Newton-Okounkov bodies on projective bundles over curves
aa r X i v : . [ m a t h . AG ] O c t NEWTON-OKOUNKOV BODIES ON PROJECTIVEBUNDLES OVER CURVES
Pedro
Montero
Abstract . —
In this article, we study Newton-Okounkov bodies on projectivevector bundles over curves. Inspired by Wolfe’s estimates used to compute thevolume function on these varieties, we compute all Newton-Okounkov bodieswith respect to linear flags. Moreover, we characterize semi-stable vectorbundles over curves via Newton-Okounkov bodies.
Contents
1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1. Numerical classes and positivity . . . . . . . . . . . . . . . . . . . . . . 42.2. Flag varieties and Schubert cells. . . . . . . . . . . . . . . . . . . . . . 53. Newton-Okounkov bodies and Semi-stability . . . . . . . . . . . . . . 63.1. Newton-Okounkov bodies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2. Semi-stability and Harder-Narasimhan filtrations . . . . . 104. Divisors on projective bundles over curves. . . . . . . . . . . . . . . . . 125. Newton-Okounkov bodies on projective bundles over curves 165.1. Ruled surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.2. Some reductions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.3. A toric computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.4. Proofs of main results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Mathematics Subject Classification . —
PEDRO
MONTERO
1. Introduction
Let C be a smooth projective curve over an algebraically closed field ofcharacteristic zero and let E be vector bundle over C of rank r ≥
2. It is well-known since Hartshorne’s work [Har71] that numerical information comingfrom semi-stability properties of E can be translated into positivity conditions.Namely, the shape of the nef and pseudo-effective cone of the projective vectorbundle P ( E ) were determined by Hartshorne [Har71] (cf. [Laz04, §6.4.B])and Nakayama [Nak04, Chapter IV] (cf. [MDS15]) in terms of the Harder-Narasimhan filtration of E . More precisely, if we denote by ξ the class of thetautological line bundle O P ( E ) (1) and by f the class of a fiber of π : P ( E ) → C then we have that for t ∈ R the class ξ − tf is nef (resp. pseudo-effective) ifand only if t ≤ µ min ( E ) (resp. t ≤ µ max ( E )), where µ min ( E ) (resp. µ max ( E ))is the minimal (resp. maximal) slope of E . In particular, the nef and pseudo-effective cone of P ( E ) coincide if and only if E is semi-stable (cf. [Miy87, Theo.3.1] and [Ful11]).Indirectly, the pseudo-effective cone can be also deduced from the work ofWolfe [Wol05] and Chen [Che11], who explicitly computed the volume functionon P ( E ). In fact, they showed that for every t ∈ R the volume of the class ξ − tf on P ( E ) can be expressed in terms of numerical information comingfrom the Harder-Narasimhan filtration of E ,HN • ( E ) : 0 = E ℓ ⊆ E ℓ − ⊆ · · · ⊆ E ⊆ E = E with successive semi-stable quotients Q i = E i − /E i of rank r i and slope µ i .More precisely, if we consider σ ≥ . . . ≥ σ r to be the ordered slopes of E counted with multiplicities equal to the rank of the corresponding semi-stablequotient (1) , their results can be summarized as follows. Theorem 1.1 . —
Let C and E as above. Then, vol P ( E ) ( ξ − tf ) = r ! · Zb ∆ r − max r X j =1 σ r +1 − j λ j − t, dλ, where b ∆ r − ⊆ R r is the standard ( r − -simplex with coordinates λ , . . . , λ r and dλ is the standard induced Lebesgue measure for which b ∆ r − has volume r − . Following the idea that numerical information encoded by the Harder-Narasimhan filtration of E should be related to asymptotic numerical invari-ants of P ( E ), we study the geometry of Newton-Okounkov bodies on P ( E ).
