linear series

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linear series

Thomas Bauer, Cristiano Bocci, Susan Cooper, Sandra Di Rocco, Marcin Dumnicki, Brian Harbourne, Kelly Jabbusch,

Andreas L. Knutsen, Alex Küronya, Rick Miranda, Joaquim Roé, Hal Schenck, Tomasz Szemberg, and Zach Teitler

1 Introduction

In the week of October 3–9, 2010, the Mathematisches Forschungsinstitut at Oberwol- fach hosted the Mini-Workshop “Linear Series on Algebraic Varieties.” These notes contain a variety of interesting problems which motivated the participants prior to the event, and examples, results and further problems which grew out of discussions during and shortly after the workshop. Many arguments presented here are scattered in the literature or constitute “folklore.” It was one of our aims to have a usable and easily accessible collection of examples and results.¹

2 Original problems

We begin with a list of problems which were suggested by the participants for the Mini-Workshop. This list was discussed for three months before the workshop began.

2.1 Asymptotic effectivity (B. Harbourne). LetS D fp₁; : : : ; p_rgbe distinct points in P^N, over an algebraically closed ground fieldk of arbitrary characteristic. Let fW X ! P^N be the morphism obtained by blowing upp₁; : : : ; p_r, and denote the exceptional divisors byE₁; : : : ; E_r. LetH Df.O_P^N.1//and letL.d; m/DdH m.E₁C CE_r/. Waldschmidt [56] introduced and showed the existence of the following quantity:

e.S /WD lim

m!1

a.S; m/

m where a.S; m/ WD min˚

d W h⁰.X; L.d; m// > 0

. It follows from the proof that me.S /6a.S; m/for allm>1.

Problem 2.1.1. Develop computational or conceptual methods for evaluating, estimat- ing or boundinge.S /.

It is trivial to show thata.S; m/ Drmand hencee.S / D r whenN D 1. It is an open problem in general to computee.S /whenN > 1. For example, forr > 9 generic pointsp_i 2 P², a still open conjecture of Nagata [40, Conjecture, p. 772] is equivalent to havinge.S /Dp

r.

Using complex analytic methods, Waldschmidt [56] and Skoda [47] showed that a.S; 1/

N 6e.S /: (1)

1Cooper’s participation was supported by the “US Junior Oberwolfach Fellows” joint NSF-MFO program under NSF grant DMS-0540019. Partial support during this project is kindly acknowledged as follows: Bocci by Italian PRIN funds; Di Rocco by Vetenskapsådets grant NT:2006-3539; Küronya by DFG-Forschergruppe 790 “Classification of Algebraic Surfaces and Compact Complex Manifolds” and the OTKA grants 77476 and 77604 by the Hungarian Academy of Sciences; Roé by Spanish Ministerio de Educación y Ciencia, grant MTM2009-10359; Schenck by NSF 07–07667, NSA 904-03-1-0006; Szemberg by MNiSW grant N N201 388834

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Givenk>0, by [28, Theorem 1.1(a)], for any finite subsetS P^N and allm>1we

have a.S; kC1/

kCN 6 a.S; m.N Ck//

m.kCN / and hence

a.S; kC1/

kCN 6e.S /6 a.S; kC1/

kC1 :

Takingk D 0recovers the Waldschmidt-Skoda bound (1). Taking large values ofk gives one a way of computing arbitrarily accurate estimates ofe.S /[24], but computing a.S; k/for largekis difficult to do.

Chudnovsky [8] conjectured (and proved forN D2) the stronger bound Conjecture 2.1.2.

a.S; 1/CN1

N 6e.S /:

(For the proof whenN D2, reduce to the case thatr D _aC1

2

, wherea Da.S; 1/, and use the fact that thenL.a; 1/is nef.)

Two examples are known where equality in (2.1.2) holds: when the points lie in a hyperplane, or whenS is astar configuration[4] (i.e., given a set ofshyperplanes in P^N such that at mostN of these hyperplanes meet at any single point,S is the set of r D_s

N

points at whichN of the hyperplanes meet).

Problem 2.1.3. If for somemwe have a.S; m/

m D a.S; 1/CN 1 N

is it true thatSis either contained in a hyperplane or is a star configuration? What if e.S /D a.S; 1/CN1

N ‹

Bocci and Chiantini [3] show forN D2that ^a.S;2/₂ D ^a.S;1/C1₂ implies thatS is either a set of collinear points or a star configuration.

2.2 Semi-effectiveness (B. Harbourne). Again let the ground field k be an algebraically closed field of arbitrary characteristic.

Definition 2.2.1. LetXbe an algebraic variety andLa line bundle onX. We say that Lissemi-effective, if there existsn > 0such thath⁰.nL/ > 0.

Letp₁; : : : ; p_rbe distinct points inP^N. LetfW X!P^N be obtained by blowing upp₁; : : : ; p_r with the exceptional divisors beingE₁; : : : ; E_r. Let

D DdHm₁E₁ m_rE_r; whereH is the pullback viaf of a general hyperplane.

The following question was raised by M. Velasco and D. Eisenbud (in an email from Velasco to Harbourne, November 2009).

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Problem 2.2.2. Is there a way to determine ifDis semi-effective?

For a specific problem consider

DD13L5E₁4E₂ 4E₁₀

forgeneric pointsp₁; : : : ; p₁₀ 2 P². M. Dumnicki and J. Roé have independently shown that this system is not semi-effective (see Section 3.10). But there are infinitely many more similar examples for which semi-effectivity is still not known. For example, given generic pointsp_i 2P², considerD D111L36E₁35E₂ 35E₁₀or more generallyDDdL.mC1/E₁mE₂ mE₁₀, wheredD.b²Ca²/=2, mC1 D ba, m D .b²a²/=6and where 0 < a < bare odd integers satisfying .aC3b/² 10b² D 6. Note thatD² D 0butD cannot be reduced by Cremona transformations. According to the SHGH Conjecture [46], [23], [20], [27], we expect that none of theseDare semi-effective.

