On the Seshadri constants of adjoint line bundles
aa r X i v : . [ m a t h . AG ] N ov On the Seshadri constants of adjoint line bundles
Thomas Bauer and Tomasz SzembergNovember 19, 2010
Seshadri constants are interesting invariants of ample line bundles on algebraic va-rieties. They were introduced by Demailly in [Dem] and may be thought of ascapturing the local positivity of a given line bundle. A nice introduction to thiscircle of ideas is given in [PAG, Sect. 5], an overview of recent results can be foundin [PSC]. Here we merely recall the basic definition:Let X be a smooth projective variety, L an ample line bundle on X , and x ∈ X a point on X . The number ε ( L, x ) := inf C ∋ x L · C mult x C is the Seshadri constant of L at x , whereas ε ( L ) := inf x ∈ X ε ( L, x )is the
Seshadri constant of L .While ε ( L ) is always a positive number, Miranda [PAG, Example 5.2.1] showedthat there is no uniform positive lower bound for Seshadri constants of ample linebundles on varieties of fixed dimension. The purpose of the present note is toshow that for adjoint line bundles, Seshadri constants exhibit surprisingly regularbehavior.Here is a more detailed description of the content of this paper:(1) While every positive rational number occurs as a Seshadri constant of some(integral) ample line bundle (Proposition 2.1), we show that there exists auniform lower bound in the adjoint setting, i.e., for ample bundles K X + L ,where L is nef (Theorem 3.2).(2) On surfaces we show that the potential values that ε ( K X + L, x ) can assumein the interval (0 ,
1) form a monotone increasing sequence with limit 1 (The-orem 4.1).(3) Still on surfaces, we prove that in the ‘hyper-adjoint’ setting no values below1 occur for ε ( K X + L, x ), and we classify the borderline case (Theorem 4.6).(4) We complete the picture by looking at the multi-point setting, where we pro-vide a uniform lower bound for adjoint bundles (Proposition 5.6). On surfaceswe answer the question corresponding to (2) by showing that there are onlyfinitely many possible values (Theorem 5.7).We work throughout over the field of complex numbers.
Acknowledgement.
We would like to thank C. Ciliberto who sparked our interestin studying Seshadri constants in the adjoint setting. Also, we thank G. Heier forhelpful remarks on an earlier version of our paper.The second author was partially supported by a MNiSW grant N N201 388834.
The first observation is that in general every positive rational number occurs as aSeshadri constant:
Proposition 2.1
For every rational number q > there exists a smooth projectivesurface X , an ( integral ) ample line bundle L on X , and a point x ∈ X such that ε ( L, x ) = q .
Proof.
We write q = dm . In the proof we follow closely Miranda’s idea (cf. [Laz,Prop. 5.2]). We construct X as a blow-up of the projective plane, but in fact ananalogous argument using [Bau, Lemma 3.5] would work on a suitable blow-up ofan arbitrary smooth projective surface. For a suitably large integer k , the followingholds true:(i) There exists an irreducible plane curve C of degree k with a point x of mul-tiplicity m .(ii) There exists another curve C of the same degree k such that • C and C intersect transversally in k distinct points, and • all curves in the pencil V generated by C and C are irreducible.The existence of C and C is basically a dimension count on sections of O P ( k ) plusBertini’s theorem. Let now f : X −→ P be the blow-up of P at the intersectionpoints of C and C , with exceptional divisors E , . . . , E k . Thus, by (ii), the surface X is fibred over P by the irreducible curves from the pencil V . It is easy to verifythat the line bundle L = E + 2 C is ample, where C denotes the class of the fiber on X and E is a fixed exceptional divisor. With a slight abuse of notation, we denotethe preimage of x on X again by x . Then we have by (i) L · f C mult x f C = 1 m for the proper transform f C of C , so that in any case ε ( L, x ) m . But for anyirreducible curve D on X passing through x (hence different from E ) and differentfrom f C we have L · D = ( E + 2 f C ) · D > m · mult x D , (1)so that in fact ε ( L, x ) = m . Replacing L by dL we get ε ( dL, x ) = dm , as claimed. (cid:3) Remark 2.2
One can easily generalize this construction to arbitrary dimension n +2 >
3, following the idea of [PAG, Example 5.2.2]: to this end, let Y = X × P n , where X is the surface constructed in the proof of Proposition 2.1, let M := pr ∗ L × pr ∗ H ,where L is the line bundle from the previous proof and H is the hyperplane bundleon P n . Furthermore, let p ∈ P n be a fixed point and Y p = X × { p } . (We identify Y p with X , in particular we view now the curve f C as a subvariety of Y p .) Then ε ( M, ( x, p )) = 1 m . In fact, it follows from the projection formula that M · f C = L · f C = 1 , so that in the point ( x, p ) we have in any case ε ( M, ( x, p )) m . Let now D be another curve passing through the point ( x, p ). If D is not containedin Y p , then M · D > pr ∗ H · D > mult ( x,p ) D , which shows that D cannot give a lower Seshadri quotient than f C . If on the otherhand D is contained in Y p , then we conclude the same exactly as in (1).Thus we saw that every positive rational number appears as the Seshadri con-stant of some ample line bundle on a variety of dimension >
2. On the other hand itis not known – and it would be extremely interesting to know – whether there existirrational Seshadri constants [PAG, Remark 5.1.13].