1. In other words, µ max ( E ) = σ ≥ . . . ≥ σ r = µ min ( E ) can be viewed as the componentsof the vector σ = ( σ , . . . , σ r ) = ( µ ℓ , . . . , µ ℓ | {z } r ℓ times , µ ℓ − , . . . , µ ℓ − | {z } r ℓ − times , . . . , µ , . . . , µ | {z } r times ) ∈ Q r . EWTON-OKOUNKOV BODIES ON PROJECTIVE BUNDLES OVER CURVES These compact convex bodies were introduced by Okounkov in his original ar-ticle [Oko96] and they were studied later on by Kaveh and Khovanskii [KK12]and Lazarsfeld and Mustaţă [LM09], who associated to any big divisor D ona normal projective variety X of dimension r , and any complete flag of sub-varieties Y • on X satisfying suitable conditions, a convex body ∆ Y • ( D ) ⊆ R r depending only on the numerical equivalence class of D . Moreover, there existsa global Newton-Okounkov body ∆ Y • ( X ) ⊆ R r × N ( X ) R such that the slice of∆ Y • ( X ) over any big rational class η ∈ N ( X ) Q is given by ∆ Y • ( η ) ⊆ R r × { η } .Newton-Okounkov bodies of big divisors on geometrically ruled surfaceswith respect to linear flags (see Definition 5.1) can be computed via Zariskidecomposition (see Example 5.2). In higher dimension, we will use methodssimilar to those used by Wolfe to compute the volume function in [Wol05].More precisely, we will first reduce ourselves to the case of the nef and bigclass ξ − µ min ( E ) f (see Lemma 5.3). Afterwards, we will need to understandthe Harder-Narasimhan filtration of the symmetric products S m E for m ≥ Y • be a complete linear flag which is compatible with the Harder-Narasimhan filtration of E (see Definition 5.6). With the notation of Theorem1.1 above, define for each real number t ∈ R the following polytope inside thefull dimensional standard simplex ∆ r − in R r − (see Notation 4.7) (cid:3) t = ( ( ν , . . . , ν r ) ∈ ∆ r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r X i =2 σ i − ν i + σ r − r X i =2 ν i ! ≥ t ) . Then, we prove the following result (see Corollary 5.9).
Theorem A . —
Let C be a smooth projective curve and let E be a vectorbundle over C of rank r ≥ . Then, for every real number t < µ max ( E ) = µ ℓ we have that ∆ Y • ( ξ − tf ) = n ( ν , . . . , ν r ) ∈ R r ≥ | ≤ ν ≤ µ ℓ − t, ( ν , . . . , ν r ) ∈ (cid:3) t + ν o . In particular, the global Newton-Okounkov body ∆ Y • ( P ( E )) is a rational poly-hedral cone and it depends only on gr(HN • ( E )) , the graded vector bundle as-sociated to the Harder-Narasimhan filtration of E . Moreover, we obtain the following characterization of semi-stability in termsof Newton-Okounkov bodies.
Proposition B . —
Let C be a smooth projective curve and let E be a vectorbundle over C of rank r ≥ . The following conditions are then equivalent: (1) E is semi-stable. (2) For every big rational class η = a ( ξ − µ ℓ f ) + bf on P ( E ) and every linearflag Y • on P ( E ) we have that ∆ Y • ( η ) = [0 , b ] × a ∆ r − ⊆ R r . PEDRO
MONTERO
Here, a ∆ r − = { ( ν , . . . , ν r ) ∈ R r − ≥ | P ri =2 ν i ≤ a } is the full dimensionalstandard ( r − -simplex with side length a . Outline of the article . — First of all, we establish some notation and recallsome basic facts in §
2. Secondly, we recall in § § P ( E )as well as results concerning their volume and restricted volume. Finally, weprove both Theorem A and Proposition B in § Acknowledgements . — I would like to express my gratitude to my thesissupervisors, Stéphane
Druel and Catriona
Maclean , for their advice, helpfuldiscussions and encouragement throughout the preparation of this article.I also thank Bruno
Laurent , Laurent
Manivel and Bonala
NarasimhaChary for fruitful discussions. Finally, I would like to thank the anonymousreferee for a very helpful and detailed report.