Limits like those of Waldschmidt are relevant to Problem 2.2.2. Leta.P_r

iD1m_ip_i/ be the leasttsuch thath⁰.tLP

im_iE_i/ > 0and define e X

i

m_ip_i

WDrlim

t

a.tP

im_ip_i/ tP

im_i :

ThenD DdLP_r

iD1m_i is semi-effective ife.P

im_ip_i/=r < d=P

im_i, and it is not semi-effective ife.P

im_ip_i/=r > d=P

im_i. It is not clear whetherD is semi- effective whene.P

im_ip_i/=rDd=P

im_i. But, if the SHGH Conjecture is true, this boundary case is precisely the situation of the examples in the preceding paragraph.

2.3 Stability of speciality (T. Szemberg). Letp₁; : : : ; p_rbe points in the projective plane. Let fW X ! P² be the blow-up of p₁; : : : ; p_r with exceptional divisors E₁; : : : ; E_r. LetH Df.O_P².1//, and let

D DdHm₁E₁ m_rE_r:

Assume that the divisorDis special (i.e.,Dis effective withh¹.X; D/ > 0). Is it then true that

nDis special for all n > 1‹

A somewhat more demanding problem is to determine whether the asymptotic cohomology functionhO¹as defined in [33] is positive. This is not true if the points are arbitrary (or less than10?). Indeed, letp₁; : : : ; p₉be intersection points of two cubics and letLD3H P₉

iD1E_ibe the anti-canonical pencil. Then h⁰.nL/DnC1; h¹.nL/Dn and h².nL/D0;

so that allnLare special but all asymptotic cohomology functions vanish.

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2.4 Regularity for generic monomial zero-schemes (J. Roé). LetI kŒx; ybe a monomial ideal, and letZDSpeckŒx; y=I. LetnDdim_kkŒx; y=Ibe the length of Z, which we assume to be finite (in this case,I is.x; y/-primary, andZis supported at the origin). For each (irreducible smooth projective) surfaceS defined overkthere is an irreducible constructible subset Hilb_IS of HilbⁿS whose closed points are the subschemesY ofSisomorphic toZ.

Of course, ifI D .x; y/^m then Z is just an m-fold point, and Hilb_IS Š S is the set of points ofS taken with multiplicity m. IfI D .y²; yx²; x³/thenZ is a cusp scheme, which means that curves containingZ have at least a cusp at the point supportingZ(i.e., they have a cusp or a more special singularity, and generically it is a cusp). Since the cusp scheme marks the tangent direction to the cusp, Hilb_IS is in this case naturally isomorphic to the (projectivized) tangent bundle ofS. Other monomial ideals correspond to other singularity “types.”

Problem 2.4.1. Describe Hilb_IS. Is it locally closed? What adjacencies are there between such subsets of the Hilbert scheme?

As Hilb_IS is irreducible, it makes sense to consider general (or very general, or generic) subschemesY ofS isomorphic toZ. For each divisorDwe have an exact sequence

0!Y ˝OS.D/!OS.D/!OY.D/!0 inducing the usual exact sequence in cohomology.

Problem 2.4.2. What can we say (or conjecture) about the cohomology of Y ˝ OS.D/? To be more precise, it would be nice to have conditions on S, I and D implying that h⁰ D maxf0; h⁰.OS.D//ng, in which case h¹ and h² are easily computed by Riemann–Roch and duality.

Assuming the previous problem is understood, one can then further ask about the base locus of global sections ofY˝OS.D/, and ask if they cut out the schemeZ. Such questions are of interest in the construction of curves with imposed singularities, and have been studied mainly for schemes of small multiplicity (the multiplicity ofZis the maximummsuch thatI .x; y/^m). Note that multiplicity 1 schemes are curvilinear and well known; multiplicity 2 monomial schemes are also quite well understood [43].

The same kind of questions arise as auxiliary problems for the induction arguments of differential Horace methods [15], even when one is primarily interested in linear series defined by ordinary multiple points.

Note also that the scheme defined byI D.y; x^r/^mis a specialization (or collision) ofrdistinctm-fold points; thus the dimension of the linear seriesH⁰.Y ˝OS.D//

is by semicontinuity a bound for the dimension of the linear series determined by a set ofr generalm-fold points. This gives a link with the Nagata conjecture [40] and the SHGH conjecture [46], [23], [20], [27]. More precisely:

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Conjecture 2.4.3. LetS DP²andI D.y; x^r/^m, whererandmare natural numbers withr > 9. Then for alld > 0, and generalY 2Hilb_IS,

H⁰.Y ˝O_P2.D//Dmax

´

0; dC2 2

!

r mC1 2

!μ :

Conjecture 2.4.4. LetS DP²andI D.y; x^r/^m, whererandmare natural numbers withr > 9. Then for alld 6mp

r, and generalY 2Hilb_IS,H⁰.Y˝O_P2.D//D0.

Conjecture 2.4.3 implies the uniform Harbourne–Hirschowitz conjecture, and Con- jecture 2.4.4 implies Nagata’s conjecture. It is also clear that Conjecture 2.4.3 implies Conjecture 2.4.4.

A conjecture in terms of monomial ideals implying the general Harbourne–Hirscho- witz conjecture (without uniformity assumptions on the multiplicities) can be stated similarly; we skip it here to avoid introducing the necessary notations, which would lengthen this section unnecessarily, and refer to Hirschowitz’s description of the “col- lisions de front” in [26] instead.

2.5 Bounding cohomology (B. Harbourne). There are various equivalent versions of the SHGH Conjecture [46], [23], [20], [27]. Here’s one:

Conjecture 2.5.1(SHGH). LetC X be a prime divisor whereX ! P² is the blow-up of generic pointsp₁; : : : ; p_s. Thenh¹.X;OX.C //D0.

Problem 2.5.2. How can we remove the assumption about the points being generic in the SHGH Conjecture?