Now we show that there exists a uniform lower bound on Seshadri constants ofadjoint line bundles. This is a direct consequence of the following result of Angehrnand Siu [AS, Theorem 0.1], but it seems that it has not been explicitly noticed sofar.
Theorem 3.1 (Angehrn-Siu)
Let X be a smooth projective variety of dimension n and let A be an ample divisor on X . Assume that ( A d · Z ) > (cid:18)(cid:18) n + 12 (cid:19) + 1 (cid:19) d for every irreducible subvariety Z ⊂ X of positive dimension d . Then the adjointline bundle K X + A is globally generated. Theorem 3.2
Let X be a smooth projective variety of dimension n . Let L be a nefline bundle on X and assume that the adjoint line bundle K X + L is ample. Then ε ( K X + L ) > n + n + 4 . Proof.
We claim that m ( K X + L ) is globally generated for m > (cid:18) n + 12 (cid:19) + 2 . In fact, take an integer m > (cid:0) n +12 (cid:1) + 2 and let A := ( m − K X + L ) + L . This linebundle is ample and it satisfies the inequality( A d · Z ) > ( m − d for any subvariety Z ⊂ X of positive dimension d . Therefore the numerical conditionin Theorem 3.1 is satisfied, and hence the adjoint bundle K X + A = m ( K X + L )is globally generated.Now, Seshadri constants of globally generated ample line bundles are at least 1(see [PAG, Example 5.1.18]), and this implies the assertion after dividing by m . (cid:3) Remark 3.3
One can obtain an improved bound for ε ( K X + L ) by using Heier’sresult [Hei], which says that for any nef line bundle N and any integer m > ( e + ) n + n + 1 the bundle K X + mL + N is base-point free. Writing m ( K X + L ) = K X + ( m − K X + L ) + L and arguing as in the proof of Theorem 3.1, we get ε ( K X + L ) > e + ) n + n + 2 . Remark 3.4
It is quite unlikely that the particular bounds on ε ( K X + L ) given byTheorem 3.2 and Remark 3.3 are sharp. The important observation is that Seshadriconstants of adjoint line bundles are bounded from below by a universal numberdepending only on the dimension of the underlying variety.There are two important classes of varieties where all ample line bundles can bewritten as adjoints of ample bundles. On these varieties we have universal lowerbounds valid for all ample line bundles in all points. In particular we have in thesecases a positive answer to the following problem raised by Demailly [Dem, Question6.9]. Question 3.5
Let ε ( X ) be the infimum of the numbers ε ( L ) taken over all integralample line bundles on X . Is the number ε ( X ) positive, and if so, is there an effectivelower bound on ε ( X )? Corollary 3.6
Let X be a variety of dimension n with nef anti-canonical divisor.Then ε ( X ) > n + n + 4 . So in particular there is a universal lower bound for Seshadri constants on ( weak ) Fano varieties and varieties with numerically trivial canonical divisor.