2. Preliminaries
Throughout this article all varieties will be assumed to be reduced andirreducible schemes of finite type over a fixed algebraically closed field ofcharacteristic zero (2) k . We denote by N ( X ) the groupof numerical equivalence classes of Cartier divisors on X , and we defineN ( X ) k = N ( X ) ⊗ Z k for k = Q or R . All the R -divisors that we consider are R -Cartier. Dually, we denote by N ( X ) the group of numerical equivalenceclasses of 1-cycles on X . Inside N ( X ) R = N ( X ) ⊗ Z R we distinguish the Mori cone
NE( X ) ⊆ N ( X ) R , which is the closed and convex cone generatedby numerical classes of 1-cycles with non-negative real coefficients.Let E be a locally free sheaf on a variety X . We follow Grothendieck’sconvention and we define the projectivization P X ( E ) = P ( E ) of E to be Proj O X ⊕ m ≥ S m E , where S m E denotes de m th symmetric power of E . Thisvariety is endowed with a natural projection π : P ( E ) → X and a tautologicalline bundle O P ( E ) (1).Let X be a normal projective variety. Following [Laz04], we say that anumerical class η ∈ N ( X ) R is big if there exists an effective R -divisor E suchthat η − E is ample. We denote by Big( X ) ⊆ N ( X ) R the open convex cone ofbig numerical classes. A numerical class η ∈ N ( X ) R is called pseudo-effective
2. In positive characteristic one should take into account iterates of the absolute Frobeniusas in [BP14, BHP14]. It is worth mentioning that in positive characteristic the semistabilityof a vector bundle over a curve does not imply the semi-stability of its symmetric powers.
EWTON-OKOUNKOV BODIES ON PROJECTIVE BUNDLES OVER CURVES if it can be written as the limit of classes of effective R -divisors. The pseudo-effective cone is the closure of the big cone: Big( X ) = Psef( X ) (see [Laz04,Theo. 2.2.26], for instance). Moreover, a numerical class η ∈ N ( X ) R is nef if η · [ C ] ≥ C ] ∈ NE( X ), and is ample if it is the numerical class ofan R -divisor that can be written as a finite sum of ample Cartier divisors withpositive real coefficients. The cone Nef( X ) ⊆ N ( X ) R of nef classes is closedconvex, and its interior Amp( X ) is the cone of ample classes, by Kleiman’sampleness criterion. A line bundle L is big (resp. pseudo-effective, ample, nef)if and only if its numerical class c ( L ) ∈ N ( X ) R is big (resp. pseudo-effective,ample, nef).Let us recall that the stable base locus of a Q -divisor D on X is the closedset B ( D ) = \ m> Bs( mD )where Bs( mD ) is the base locus of the complete linear series | mD | . Following[ELM + augmented base locus of D to be the Zariski closedset B + ( D ) = \ A B ( D − A ) , where the intersection runs over all ample Q -divisors A . Similarly, the re-stricted base locus of D is defined by B − ( D ) = [ A B ( D + A ) , where the union runs over all ample Q -divisors A . The restricted base locus B − ( D ) consist of at most a countable union of subvarieties whose Zariskiclosure is contained in B + ( D ) (see [ELM +
06, Rema. 1.13] and [Les14]). By[ELM +
06, Prop. 1.4, Exam. 1.8, Prop. 1.15, Exam. 1.16], both B − ( D ) and B + ( D ) depend only on the numerical class of D , there is an inclusion B − ( D ) ⊆ B + ( D ), and for any rational number c > B − ( cD ) = B − ( D ) and B + ( cD ) = B + ( D ). Moreover by [ELM +
06, Exam. 1.7, Exam. 1.18] we havethat B + ( D ) = ∅ if and only if D is ample, and that B − ( D ) = ∅ if and only if D is nef. Let us denote by F r the fullflag variety parametrizing all complete linear flags on P r − . Recall that if wefix a reference complete linear flag Y • on P r − , then there is a decompositionof F r into Schubert cells F r = a w ∈ S r Ω w . PEDRO
MONTERO
More explicitly, if we consider homogeneous coordinates [ x : . . . : x r ] on P r − ,we assume that for every i = 1 , . . . , r − Y i = { x = . . . = x i = 0 } ⊆ P r − and we regard the permutation group S r as a subgroup of PGL r ( k ) via itsnatural action on the standard basis points e , . . . , e r ∈ P r − then we havethat Ω w is the orbit B · Y w • , where B denotes the (Borel) subgroup of PGL r ( k )that fixes the flag Y • and Y w • is the complete linear flag such that for every i = 1 , . . . , r − Y wi = { x w (1) = . . . = x w ( i ) = 0 } ⊆ P r − . We refer the reader to [Bri05, § .
2] and [Man98, § .
6] for further details.
3. Newton-Okounkov bodies and Semi-stability
In this section, we review the construction of Newton-Okounkov bodies andsemi-stability of vector bundles over smooth projective curves.