The following conjecture arose out of discussions between Harbourne, J. Roé, C. Ciliberto and R. Miranda. [NB: Corollary 3.1.2 gives a counterexample. See also Proposition 3.1.3.]

Conjecture 2.5.3. LetXbe a smooth projective surface (either rational or assume the characteristic is 0). Then there exists a constantc_X such that for every prime divisor Cwe haveh¹.X; C /6c_Xh⁰.X; C /.

The SHGH Conjecture is thatc_X D0whenX is obtained by blowing up generic points ofP².

2.6 Algebraic fundamental groups and Seshadri numbers (J.-M. Hwang). Denote byy₁.Y /the algebraic fundamental group of an irreducible varietyY. Following [32, Definition (2.7.1)], we say that a projective manifoldXhas large algebraic fundamental group if for every irreducible varietyZ X and its normalizationW xZ ! X, the image of the induced homomorphism on the algebraic fundamental groups

W y₁.Z/x ! y₁.X /

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is infinite. This is equivalent to saying that the algebraic universal cover ofXdoes not contain a complete subvariety. The proof of [32, Lemma 8.2] gives the following.

Proposition 2.6.1. LetN be a positive number. LetX be a projective manifold with large algebraic fundamental group and letLbe an ample line bundle onX. Then there exists afinite étale coverpWX⁰ ! X such that any irreducible subvarietyW inX⁰ satisfies.pL/^{dim.W /}W > N.

One can then ask the following:

Problem 2.6.2. LetXbe a projective manifold with large algebraic fundamental group and letLbe an ample line bundle onX.

(1) Given a positive numberN, does there exist a finite étale coverpWX⁰!Xsuch that the Seshadri number ofpLat any point is bigger thanN?

(2) Given a positive numberN, does there exist a finite étale coverpWX⁰!Xsuch that denoting bypQWX⁰X⁰!XXthe self-product ofpandD⁰X⁰X⁰ the diagonal, the Seshadri number ofpQ.p₁L˝p₂L/alongD⁰is bigger than N?

(3) If the answer to (1) or (2) is negative or unclear, what is the condition on the fundamental group ofXto guarantee a positive answer?

2.7 Blow-ups ofPⁿand hyperplane arrangements (H. Schenck). Let AD

[d iD1

V .˛_i/P²

be a union of lines inP²,Y the singular locus, andWX !P²the blow-up atY. Let R D CŒy₁; : : : ; y_d, and for each linear dependencyƒ D P_k

jD1c_i_j˛_i_j D 0on the lines ofA, let

f_ƒD Xk jD1

c_i_j.y_i₁ yy_i_j y_i_k/:

The ideal I generated by the f_ƒ is called the Orlik–Terao ideal, and the quotient C.A/ DR=I is called the Orlik–Terao algebra; of course,C.A/can be defined for arrangements in higher dimensional spaces. For a real, affine arrangementA, Aomoto conjectured a relationship betweenC.A/and the topology ofRⁿnA, which Orlik and Terao proved in [42].

Example 2.7.1. ConsiderADV .x₁x₂x₃.x₁Cx₂Cx₃/.x₁C2x₂C3x₃//P²: Since˛₁C˛₂C˛₃˛₄D0,

y₂y₃y₄Cy₁y₃y₄Cy₁y₂y₄y₁y₂y₃2I:

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The five lines meet in ten points, and every subset of four lines gives a similar relation, one of which is redundant. Thus,I is generated by four cubics, which turn out to be the maximal minors of a matrix of linear forms. This means thatI has a Hilbert–Burch resolution, andV .I /is a surface of degree six inP⁴; a computation showsV .I /has five singular points. Consider the divisor

D_AD4E₀ X10 iD1

E_i

onX, whereX is the blow-up of P² at the ten points ofY,E_i are the exceptional curves over the singular points, andE₀is the proper transform of a line. ThenD_Ais nef but not ample; the lines of the original arrangement are contracted to points, andI is the ideal ofXin Proj.H⁰.D_A//.

The example above is representative of the general case. In [45], it is shown that if '_AWX !P.H⁰.D_A/^_/;

thenC.A/is the homogeneous coordinate ring of'_A.X /and'_Ais an isomorphism on.P²nA), contracts the lines ofAto points, and blows upY.

The motivation for studyingC.A/arises from its connection to topology. In [41], Orlik and Solomon determined the cohomology ring of a complex, affine arrangement complementM DC^nC1nA: ADH.M;Z/is the quotient of the exterior algebra EDV

.Z^d/on generatorse₁; : : : ; e_din degree1by the ideal generated by all elements of the form@e_i₁_:::i_r WDP

q.1/^q1e_i₁ e_i_q e_i_r, for which codimH_i₁\ \H_i_r <

r. SinceAis a quotient of an exterior algebra, multiplication by an elementa 2A¹ gives a degree one differential onA, yielding a cochain complex.A; a/:

.A; a/W 0 ^//A⁰ â ^//A¹ â ^//A² â ^// â //A^` ^//0 : Thefirst resonance varietyR¹.A/consists of pointsaDP_d

iD1a_ie_i $.a₁ W Wa_d/ inP.A¹/ Š P^d1 for which H¹.A; a/ ¤ 0. Conjectures of Suciu [49] relate the fundamental group ofMtoR¹.A/. Falk showed thatR¹.A/may be described in terms of combinatorics, and conjectured thatR¹.A/is a subspace arrangement, which was shown by Cohen–Suciu ([9]) and Libgober–Yuzvinsky ([36]). The paper [45] describes a connection between combinatorics ofR¹.A/which give rise to factorizations ofD_A and corresponding determinantal equations inI.

Problem 2.7.2. Do the results for lines inP² generalize to higher dimension? For example, ifA Pⁿ, then [45] shows that the Castelnuovo–Mumford regularity of C.A/is bounded byn. Can an explicit description of the graded Betti numbers of C.A/be given in terms of the geometry ofA?