Remark 3.7
Note that the lower bound ε ( X ) > n − was proved before for Fanovarieties of dimension n > − K X be globally generated. It seems thatthe existence of a lower bound valid without any restrictions is new. Actually, in the Main Theorem of [Hei] there is no mention of a nef bundle N , but as G. Heierinformed us, his result remains true in the form needed here. For surfaces, i.e., n = 2, Theorem 3.2 gives as the lower bound. One could invokeReider’s theorem in this case to improve this number to . However, we show herethat the optimal lower bound for the Seshadri constants of an adjoint line bundleon a surface is , and we give further restrictions for the possible values in the rangebelow 1. Theorem 4.1
Let X be a smooth projective surface and L a nef line bundle suchthat K X + L is ample. If for some point x ∈ X the Seshadri constant ε ( K X + L, x ) lies in the interval (0 , , then ε ( K X + L, x ) = m − m for some integer m > .Proof. Let x be a point such that ε ( K X + L, x ) <
1. Then there exists a curve C ⊂ X such that ε ( K X + L, x ) = ( K X + L ) · C mult x ( C ) = dm . By assumption, we have d m − . (2)By the Index Theorem, we have d = (( K X + L ) · C ) > C ( K X + L ) , (3)so that in any case C d . The nefness of L and the adjunction formula implythat we have the following upper bound on the arithmetic genus of C : p a ( C ) = 1 + 12 C + 12 C · K X d + 12 C · ( K X + L ) = 1 + d ( d + 1)2 . On the other hand, a curve having a point of multiplicity m is subject to the followinginequality p a ( C ) > (cid:18) m (cid:19) = m ( m − . (4)Combining these two inequalities, we see that for m > d > m − . Together with (2) this gives the claim. (cid:3)
The following lower bound is a direct consequence of the above Proposition.
Corollary 4.2
Let X be a smooth projective surface and L a nef line bundle suchthat K X + L is ample. Then ε ( K X + L, x ) > for every point x ∈ X . Now we show that the bound in Corollary 4.2 is sharp.
Example 4.3
Let X be a general surface of degree 10 in weighted projective space P (1 , , , X is smooth, K X is ample with K X = 1, and there is a point x ∈ X such that there exists a canonical curve D ∈ | K X | with a double point in x . For details we refer to [BS, Example 1.2]. Taking L to be the trivial line bundle,we see that ε ( K X + L, x ) = ( K X + L ) · D mult x D = 12 . This example was extreme in the sense that K X was already ample and we took L to be trivial. In the next example we show that the Seshadri constant is possiblealso at the other extreme, i.e., when K X trivial and L ample. Example 4.4
Let X be a K3 surface with intersection matrix (cid:18) − (cid:19) . Such a surface exists by [Mor, Corollary 2.9]. Moreover, by [Kov, Theorem 2] thereexist effective curves Γ and E such that Γ = − E = 0 generating the Picardgroup of X . In particular, we have Γ · E = 1. The line bundle L = O X (Γ + 3 E )is ample. It intersects every curve in the pencil | E | with multiplicity 1, so thatthere are no reducible curves in the pencil. On the other hand, the elliptic fibrationdefined by | E | must have singular fibers. If E is such a singular fiber, then it hasa double point x . We have again ε ( K X + L, x ) = L · E mult x E = 12 . Remark 4.5
If both K X and L are ample, then the Seshadri constant of K X + L is at least 1. To see this, it suffices to repeat the proof of Theorem 4.1, taking intoaccount that the self-intersection of K X + L is in that case at least 4, so that theIndex Theorem as in (3) implies now C . Combining this again with the lowerbound on p a ( C ) we get a contradiction to (2).One might hope that there exist statements stronger than Corollary 4.2 for‘hyper-adjoint’ bundles, i.e., for adjoints K X + L of very ample line bundles L .This is indeed the case: Theorem 4.6
Let X be a smooth projective surface and L a very ample line bundleon X such that K X + L is ample. Then a) ε ( K X + L ) > . b) If ε ( K X + L, x ) = 1 for all x ∈ X , then either ( X, L ) = ( P , O P (4)) or X isa ruled surface. In the latter case, one has L = − C + s · f , where C is asection, f a fiber of the ruling, and s a positive integer.Proof. a) Let x ∈ X and let C ⊂ X be an irreducible curve passing through x with m := mult x C . We will show that m ( K X + L ) · C >
1. Suppose first that L · C L is very ample, the curve C is then a line or a smooth conic, and therefore m ( K X + L ) · C is an integer > L · C >