Let X be a smooth projective varietyof dimension n and let L be a big line bundle on X . A full flag of closedsubvarieties of X centered at the point p ∈ XY • : X = Y ⊇ Y ⊇ Y ⊇ · · · ⊇ Y n − ⊇ Y n = { p } is an admissible flag if codim X ( Y i ) = i , and each Y i is smooth at the point p . Inparticular, Y i +1 defines a Cartier divisor on Y i in a neighborhood of the point p . Following the work of Okounkov [Oko96, Oko03], Kaveh and Khovanskii[KK12] and Lazarsfeld and Mustaţă [LM09] independently associated to L and Y • a convex body ∆ Y • ( X, L ) ⊆ R n encoding the asymptotic properties of thecomplete linear series | L ⊗ m | . We will follow the presentation of [LM09] andwe refer the interested reader to the survey [Bou12] for a comparison of bothpoints of view.Let D be any divisor on X and let s = s ∈ H ( X, O X ( D )) be a non-zero section. We shall compute successive vanishing orders of global sectionsin the following manner: let D = D + div( s ) be the effective divisor inthe linear series | D | defined by s and set ν ( s ) = ord Y ( D ) the coefficientof Y in D . Then D − ν ( s ) Y is an effective divisor in the linear series | D − ν ( s ) Y | , and does not contain Y in its support, so we can define D = ( D − ν ( s ) Y ) | Y and set ν ( s ) = ord Y ( D ). We proceed inductivelyin order to get ν Y • ( s ) = ( ν ( s ) , . . . , ν n ( s )) ∈ N d . This construction leads to avaluation-like function ν Y • : H ( X, O X ( D )) \ { } → Z n , s ( ν ( s ) , . . . , ν n ( s )) . EWTON-OKOUNKOV BODIES ON PROJECTIVE BUNDLES OVER CURVES We then define the graded semigroup of D to be the sub-semigroup of N n × N defined byΓ Y • ( D ) = n ( ν Y • ( s ) , m ) ∈ N n × N | = s ∈ H ( X, O X ( mD ) o . Finally, we define the
Newton-Okounkov body of D with respect to Y • to be∆ Y • ( D ) = cone(Γ Y • ( D )) ∩ ( R n × { } ) , where cone(Γ Y • ( D )) denotes the closed convex cone in R n × R spanned byΓ Y • ( D ).These convex sets ∆ Y • ( D ) are compact and they have non-empty interiorwhenever D is big. Moreover, by [LM09, Theo. A], we have the followingidentity vol R n (∆ Y • ( D )) = 1 n ! · vol X ( D ) , where vol X ( D ) = lim m →∞ h ( X, O X ( mD )) m n /n ! . In particular, if D is big and nef,then vol R n (∆ Y • ( D )) = n ! D n , by the Asymptotic Riemann-Roch theorem.The Newton-Okounkov bodies of big divisors depend only on numericalclasses: if D ≡ num D ′ are big divisors then ∆ Y • ( D ) = ∆ Y • ( D ′ ) for everyadmissible flag Y • on X , by [LM09, Prop. 4.1] (see [Jow10, Theo. A] forthe converse). This fact, along with the identity ∆ Y • ( pD ) = p · ∆ Y • ( D )for every positive integer p , enables us to define an Newton-Okounkov body∆ Y • ( η ) ⊆ R n for every big rational class η ∈ Big( X ) ∩ N ( X ) Q . Moreover, by[LM09, Theo. B], there exists a global Newton-Okounkov body : a closed convexcone ∆ Y • ( X ) ⊆ R n × N ( X ) R such that for each big rational class η ∈ Big( X ) Q = Big( X ) ∩ N ( X ) Q thefiber of the second projection over η is ∆ Y • ( η ). This enables us to defineNewton-Okounkov bodies for big real classes by continuity.The above construction works for graded linear series A • associated toa big divisor D on X . A graded linear series is a collection of subspaces A m ⊆ H ( X, O X ( mD )) such that A • = ⊕ m ≥ A m is a graded subalgebra ofthe section ring R ( D ) = ⊕ m ≥ H ( X, O X ( mD )). The construction enables usto attach to any graded linear series A • a closed and convex set ∆ Y • ( A • ) ⊆ R n .This set ∆ Y • ( A • ) will be compact and will compute the volume of the linearseries under some mild conditions listed in [LM09, § restricted complete linear series of a big divisor D , namely gradedlinear series of the form A m = H ( X | F, O X ( mD )) = Im (cid:16) H ( X, O X ( mD )) rest −−→ H ( F, O F ( mD )) (cid:17) where F ⊆ X is an irreducible subvariety of dimension d ≥