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2.8 Bounds for symbolic powers (Z. Teitler). LetX be a non-singular variety of dimensionndefined over the complex numbers and letZXbe a reduced subscheme ofX with ideal sheafI D I_Z OX. Thep^th symbolic power ofI, denotedI^.p/, is the sheaf of all function germs vanishing to order > p at each point ofZ. The inclusionI^pI^.p/is clear, or in other wordsI^r I^.m/forr >m, but it is not clear when inclusion in the other direction,I^.m/ I^r, holds;m >r is necessary but not sufficient in general. Related to a result of Swanson [50], Ein–Lazarsfeld–Smith used multiplier ideals to prove that if every component ofZhas codimension6einXthen I^.re/I^rfor allr 2N; in particularI^.rn/I^r[13]. This was subsequently proved in greater generality by Hochster–Huneke using the theory of tight closure [28].

The big height of a radical idealI, denoted bight.I /, is the maximum codimension of a component ofV .I /. Harbourne raised the question whether it is possible to give an improvement of the formI^.m/ I^r whenever m > f .r/, for some function f .r/ 6 re, for all radical ideals of big height e. Bocci–Harbourne [4] showed if > 0 is such that for all radical ideals I, I^.m/ I^r whenever m > r, then > n. Furthermore, ifI^.m/ I^r holds wheneverm > r for all radical idealsI with bight.I /D e, then >e. This shows that the functionf .r/ D reappearing in the Ein–Lazarsfeld–Smith result cannot be decreased by lowering the coefficient eDbight.I /.

Harbourne asked whether a constant term can be subtracted, that is whetherI^.m/

I^r holds wheneverm>f .r/Derkfor all radical ideals of big heighte, for some k. Valuesk > edo not work (the containment fails if I is a complete intersection and – rather trivially – ifr D 1ore D 1). On the other hand, examples studied by Bocci–Harbourne suggest thatkDe1might work. Harbourne made the following:

Conjecture 2.8.1([2, Conjecture 8.4.3]). For radical idealsI of big heighte,I^.m/

I^r wheneverm>re.e1/.

A weaker result was observed by Takagi–Yoshida [52] (see also [53] for an exposi- tory account) who showed that if` <lct.I^.//thenI^.m/I^rwheneverm>re`.

Here lct.I^.//is the log canonical threshold of the graded system of idealsI^./; in particular this is always6e(so`6e1).

Harbourne–Huneke have asked if an improvement is possible on the other side of the inclusion: Instead of asking forI^.m/I^r, they raise the following:

Problem 2.8.2. Suppose.R;m/is a regular local ring of dimensionnandI Ris an ideal with bight.I /De. Then do the following hold?

1. I^.m/m^rnrI^r form>rn.

2. I^.m/m^rnr.n1/I^rform>rn.n1/.

3. I^.m/m^rerI^r form>re.

4. I^.m/m^rer.e1/I^r form>re.e1/.

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3 Progress

In this part we show several solutions to the original problems, present examples which are closely related to the problems, and provide some evidence either for positive or negative answers or forcing reformulation of the original statements.

3.1 Relating h⁰ and h¹ on surfaces. We show that Conjecture 2.5.3 is false in general. We claim (see Corollary 3.1.2) that there exists a surfaceX such that for an arbitrary positive integercthere exists a reduced, irreducible curveC Xwith

h¹.X;OX.C // > ch⁰.X;OX.C //:

It turns out that an example of Kollár (taken from [35, Example 1.5.7]) provides a counterexample to Conjecture 2.5.3. We recall briefly the construction, in which we will closely follow the exposition mentioned above. We consider an elliptic curveE without complex multiplication, and take the abelian surface

Y WDEE

to be our starting point. Divisors and the various cones onY are well understood.

The Picard number.Y /equals3, andN¹.X /_Rhas the fairly natural basis consisting of the classes ofF₁,F₂(the fibres of the two projection morphisms), and that of the diagonalEE. The intersection form onY is given by the numbers

.F₁²/D.F₂²/D.²/D0; .F₁F₂/D.F₁/D.F₂/D1:

It is known (for details see for example [35, Section 1.5.B]) that the nef and pseudo- effective cones coincide, and a classC D a₁F₁Ca₂F₂Cbis nef if and only if .C²/ >0and.C H /> 0for some ample classH. In coordinates we can express this as

a₁a₂Ca₁bCa₂b>0 and a₁Ca₂Cb>0 by choosing the ample divisorF₁CF₂CforH.

For every integern>2set

A_nWDnF₁C.n²nC1/F₂.n1/:

It is immediate to check that 1. .A²_n/D2, and

2. .A_n.F₁CF₂//Dn²2nC3 > 0.

Kollár now setsRWDF₁CF₂, and picks a smooth divisorB2 j2Rj(which exists because2Ris base point free by the Lefschetz theorem [34, Theorem 4.5.1]) to form the double coverfWX !Y branched alongB. LetD_nWDfA_n.

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Proposition 3.1.1. With notation as above,

h¹.X;OX.nD_n//>n³2n²C3n1:

Proof. We will estimateh¹

X;OX.nD_n/

from below with the help of the Leray spec- tral sequence. It is a standard fact that

E₂^p;qWDH^p

Y; R^qfOX.nD_n/

H)H^pCq

X;OX.nD_n/ :

We are interested in the casepCqD1, in which case the involved terms are H⁰

Y; R¹f.OX.nD_n/

and H¹

Y; fOX.nD_n/ :

Of these the second term survives unchanged to theE₁ term, and so we obtain an inclusion

H¹

Y; fOX.nD_n/

,!H¹

X;OX.nD_n/ : It isH¹

Y; fOX.nD_n/

that we will estimate from below. By [35, Proposition 4.1.6]

on the properties of cyclic coverings, one has

fOX DOY ˚OY.R/;

which implies

f.OX.nD_n//DOY.nA_n/˚OY.nA_nR/

via the projection formula.