3. Thisinequality implies, when arguing as in the proof of Theorem 4.1, that p a ( C ) + 32 C + 12 C ( K X + L ) d ( d + 1)2Using the inequality (4) we get d > m , and this completes the proof of a).There is an alternative, adjunction-theoretic approach for the proof of asser-tion a) as follows. The situation described in the proposition was studied bySommese and Van de Ven: In [SV, Theorem 0.1] they showed that the adjointline bundle K X + L is globally generated unless • X = P and L = O ( d ), with d equal to either 1 or 2, or • X is a smooth quadric in P and L is the hyperplane bundle, or • X is a P bundle over a smooth curve and L restricted to any fiber is O P (1).It is easy to see that under our assumptions none of the exceptional cases is possible,so that the claim follows using the fact that the Seshadri constants of ample andglobally generated line bundles are > ϕ K X + L : X → P N , which by the cited result of Sommese and Van de Ven is a morphism.Suppose first that ( K X + L ) = 1. Then the image of ϕ K X + L is P and we aredone.So it remains to consider that case that ( K X + L ) >
2. By assumption thereexists a family of curves C ⊂ X and points x ∈ X such that( K X + L ) · Cm = 1where m = mult x C (cf. [EL]). We claim first that m = ( K X + L ) · C = 1 . (5)Indeed, if we had m >
2, then by [Xu, Lemma 1] (or [KSS, Theorem A]) we wouldhave the inequality C > m ( m −
1) + 1 . Upon using the Index theorem, this implies4( m ( m −
1) + 1) C ( K X + L ) C (( K X + L ) · C ) = m and this is a contradiction, establishing (5).Next we wish to show that C = 0. In fact, applying the Index theorem again,we see that 4 C C ( K X + L ) (( K X + L ) · C ) = 1and hence C
0. The possibility that C < C are smooth and rational with K X · C = − K X · C < ( K X + L ) · C = 1, hence K X · C
0. Using this inequality,together with C = 0 and the adjunction formula0 p a ( C ) = 1 + 12 C + 12 K X · C implies the claim.In order to prove now that X is a ruled surface, we show that for some integer k > | kC | is a basepoint-free pencil. To this end, consider for k > → O X (( k − C ) → O X ( kC ) → O C ( kC ) → h ( X, ( k − C ) = h ( X, kC ), then h ( X, kC ) < h ( X, ( k − C ). Therefore there exists a k such that h ( X, kC ) > h ( X, ( k − C ) (6)and hence | kC | is a pencil. The curve C is the only possible base curve, but we seefrom (6) that it cannot be the base part of | kC | .Finally, after taking the Stein factorization and normalizing, we may assumethat the general element f of | kC | is irreducible. We then see from 0 p a ( f ) =1 + k C + C · K X = 1 − k that k = 1, and therefore L · f = 3. This implies that L is of the form that is asserted in the statement of the theorem. (cid:3) Remark 4.7 a) The example of the projective plane P and L = O P (4) shows thatthe bound in part a) of the previous proposition cannot be improved.b) One might hope that in part b) of the theorem it could suffice to ask that ε ( K X + L, x ) = 1 holds for infinitely many points x instead of requiring it on all points x . But the example of a smooth quartic surface X ⊂ P containing a line ℓ ,with L = O X (1) and x ∈ ℓ shows that this is not the case. Some applications, notably in multivariate interpolation and in Nagata andHarbourne-Hirschowitz problems require knowledge of the multi-point version ofthe Seshadri constants defined in the introduction.
Definition 5.1
Let X be a smooth projective variety and L be an ample line bundleon X . Let r be a positive integer and x , . . . , x r be arbitrary pairwise distinct pointson X . The real number ε ( L ; x , . . . , x r ) = inf C ∩{ x ,...,x r }6 = ∅ L · C P ri =1 mult x i C is the multi-point Seshadri constant of L at x , . . . , x r .It is easy to check that ε ( L ; x , . . . , x r ) > P ri =1 1 ε ( L,x i ) , (7)so that a lower bound on ε ( L ) gives an immediate lower bound on ε ( L ; x , . . . , x r ).Without any restrictions on L we can again produce examples of line bundles witharbitrary rational multi-point Seshadri constants quite along lines of Proposition 2.1: Proposition 5.2
For every rational number q > and every positive integer r there exists a smooth projective surface X , an integral ample line bundle L on X ,and points x , . . . , x r ∈ X such that ε ( L ; x , . . . , x r ) = q . Proof.