1. Under thehypothesis that F B + ( D ), the conditions listed in [LM09, § PEDRO
MONTERO by [LM09, Lemm. 2.16]. Therefore, the Newton-Okounkov body associatedto A • above, the restricted Newton-Okounkov body (with respect to a fixedadmissible flag) ∆ X | F ( D ) ⊆ R d , is compact and vol R d (∆ X | F ( D )) = 1 d ! vol X | F ( D ) , where vol X | F ( D ) = lim m →∞ dim k A m m d /d !is the restricted volume on F of the divisor D . In particular, if D is big andnef, then vol X | F ( D ) = ( D d · F ), by [ELM +
09, Cor. 2.17]. Restricted Newton-Okounkov bodies depend only on numerical classes (see [LM09, Rema. 4.25]),so it is meaningful to consider ∆ X | F ( η ) for every big rational class η such that F B + ( η ).As before, there exists a global Newton-Okounkov body ∆ Y • ( X | F ) thatenables us to define, by continuity, ∆ X | F ( η ) for any big real numerical class η such that F B + ( η ). See [LM09, Exam. 4.24] for details.Restricted Newton-Okounkov bodies can be used to describe slices ofNewton-Okounkov bodies. Theorem 3.1 ([LM09, Theo. 4.26, Cor. 4.27]). —
Let X be a normalprojective variety of dimension n , and let F ⊆ X be an irreducible and reducedCartier divisor on X . Fix an admissible flag Y • : X = Y ⊇ Y ⊇ Y ⊇ · · · ⊇ Y n − ⊇ Y n = { p } with divisorial component Y = F . Let η ∈ Big( X ) Q be a rational big class, andconsider the Newton-Okounkov body ∆ Y • ( η ) ⊆ R n . Write pr : ∆ Y • ( η ) → R for the projection onto the first coordinate, and set ∆ Y • ( η ) ν = t = pr − ( t ) ⊆ { t } × R n − ∆ Y • ( η ) ν ≥ t = pr − ([ t, + ∞ )) ⊆ R n Assume that F B + ( η ) and let τ F ( η ) = sup { s > | η − s · f ∈ Big( X ) } , where f ∈ N ( X ) is the numerical class of F . Then, for any t ∈ R with ≤ t < τ F ( η ) we have
1. ∆ Y • ( η ) ν ≥ t = ∆ Y • ( η − tf ) + t · ~e , where ~e = (1 , , . . . , ∈ N n is thefirst standard unit vector. (3)
3. In fact, this statement remains true even if we do not assume that E B + ( η ). See[KL15, Prop. 1.6]. EWTON-OKOUNKOV BODIES ON PROJECTIVE BUNDLES OVER CURVES
2. ∆ Y • ( η ) ν = t = ∆ X | F ( η − tf ) . The function t vol X ( η + tf ) is differentiable at t = 0 , and ddt (vol X ( η + tf )) | t =0 = n · vol X | F ( η ) . We will need the following observation by Küronya, Lozovanu and Maclean.
Proposition 3.2 ([KLM12, Prop. 3.1]). —
Let X be a normal projectivevariety together with an admissible flag Y • . Suppose that D is a big Q -divisorsuch that Y B + ( D ) and that D − tY is ample for some ≤ t < τ Y ( D ) ,where τ Y ( D ) = sup { s > | D − sY is big } . Then ∆ Y • ( X, D ) ν = t = ∆ Y • | Y ( Y , ( D − tY ) | Y ) , where Y • | Y : Y ⊇ Y ⊇ · · · ⊇ Y n is the induced admissible flag on Y .In particular, if Psef( X ) = Nef( X ) then we have that the Newton-Okounkovbody ∆ Y • ( D ) is the closure in R n of the following set { ( t, ν , . . . , ν n ) ∈ R n | ≤ t < τ Y ( D ) , ( ν , . . . , ν n ) ∈ ∆ Y • | Y ( Y , ( D − tY ) | Y ) } . Let us finish this section with the case of Newton-Okounkov bodies onsurfaces (see [LM09, §6.2] for details). We will use this description in Example5.2 in order to illustrate the shape of Newton-Okounkov bodies on ruledsurfaces.
