AG/0310219, 2003.[6] Alessandro Gimigliano. Regularity of linear systems of plane curves. J. Algebra,124(2):447–460, 1989.[7] Brian Harbourne. The geometry of rational surfaces and Hilbert functions of points in theplane. In Proceedings of the 1984 Vancouver conference in algebraic geometry, volume 6of CMS Conf. Proc., pages 95–111, Providence, RI, 1986. Amer. Math. Soc.[8] Andre´ Hirschowitz. Une conjecture pour la cohomologie des diviseurs sur les surfacesrationnelles ge´ne´riques. J. Reine Angew. Math., 397:208–213, 1989.[9] Steven L. Kleiman. Bertini and his two fundamental theorems. Rend. Circ. Mat. Palermo(2) Suppl., (55):9–37, 1998. Studies in the history of modern mathematics, III.[10] Alan L. Mayer. Families of K3 surfaces. Nagoya Math. J., 48:1–17, 1972.[11] Beniamino Segre. Alcune questioni su insiemi finiti di punti in geometria algebrica. In AttiConvegno Internaz. Geometria Algebrica (Torino, 1961), pages 15–33. Rattero, Turin,1962.Department of Pure Mathematics and Computeralgebra, Galglaan 2,B-9000 Ghent, BelgiumE-mail address: cdv@cage.ugent.beDipartimento di Matematica, Universita` degli Studi di Milano, Via Saldini 50,20133 Milano, ItalyE-mail address: antonio.laface@unimi.it

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