It suffices to produce examples with ε ( L ; x , . . . , x r ) = m + r − , where m is agiven positive integer. All other rational numbers can be obtained as multiples ofthese numbers.We modify slightly the construction from the proof of Proposition 2.1. In fact,keeping the notation from this proposition, we simple put x = x and take x , . . . , x r as arbitrary pairwise distinct points on f C . Then we have certainly L · f C mult x f C + . . . + mult x r f C = 1 m + 1 + . . . + 1 = 1 m + r − . Now, if D is an irreducible curve different from f C , then we have L · D = ( E + 2 f C ) · D > m · mult x D + mult x D + . . . + mult x r D ) > · r X i =1 mult x i D , and this implies that ε ( L ; x , . . . , x r ) is computed by f C . (cid:3) Remark 5.3
Of course one can again modify the proof of Proposition 5.2 to obtainexamples in arbitrary dimension, quite as in Remark 2.2.On the other hand, in the adjoint case, for X , L and K X + L as in Theorem 3.2,we see from (7) and Theorem 3.2 that one has ε ( K X + L ; x , . . . , x r ) > r · n + n + 4 (8)for all r -tuples x , . . . , x r ∈ X .Alternatively one can invoke the following generalization of Theorem 3.1 from[AS, Theorem 0.3]. Theorem 5.4 (Angehrn-Siu)
Let r be a positive integer. If ( L d · Z ) > (cid:18) n ( n + 2 r −
1) + 1 (cid:19) d for all irreducible subvarieties Z ⊂ X of positive dimension d > , then K X + L separates any set of arbitrary r distinct points. Combining this with the following Lemma leads to the improved lower boundexpressed in Proposition 5.6.
Lemma 5.5
Let r be a positive integer and let M be a line bundle such that thelinear series | M | separates any set of r + 1 distinct points. Then ε ( M ; x , . . . , x r ) > for all r -tuples x , . . . , x r . Proof.
Let C be a curve passing through at least one of the points x , . . . , x r andhaving multiplicities m , . . . , m r at these points. Furthermore let y be a point on C distinct from x , . . . , x r . Then, by the assumption on point separation, there existsa divisor D ∈ | M | which contains points x , . . . , x r in its support and which avoids y . So it intersects C properly, from which we get M · C = D · C > r X i =1 m i , and the assertion follows. (cid:3) Proposition 5.6
Let X , L and K X + L be as in Theorem 3.2. Then ε ( K X + L ; x , . . . , x r ) > n + (2 r + 1) n + 1 . This bound is better than (8), but still it is quite unlikely that it is sharp. Asbefore we turn now our attention to surfaces, where further restrictions are betteraccessible.Corollary 4.2 together with (7) implies that ε ( K X + L ; x , . . . , x r ) > r . Onthe other hand it is easy to construct examples of surfaces of arbitrary Kodairadimension, adjoint ample line bundles on them and r -tuples x , . . . , x r such that ε ( K X + L ; x , . . . , x r ) = r . A sample list of these is the following: • κ ( X ) = −∞ : Take X = P , L = O P (1) and r points on a line, • κ ( X ) = 0: Take a product X = E × E of two elliptic curves, L = E + E and r points on E , • κ ( X ) = 1: Take a product X = E × C of an elliptic curve E and a smoothcurve C of genus >
2, with L = E + C and r points on E , • κ ( X ) = 2: Take the surface X from Example 4.3, L = K X and r points on acanonical curve.So the interesting question is what values are possible in the range from r to r .We show: Theorem 5.7