Example 3.3 (Surfaces). —
Let S be a smooth projective surface togetherwith a flag Y • : Y = S ⊇ Y = C ⊇ Y = { p } , where C ⊆ S is a smooth curveand p ∈ C .Let D be a big Q -divisor on S . Any such divisor admits a Zariski decompo-sition , that is we can uniquely write D as a sum D = P ( D ) + N ( D )of Q -divisors, with P ( D ) nef and N ( D ) either zero or effective with nega-tive definite intersection matrix. Moreover, P ( D ) · Γ = 0 for every irre-ducible component Γ of N ( D ) and for all m ≥ ( S, O S ( ⌊ mP ( D ) ⌋ )) ∼ = H ( S, O S ( ⌊ mD ⌋ )). In this decomposition P ( D ) iscalled the positive part and N ( D ) the negative part . See [Laz04, §2.3.E] andreferences therein for proofs and applications.With the above notation, we have that∆ Y • ( D ) = n ( t, y ) ∈ R | ν ≤ t ≤ τ C ( D ) , α ( t ) ≤ t ≤ β ( t ) o where1. ν ∈ Q the coefficient of C in N ( D ),2. τ C ( D ) = sup { t > | D − tC is big } ,3. α ( t ) = ord p ( N t · C ), PEDRO
MONTERO β ( t ) = ord p ( N t · C ) + P t · C ,where D − tC = P t + N t is a Zariski decomposition, P t being the positive and N t the negative part. Moreover, these bodies are finite polygons, by [KLM12,Theo. B]. Through-out this section, C is a smooth projective curve and E is a locally free sheafon C of rank r > d = deg( E ) = deg( c ( E )). Given such a bundlewe call the rational number µ ( E ) = dr the slope of E . Definition 3.4 (Semi-stability). —
Let E be a vector bundle on C of slope µ . We say that E is semi-stable if for every non-zero sub-bundle S ⊆ E , wehave µ ( S ) ≤ µ . Equivalently, E is semi-stable if for every locally-free quotient E ։ Q of non-zero rank, we have µ ≤ µ ( Q ). A non semi-stable vector bundlewill be called unstable.Following [LP97, Prop. 5.4.2], there is a canonical filtration of E withsemi-stable quotients. Proposition 3.5 . —
Let E be a vector bundle on C . Then E has an increas-ing filtration by sub-bundles HN • ( E ) : 0 = E ℓ ⊆ E ℓ − ⊆ · · · ⊆ E ⊆ E = E where each of the quotients E i − /E i satisfies the following conditions: Each quotient E i − /E i is a semi-stable vector bundle; µ ( E i − /E i ) < µ ( E i /E i +1 ) for i = 1 , . . . , ℓ − .This filtration is unique. The above filtration is called the
Harder-Narasimhan filtration of E . Notation 3.6 . —
Let E be a vector bundle on a smooth projective curve C . We will denote by Q i = E i − /E i the semi-stable quotients of the Harder-Narasimhan filtration of E , each one of rank r i = rank( Q i ), degree d i =deg( c ( Q i )) and slope µ i = µ ( Q i ) = d i /r i . With this notation, µ and µ ℓ correspond to the minimal and maximal slopes, µ min ( E ) and µ max ( E ),respectively.From a cohomological point of view, semi-stable vector bundles can be seenas the good higher-rank analogue of line bundles. For instance, we have thefollowing classical properties (see [RR84] or [But94, Lemm. 1.12, Lemm. 2.5]). Lemma 3.7 . —