It follows that H¹

Y; f.OX.nD_n//

DH¹

Y;OY.nA_n/

˚H¹

Y;OY.nA_nR/

:

We can determine the second term of the sum from the Riemann–Roch theorem. On the abelian surfaceY DEEone has .OY/D0andK_Y DOY, hence Riemann–Roch has the particularly simple form

.OY.nA_nR//D 1

2.nA_nR/²: We compute

.nA_nR/²Dn²A²_n2n.A_nR/CR² D2n²2n.n²2nC3/C2 D 2n³C6n²6nC2 D 2.n1/³

< 0;

therefore neithernA_nRnor its negative is effective, resulting in H⁰

Y;OY.nA_nR/

D0

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and

H²

Y;OY.nA_nR/

DH⁰

Y;OY.RnA_n/ D0:

Hence we can conclude that h¹

X;OX.nD_n/

>h¹

Y;OY.nA_nR/

Dn³2n²C3n1 > 0:

Corollary 3.1.2.With notation as above, for an arbitrary positive integercthere exists a prime divisorConXsuch that

h¹

X;OX.C /

> ch⁰

X;OX.C / :

Proof. This is in fact a corollary of the proof of the Proposition 3.1.1. For n > 2 the linear systemjnA_njis globally generated by the Lefschetz theorem [34, Theorem 4.5.1] and it is not composed with a pencil. The same holds true forjnD_nj, therefore the base-point free Bertini theorem implies that the general element ofjnD_njis reduced and irreducible. LetC_nbe such an element. Then

h⁰

X;OX.C_n/ Dh⁰

Y;OY.nA_n/ D 1

2..nA_n/²/Dn² by the Riemann–Roch theorem. On the other hand,

h¹

X;OX.C_n/

>n³3n²C3n1 > cn² for large enoughn.

The surfaceX studied above is of general type. It still could be true that Conjec- ture 2.5.3 holds when restricted torationalsurfaces, in any characteristic. For some evidence in this direction we now prove a particularly strong form of the conjecture in the case of smooth projective rational surfaces with an effective anticanonical divisor, and hence in particular for smooth projective toric surfaces.

Proposition 3.1.3. LetX be a smooth projective rational surface having an effective anticanonical divisorD 2 j K_Xj. Then there exists a constantc_X such that for every prime divisorC X we haveh¹.X; C / 6c_X. In fact, the maximum value of h¹.X; C /is either 0 or 1, or it occurs whenC is a component ofD.

Proof. LetCbe a prime divisor onX. Clearly, we can assume thatCis not a component ofD. ThusK_X C DDC >0, and ifK_XC D0, thenC is disjoint fromD.

SupposeK_X C > 0. IfC² >0, thenh¹.X; C / D0by [22, Theorem III.1(a, b)]. IfC² < 0, then0 > C²D2p_a.C /2K_X C > 2p_a.C /2by adjunction, hencep_a.C /D0andC is smooth and rational withC²D 1. Consider

0!OX !OX.C /!OC.C /!0: .?/

SinceX is rational we haveh².X;OX/ D h¹.X;OX/D 0. SinceC is smooth and rational withC² D 1, we haveh¹.C;OC.C //D0. Thush¹.X;OX.C //D0.

We are left with the case thatDC D K_X C D0, henceC is disjoint fromD.

ThusOC.C /DK_C by adjunction, soh¹.C;OC.C //D1, and therefore from.?/we haveh¹.X; C /D1.

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3.2 Speciality on blow-ups ofP². In this section we look at the speciality of nef linear systems on blow-ups ofP². We try to formulate a statement, which is valid without assuming that the points we blow up are in general position. The examples presented here suggest that one needs to reformulate the problem stated in Section 2.3.

The first example shows that multiples of a special effective nef linear system might no longer be special.

Example 3.2.1. This example is based on the existence of very ample but special linear systems. Letp₁; : : : ; p₂₅ be transversal intersection points of two smooth curves of degree5. Letf WX ! P² be the blow-up of the plane at these 25 points. By [19, p. 796], for16r 62and anym > 0, the linear system

LD.5mCr/H m.E₁C CE₂₅/;

whereH is the class of the line andE_iare the exceptional divisors off, is very ample and special, but Serre vanishing implies that some multiple ofLis no longer special.

LetX⁰!Xbe the blow-up ofXat an additional point, and letL⁰be the pullback to X⁰ofL. Then we have an example of a linear systemL⁰which is special and nef but not ample and for whichsL⁰is non-special fors0.

The second example shows that speciality might persist.

Example 3.2.2. LetC andDbe smooth plane curves of degreed >3intersecting transversally inp₁; : : : ; p_d2. LetfWX ! P²be the blow-up of the plane at these points. The linear system

LDdH

d²

X

iD1

E_i

is special (again because its virtual dimension is negative). The same remains true for all multiples ofL. This follows from the restriction sequence

0!mL!.mC1/L!.mC1/Lj_C !0:

Indeed, by Serre duality we haveh².mL/D0for allm>1. Also, asLis a pencil of disjoint curves, we haveLj_C D OC. Then taking the long cohomology sequence of the restriction sequence we have the mapping

!H¹.X; .mC1/L/!H¹.C;OC/!0:

The assumptiond>3guarantees thatCis non-rational, henceh¹.C;OC/Dg.C / > 0.

The next example shows that speciality may increase linearly while the number of global sections remains fixed.

Example 3.2.3. LetC P²be a quartic curve with a simple node atp₀and smooth otherwise. Such a curve exists by a simple dimension count. Then we can take

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12pointsp₁; : : : ; p₁₂onC in such a way thatCis the unique quartic passing through p₀ with multiplicity2and throughp₁; : : : ; p₁₂with multiplicities all equal to1. Let fWX !P² be the blow-up of the plane at the pointsp₀; : : : ; p₁₂. We consider the linear systemLD4H2E₀E₁ E₁₂onX. By a slight abuse of notation we writeC also for the proper transform ofC onX. It is a smooth curve of genus2.