We fix an integer r > . Let X be a smooth projective surface andlet L be a nef line bundle on X such that K X + L is ample. If for some distinctpoints x , . . . , x r ∈ X the Seshadri constant ε ( K X + L ; x , . . . , x r ) lies in the interval (0 , r ) , then ε ( K X + L ; x , . . . , x r ) = 1 r + 1 or r + 2 , unless r = 2 and ε ( K X + L ; x , x ) = .Proof. The proof is quite similar to that of Theorem 4.1. Let C be a curve on X passing through x , . . . , x r with multiplicities m , . . . , m r and such that( K X + L ) · Cm + . . . + m r = dm < r . Then as in the proof of Theorem 4.1 we have p a ( C ) d ( d + 1)2 . (9)1On the other hand, there is the lower bound p a ( C ) > (cid:18) m (cid:19) + . . . + (cid:18) m r (cid:19) = 12 r X i =1 m i − r X i =1 m i ! > (cid:18) r m − m (cid:19) . (10)Using the assumption rd m − r ( m − rm − r ) ( m − · ( m + r − . This implies that either(i) m r , or(ii) r = 2 and m = 5.In case (ii) we get the exceptional value . In case (i) we must have d = 1, and using(9) we get p a ( C )
2. Therefore there can be at most two double points amongthe x i , and hence m is bounded by r + 2. This implies the assertion. (cid:3) We conclude by showing that both main cases in the preceding theorem actuallyoccur. To obtain r +1 as a Seshadri constant is easy. Indeed, we can either startfrom Example 4.3 or Example 4.4 and take r − D or E respectively. To get r +2 requires a little bit more work. The idea isto construct a surface X as in Example 4.3 containing a canonical curve with twodouble points: Example 5.8
In the weighted projective plane H = P (1 , ,
5) with variables y, z, w let C be the curve that is defined by the homogeneous equation of degree 10 f ( y, z, w ) = w + z · ( z + y ) · ( z − y ) . Note that the curve C omits both singular points P = (0 : 1 : 0) and P = (0 : 0 : 1)of H . It follows that C is irreducible. Indeed, it is elementary to check that allpolynomials of degree P or P . The curve C has arithmeticgenus 2 and two double points at x := (1 : 0 : 0) and x := (1 : − C as the hyperplane section H ∩ X of a surface X ⊂ P (1 , , ,
5) ofdegree 10. To this end let D be a curve in H defined by a homogeneous polynomial g ( y, z, w ) of degree 9 intersecting C transversally. Then we let X be the surfacedefined by the equation F ( x, y, z, w ) := f ( y, z, w ) + x · g ( y, z, w ) = 0 . We claim that X is smooth. Taking the partial derivative of F with respect to x we see that the only singular points of X could be the intersection points of C and D . Since the intersection is transversal, we obtain a local coordinate system at eachof the intersection points and this shows that X is smooth. For details cf. [Bau,Lemma 2.2], where an analogous construction in P is carried out.Now, taking x , . . . , x r on C pairwise different and different from x and x , weget for the canonical bundle K X = O X (1) the equation K X · C P ri =1 mult x i C = 1 r + 2 , as desired.We don’t know if the exotic value can be actually obtained as a two-pointSeshadri constant.2 References [AS] Angehrn, U., Siu, Y. T.: Effective freeness and point separation for adjoint bundles. Invent.Math. 122 (1995), 291–308[Bau] Bauer, Th.: Seshadri constants on algebraic surfaces. Math. Ann. 313 (1999), 547–583[BS] Bauer, Th., Szemberg, T.: Seshadri constants on surfaces of general type, ManuscriptaMath. 126 (2008), 167–175[PSC] Bauer, Th., et al.: A primer on Seshadri constants. Contemporary Mathematics 496 (2009),33–70[Dem] Demailly, J.-P.: Singular Hermitian metrics on positive line bundles. Complex algebraicvarieties (Bayreuth, 1990), Lect. Notes Math. 1507, Springer-Verlag, 1992, pp. 87–104[EL] Ein, L., Lazarsfeld, R.: Seshadri constants on smooth surfaces. In Journ´ees de G´eom´etrieAlg´ebrique d’Orsay (Orsay, 1992). Ast´erisque No. 218 (1993), 177–186[Hei] Heier, G.: Effective freeness of adjoint line bundles. Doc. Math. 7 (2002), 31–42[KSS] Knutsen, A., Syzdek, W., Szemberg, T.: Moving curves and Seshadri constants.Math.Res.Lett. 16 (2009), no. 4, 711–719[Kov] Kov´acs, S. J.: The cone of curves of a K K Thomas Bauer, Fachbereich Mathematik und Informatik, Philipps-Universit¨at Marburg,Hans-Meerwein-Straße, D-35032 Marburg, Germany.
E-mail address: [email protected]
Tomasz Szemberg, Instytut Matematyki UP, PL-30-084 Krak´ow, Poland.
E-mail address: [email protected]