Let E and F be vector bundles on C and m ∈ N . Then, EWTON-OKOUNKOV BODIES ON PROJECTIVE BUNDLES OVER CURVES µ max ( E ⊗ F ) = µ max ( E ) + µ max ( F ) . µ min ( E ⊗ F ) = µ min ( E ) + µ min ( F ) . µ max ( S m E ) = mµ max ( E ) . µ min ( S m E ) = mµ min ( E ) . If µ max ( E ) < , then dim k H ( C, E ) = 0 . If µ min ( E ) > g − , then dim k H ( C, E ) = 0 .In particular, if E and F are semi-stable then S m E and E ⊗ F are semi-stable. If E , . . . , E ℓ are vector bundles on C and m , . . . , m ℓ be non-negativeintegers. By the splitting principle [Ful84, Rema. 3.2.3] we can prove thefollowing formula: µ ( S m E ⊗ · · · ⊗ S m ℓ E ℓ ) = ℓ X i =1 m i µ ( E i ) . Moreover, we have that for every m ≥ S m E can be computed in terms of the one for E (see [Che11, Prop. 3.4] and [Wol05, Prop. 5.10], for instance). Proposition 3.8 . —
Let E be a vector bundle on C with Harder-Narasimhanfiltration HN • ( E ) : 0 = E ℓ ⊆ E ℓ − ⊆ · · · ⊆ E ⊆ E = E and semi-stable quotients Q i = E i − /E i with slopes µ i = µ ( Q i ) , for i =1 , . . . , ℓ . For every positive integer m ≥ , let us consider the vector bundle S m E with Harder-Narasimhan filtration HN • ( S m E ) : 0 = W M ⊆ W M − ⊆ · · · ⊆ W ⊆ W = S m E and semi-stable quotients W j − /W j with slopes ν j = µ ( W j − /W j ) , for j = 1 , . . . , M . Then, for every j = 1 , . . . , M we have that W j = XP i m i µ i ≥ ν j +1 S m E ⊗ · · · ⊗ S m ℓ E ℓ − and W j − /W j ∼ = MP i m i µ i = ν j S m Q ⊗ · · · ⊗ S m ℓ Q ℓ , where the sums are taken over all partitions m = ( m , . . . , m ℓ ) ∈ N ℓ of m , and S m E ⊗ · · · ⊗ S m ℓ E ℓ − denotes the image of the composite E ⊗ m ⊗ · · · ⊗ E ⊗ m ℓ ℓ − → E ⊗ m → S m E. In particular, there is a refinement F • of HN • ( S m E ) of length L = L ( m ) and whose respective successive quotients are of the form F i − /F i ∼ = Q m ( i ) = S m Q ⊗ · · · ⊗ S m ℓ Q ℓ PEDRO
MONTERO for some partition m ( i ) = ( m , . . . , m ℓ ) ∈ N ℓ of m , and such that for every i ∈ { , . . . L } we have µ ( Q m ( i ) ) ≤ µ ( Q m ( i +1) ) . Moreover, given any partition m ∈ N ℓ of m , there is one and only one i ∈ { , . . . , L } such that m ( i ) = m .
4. Divisors on projective bundles over curves
Let E be a vector bundle on a smooth projective curve C , of rank r ≥ d . In this section we study divisors on the projective bundle π : P ( E ) → C of one-dimensional quotients. Let us recall that in this case theNéron-Severi group of P ( E ) is of the formN ( P ( E )) = Z · f ⊕ Z · ξ, where f is the numerical class of a fiber of π and ξ = ξ E is the numericalclass of a divisor representing the tautological line bundle O P ( E ) (1). Moreover,if [pt] denotes the class of a point in the ring N ∗ ( P ( E )) then we have thefollowing relations: f = 0 , ξ r − f = [pt] , ξ r = d · [pt] . The cone of nef divisors can be described via Hartshorne’s characterizationof ample vector bundles over curves [Har71, Theo. 2.4] (cf. [Ful11, Lemm.2.1]).
Lemma 4.1 . —
Nef( P ( E )) = h ξ − µ min f, f i . The cone of pseudo-effective divisors was obtained by Nakayama in[Nak04, Cor. IV.3.8], and it was indirectly computed by Wolfe [Wol05] andChen [Che11] who independently obtained the volume function vol P ( E ) onN ( P ( E )) R . A more general result on the cone of effective cycles of arbitrarycodimension can be found in [Ful11, Theo. 1.1]. Lemma 4.2 . —
Psef( P ( E )) = h ξ − µ max f, f i . In particular, we recover a result of Miyaoka [Miy87, Theo. 3.1] on semi-stable vector bundles over curves that was generalized by Fulger in [Ful11,Prop. 1.5].