If the pointsp₁; : : : ; p₁₂ are generic enough, thenmLj_C has no global sections for allm>1and by Riemann–Roch onC we haveh¹.C; mLj_C/D1. Using again the restriction sequence we get

h⁰.X; mL/D1 and h¹.X; mL/Dm for all m>1:

These examples suggest the following problem.

Problem 3.2.4. LetXbe the blow-up ofP²atr distinct points (r is arbitrary and also the position of the points is arbitrary but we require the points to be distinct) and letL be an effective nef divisor onX. Then is it true that either

a) there existsm>1such thath¹.mL/D0(and thismshould be1if the points are in general position); or

b) there is a (not necessarily irreducible) curveC onX such thatjLCj ¤ ¿, p_a.C / > 0andLC D0?

3.3 Bounded negativity. In the course of the discussions, we investigated the following problem, see [21, Conjecture 1.2.1].

Conjecture 3.3.1(Bounded negativity). LetXbe a smooth projective surface in characteristic zero. There exists a positive constantb.X /bounding the self-intersection of reduced, irreducible curves onX, i.e.,

C²>b.X / for every reduced, irreducible curveC X.

The restriction to characteristic zero is in general essential (see Example 3.3.3) and of course so is the hypothesis that the curves be reduced, but it is not necessarily essential that they be irreducible. See Section 3.8 for further discussion.

One situation where bounded negativity holds is when the anti-canonical divisor is Q-effective (see [21, I.2.3]):

Proposition 3.3.2. LetX be a smooth projective surface such that for some integer m > 0the pluri-anti-canonical divisormK_Xis effective. Then there exists a positive constantb.X /such thatC²>b.X /for every irreducible curveConX.

Proof. AsmK_X is effective, there exist only finitely many irreducible curvesCsuch thatK_X C < 0. Hence apart from these finitely many prime divisors, we have K_X C >0, in which case by the adjunction formula

C² D2p_a2K_X C >2p_a2>2:

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Note that the hypothesis ofQ-effectivity holds for instance on toric surfaces. An- other case where the conjecture is clearly true is whenK_X is nef (with the same argument as that in the proof of Proposition 3.3.2).

The following example (see also [21, Remark 1.2.2]) shows that the restriction on the characteristic in Conjecture 3.3.1 cannot be avoided in general (but perhaps it can be avoided by restrictingXto be, for example, rational).

Example 3.3.3(Exercise V.1.10, [25]). LetCbe a smooth curve of genusg>2defined over a field of characteristicp > 0and letXbe the product surfaceXDCC. The graph_qof the Frobenius morphism defined by takingqDp^r-th powers is a smooth curve of genusgand self-intersection_q² Dq.22g/. Withr going to infinity, we obtain a sequence of smooth curves of fixed genus with self-intersection going to minus infinity.

In view of this example it is interesting to ask if at least one of the following is true.

Conjecture 3.3.4(Weak bounded negativity). LetXbe a smooth projective surface in characteristic zero and letg>0be an integer. There exists a positive constantb.X; g/

bounding the self-intersection of curves of geometric genusgonX, i.e., C² >b.X; g/

for every reduced, irreducible curveC X of geometric genusg(i.e., the genus of the normalization ofC).

Conjecture 3.3.5 (Very weak bounded negativity). Let X be a smooth projective surface in characteristic zero and let g > 0 be an integer. There exists a positive constantb_s.X; g/bounding the self-intersection of smooth curves of geometric genus gonX, i.e.,

C²>b_s.X; g/

for every irreducible smooth curveC Xof geometric genusg.

It would be interesting to know if a bound of this type extends to a family of surfaces, i.e., if the following question has an affirmative answer.

Problem 3.3.6. LetfW Y !Bbe a morphism from a smooth projective threefoldY to a smooth curveBsuch that the general fibre is a smooth surface. Is there a constant b.Y; g/such that

C²>b.Y; g/

for all vertical curvesC Y (i.e.,f .C /Da point) of geometric genusg? (Here the self-intersection is computed in the fibre off containingC.)

We show that to a large extent both Conjectures 3.3.4 and 3.3.5 are true. The key ingredient of the proofs is the following vanishing result, [14, Corollary 6.9]. We recall the basic properties of log differential forms in Appendix 3.12.

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Theorem 3.3.7(Bogomolov–Sommese Vanishing). LetXbe a smooth projective va- riety defined over an algebraically closedfield of characteristic zero,La line bundle, Ca normal crossing divisor onX. Then

H⁰.X; _X^a.logC /˝L¹/D0 for alla < .X; L/.

Corollary 3.3.8. LetX be a smooth projective surface defined over an algebraically closedfield of characteristic zero,Ca normal crossing divisor onX. Then_X¹.logC / contains no big line bundles.

Proof. Note thatLbeing big means.X; L/D2. An inclusionL ,!_X¹.logC /gives rise to a non-trivial section ofH⁰.X; _X¹.logC /˝L¹/, which vanishes according to Theorem 3.3.7.

Remark 3.3.9. This kind of result was first observed by Bogomolov for the cotangent bundle of a surface itself:

If L_X¹ is a line bundle, thenLis not big. (2) We refer to [54, Proposition 2.2] for a nice and detailed proof.

Remark 3.3.10. Theorem 3.3.7 and statement (2) are known to be false in positive characteristic, see [31, Remark 7.1].

3.4 Partial proof of Conjecture 3.3.5: Very weak bounded negativity. Here we prove Conjecture 3.3.5 on surfaces with Kodaira dimension.X />0.

LetXbe a surface as in Theorem 3.3.7 and letF be a coherent sheaf of rankr on X. Recall that thediscriminant.F/is defined as

.F/WD2rc₂.F/.r1/c₁.F/²: IfF is a rank2vector bundle, then this reduces to

.F/D4c₂.F/c₁.F/²:

The interest in the discriminant of a vector bundle stems in part from the following useful numerical criterion for stability of vector bundles on surfaces. Recall first Definition 3.4.1. LetF be a rank2vector bundle on a smooth projective surfaceX.

We callF unstableif there exist line bundlesAandB onX and a finite subscheme ZX(possibly empty) such that the sequence

0!A!F !B˝Z !0 is exact and theQ-divisor

P WDA 1 2c₁.F/

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satisfies the conditions

P² > 0 and PH > 0 for all ample divisorsH onX.

Remark 3.4.2. The conditions in the definition imply thatP is a big divisor.

The fundamental result of Bogomolov [5] is the following numerical criterion Theorem 3.4.3(Bogomolov). LetF be a rank2vector bundle on a smooth projective surfaceX. If.F/ < 0, thenF is unstable.

As a corollary we prove very weak bounded negativity for smooth curves on surfaces with.X />0.

Proposition 3.4.4. LetXbe a smooth projective surface with.X />0and letC X be a smooth curve of genusg.C /. Then

C² >c₁².X /4c₂.X /4g.C /C4:

Proof. We consider the following exact sequence which arises via an elementary trans- formation of_X¹ along¹_C (see [29, Example 5.2.3])

0!W WD_X¹.logC /˝C !_X¹ !¹_C !0:

By [29, Proposition 5.2.2] we have

c₁.W /D.K_XCC /2C DK_XC and c₂.W /Dc₂._X¹/Cdeg.¹_C/K_XC:

Hence

.W /D4c₂.X /c₁².X /C4g4CC²:

If we assume thatC² < c₁².X /4c₂.X /4g.C /C4, then.W / < 0 andW is unstable. According to Definition 3.4.1 there exists then a line bundleA W such that

A1

2c₁.W /DAC 1

2.C K_X/ is big.

It follows that

ACC _X¹.logC / and since.X />0andCis effective

ACC D

AC1 2C 1

2K_X

C1

2K_X C1 2C

is big as well. However, this contradicts the Bogomolov–Sommese Vanishing (Theo- rem 3.3.7).

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Remark 3.4.5. Note that a statement as in the above Proposition cannot hold on ruled surfaces, i.e., there is no lower bound onC² withC smooth depending only on the genus ofC and Chern numbers ofX. Indeed, all Hirzebruch surfacesF_n D P.O_P¹˚O_P¹.n//have the same invariantsc₁².F_n/D8,c₂.F_n/D4and eachF_n contains a smooth rational curve of self-intersectionn. This observation is of course not a counterexample either to Conjecture 3.3.1 or to Conjecture 3.3.5. It merely means that the bound forC²on ruled surfaces must depend on something else. A reasonable possibility, in the case whereXis obtained by blowing uprpoints of a surfaceY when b.Y /exists, is to try to show thatb.X /can be defined in terms ofb.Y /andr. 3.5 Partial proof of Conjecture 3.3.4: Weak bounded negativity. The main auxiliary ingredient in this part is the logarithmic version of the Miyaoka–Yau inequality.

Theorem 3.5.1(Logarithmic Miyaoka–Yau inequality). LetXbe a smooth projective surface andC a smooth curve onX such that the adjoint line bundle K_X CC is Q-effective, i.e., there is an integerm > 0such thath⁰.m.K_X CC // > 0. Then

c²₁._X¹.logC //63c₂._X¹.logC //;

equivalently.K_XCC /² 63 .c₂.X /2C2g.C //.

We refer to Appendix 3.12 for the proof. Note that our statement of this inequality is not the most general, but it suffices for our needs. We also need the following elementary lemma.

Lemma 3.5.2. LetX be a smooth projective surface,C X a reduced, irreducible curve of geometric genusg.C /,P 2C a point withmult_PC >2. LetW zX !X be the blow-up ofXatP with the exceptional divisorE. LetCz D.C /mEbe the proper transform ofC. Then the inequality

Cz² >c²₁.X /z 3c₂.X /z C22g.C /z implies

C² >c²₁.X /3c₂.X /C22g.C /:

Proof. We have

C²D zC²Cm²; c₁².X /Dc₁².X /z C1; c₂.X /Dc₂.X /z 1 and g.C /Dg.C /:z Hence

C²Dm²C zC²

>m²Cc₁².X /z 3c₂.X /z C22g.C /z

Dm²Cc₁².X /13c₂.X /3C22g.C /

>c₁².X /3c₂.X /C22g.C /:

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Proposition 3.5.3. LetX be a smooth projective surface with.X / > 0. Then for every reduced, irreducible curveC Xof geometric genusg.C /we have

C² >c²₁.X /3c₂.X /C22g.C /: (3) Proof. The idea is to reduce the statement to a smooth curve and use Theorem 3.5.1.

We blow upfW zXDX_N !X_N₁ ! !X₀ DXresolving step-by-step the singularities ofC. The proper transform ofC inXzis then a smooth irreducible curve Cz. Applying Lemma 3.5.2 recursively to every step in the resolutionf we see that it is enough to prove inequality (3) forC smooth.

This follows easily from the logarithmic Miyaoka–Yau inequality 3.5.1. Note that our assumption.X />0implies thatK_X_zC zC isQ-effective. Hence

c₁².X /C2C .K_X CC /C²Dc₁²._X¹.logC //63c₂._X¹.logC //

D3c₂.X /6C6g.C /:

By adjunctionC.K_XCC /D2g.C /2and rearranging terms we arrive at (3).

Remark 3.5.4. Note that Proposition 3.5.3 applies in particular to smooth curves and provides in general a better bound than that in Proposition 3.4.4. We do not pursue the optimality problem here, and we found it instructive to provide two possible proofs for the Very Weak Bounded Negativity Conjecture.

3.6 Bounded Negativity Conjecture and Seshadri constants. We next point out an interesting connection between bounded negativity and a question on Seshadri constants posed by Demailly in [10, Question 6.9]:

Problem 3.6.1. Is the global Seshadri constant

".X /WDinff".L/ L2Pic.X /ampleg positive for every smooth projective surfaceX?

At present, this is unknown. In fact, it is unknown whether for every fixedx2X the quantity

".X; x/Dinff".L; x/ L2Pic.X /ampleg

is always positive. The latter, however, would be a consequence of the Bounded Negativity Conjecture:

Proposition 3.6.2. If the Bounded Negativity Conjecture is true, then

".X; x/ > 0

for every smooth projective surfaceXin characteristic zero, and everyx2X.

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The proof below actually gives an effective lower bound on".X; x/: IfY DBl_x.X / is the blow-up ofXatx, then

".X; x/> 1 pb.Y /C1

So if we knew that the constantb.Y /is the same for every one-point blow-up ofX(or at least bounded from below by a constant that is independent ofx), then we would get a lower bound on".X /.

Proof of the proposition. LetC X be an irreducible curve of multiplicitymatx, and letCz Y be its proper transform on the blow-upY ofXinx. Then

C²m² D.fCmE/²D zC² >b.Y /:

Consider first the case wherem6p

b.Y /. Then LC

m > LC

pb.Y / > 1 pb.Y / In the alternative case, wherem >p

b.Y /, we have C² >m²b.Y / > 0 and hence, using the Index Theorem, we get

LC m >

p L²p

C² m >

r

1 b.Y / m² >

s

1 b.Y /

b.Y /C1 D 1 pb.Y /C1: An alternative argument for the proof of the proposition goes as follows. Suppose that".X; x/D0. Then there is a sequence of ample line bundlesL_nand curvesC_n such that

L_nC_n m_n !0;

wherem_ndenotes the multiplicity ofC_natx. We use now that the blow-upY DBl_x.X / has bounded negativity, so that for the proper transformCz_nofC_nwe have

.Cz_n/² >b.Y /²:

But.Cz_n/² D .fC_nm_nE/² DC_n²m²_n, which, upon using the Index Theorem, tells us that

m²_nb.Y /6C_n² 6 .L_nC_n/² L²_n : So we see that

1b.Y / m²_n 6C_n²

m²_n 6.L_nC_n/² m²_n 1

L²_n 6

L_nC_n m_n

₂ :

Now, the left hand side of this chain of inequalities tends to one, whereas the right hand side goes to zero, a contradiction.

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3.7 The Weighted Bounded Negativity Conjecture. The following conjecture is yet another variant of the Bounded Negativity Conjecture 3.3.1:

Conjecture 3.7.1(Weighted bounded negativity). LetXbe a smooth projective surface in characteristic zero. There exists a positive constantb_w.X /such that

C² >b_w.X /.H C /²

for every irreducible curveC X and every big and nef line bundleH satisfying HC > 0.

The interest in this conjecture is due to the fact that this conjecture is sufficient to imply the conclusion of the previous proposition, by the following result.

Proposition 3.7.2. If the Weighted Bounded Negativity Conjecture is true, then

".X; x/ > 0

for every smooth projective surfaceXin characteristic zero, and everyx2X.

Proof. As above, there are two proofs, and one of these yields the effective lower bound

".X; x/> 1 pb_w.Y /C1;

whereY DBl_x.X /is the blow-up ofXatx. We first look at this proof.

LetC X be an irreducible curve of multiplicitym atx andL an ample line bundle onX. LetCz Y be the proper transform ofC onY. SincefLis big and nef andfL zC D LC > 0, the Weighted Bounded Negativity Conjecture onY yields the existence of a constantb_w.Y /such that

C²m² D.fCmE/²D.C /z ²>b_w.Y /.fL zC /²Db_w.Y /.LC /²: Then, using the Index Theorem, we obtain

.LC /²

m² >L²C² m² >L²

1b_w.Y /.LC /² m² ;

that is, LC

m

2

1Cb_w.Y /L² >L²: This yields

LC m >

s

L²

1Cb_w.Y /L² D

s 1

1

L²Cb_w.Y / >

s 1

1Cb_w.Y /D 1 p1Cb_w.Y /; as asserted.

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The second version of the proof goes as follows: Suppose that".X; x/D0. Then there is a sequence of ample line bundlesL_nand curvesC_nsuch that

L_nC_n m_n !0

wherem_n denotes the multiplicity ofC_n at x. Letf WY D Bl_x.X / ! X be the blow-up ofXatxand denote byCz_nthe proper transform ofC_n. SincefL_nis big and nef andfL_n zC_nDL_nC_n> 0, the Weighted Bounded Negativity Conjecture onY yields the existence of a constantb_w.Y /such that

.Cz_n/²>b_w.Y /.fL_n zC_n/² D b_w.Y /.L_nC_n/²: But.Cz_n/²D.fC_nm_nE/² DC_n²m²_n, which yields

C_n²

m²_n 1>b_w.Y /.L_nC_n/² m²_n :

Combining with the Index Theorem, we obtain 1b_w.Y /

L_nC_n m_n

2 6 C_n²

m²_n 6L_nC_n m_n

2 1 L²_n:

But the left hand side of this chain of inequalities tends to one, whereas the right hand side tends to zero, a contradiction.

Note that the second proof does not give an effective lower bound for".X; x/.

3.8 Bounded negativity for reducible curves. We now consider a reducible version of Conjecture 3.3.1:

Conjecture 3.8.1(Reducible bounded negativity). LetX be a smooth projective surface in characteristic zero. Then there exists a positive constantb⁰.X /bounding the self-intersection of reduced curves onX, i.e.,

C²>b⁰.X /

for every reduced (but not necessarily irreducible) curveC X.

Conjecture 3.3.1 implies Conjecture 3.8.1 in a very explicit way.

Proposition 3.8.2. LetX be a smooth projective surface(in any characteristic)for which there is a constantb.X /such thatC² >b.X /for every reduced, irreducible curveC X. ThenC² >.X /b.X /db.X /=2efor every reduced curveC X, where.X /is the Picard number ofX(i.e., the rank of the Néron–Severi groupNS.X / ofX).

linear series

linear series

Contents

1 Introduction

2 Original problems

3 Progress