Corollary 4.3 . —
A vector bundle E on a smooth projective curve C issemi-stable if and only if Nef( P ( E )) = Psef( P ( E )) . We finish this section by recalling Wolfe’s computation of the volume func-tion on N ( P ( E )). See also [Che11, Theo. 1.2]. EWTON-OKOUNKOV BODIES ON PROJECTIVE BUNDLES OVER CURVES Notation 4.4 . —
Let d ≥ standard d -simplex b ∆ d with d + 1 vertices in R d +1 to be b ∆ d = n ( x , . . . , x d +1 ) ∈ R d +1 (cid:12)(cid:12)(cid:12) P d +1 i =1 x i = 1 and x i ≥ i o . By projecting b ∆ d onto the hyperplane x = 0, we can identify b ∆ d with the full dimensional standard d -simplex (or just d -simplex ) in R d given by∆ d = n ( x , . . . , x d +1 ) ∈ R d (cid:12)(cid:12)(cid:12) P d +1 i =2 x i ≤ x i ≥ i o . Via the previous identification, we will denote by λ the Lebesgue measure on b ∆ d induced by the standard Lebesgue measure on ∆ d ⊆ R d . In particular, wewill have λ ( b ∆ d ) = d ! .Given a positive real number a >
0, we define the d -simplex with side length a by a ∆ d = (cid:8) ( x , . . . , x d +1 ) ∈ R d (cid:12)(cid:12) P d +1 i =2 x i ≤ a and x i ≥ i (cid:9) . Similarfor a b ∆ d ⊆ R d +1 , the standard d -simplex with side length a . Theorem 4.5 ([Wol05, Theo. 5.14]). —
Let E be a vector bundle withHarder-Narasimhan filtration of length ℓ and semi-stable quotients Q i of ranks r i and slopes µ i . Then, for any t ∈ R vol P ( E ) ( ξ − tf ) = r ! · Zb ∆ ℓ − max ( ℓ X i =1 µ i β i − t, ) β r − · · · β r ℓ − ℓ ( r − · · · ( r ℓ − dβ where b ∆ ℓ − ⊆ R ℓ is the standard ( ℓ − -simplex with coordinates β , . . . , β ℓ ,and β be the standard induced Lebesgue measure. Remark 4.6 . —
Alternatively, Chen computed in [Che11, Theo. 1.2] asimilar volume formula, but slightly simplified by integrating in R r insteadof R ℓ (cf. [Che11, Prop. 3.5]). More precisely, with the same notation asabove vol P ( E ) ( ξ − tf ) = r ! · Zb ∆ r − max r X j =1 s j λ j − t, dλ where b ∆ r − ⊆ R r is the standard ( r − -simplex with coordinates λ , . . . , λ r , dλ is the standard induced Lebesgue measure (4) , and s = ( s , . . . , s r ) is a vectorin R r such that the value µ i appears exactly r i times in the coordinates of s asin Notation 4.7 below.
4. Unlike Chen, we do not normalize the measure in order to have λ (∆ r − ) = 1. PEDRO
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Notation 4.7 . —
Fix ℓ ≥ r ≥ r , . . . , r ℓ ) ∈ N ℓ apartition of r and t ∈ R . We define for ( µ , . . . , µ ℓ ) ∈ Q ℓ the followingpolytopes: b (cid:3) t = ( ( β , . . . , β ℓ ) ∈ b ∆ ℓ − ⊆ R ℓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℓ X i =1 µ i β i ≥ t ) and (cid:3) t = ( ( λ , . . . , λ r ) ∈ b ∆ r − ⊆ R r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r X i =1 s i λ i ≥ t ) , where s = ( µ , . . . , µ | {z } r times , µ , . . . , µ | {z } r times , . . . , µ ℓ , . . . , µ ℓ | {z } r ℓ times ) ∈ Q r . Similarly, for every permutation w ∈ S r we define (cid:3) wt = ( ( λ , . . . , λ r ) ∈ b ∆ r − ⊆ R r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r X i =2 σ w ( i − λ i + σ w ( r ) λ ≥ t ) , where σ = ( µ ℓ , . . . , µ ℓ | {z } r ℓ times , µ ℓ − , . . . , µ ℓ − | {z } r ℓ − times , . . . , µ , . . . , µ | {z } r times ) ∈ Q r . By abuse of notation, we will also denote by (cid:3) t and (cid:3) wt the full dimensionalpolytopes in R r − obtained via the projection of b ∆ r − ⊆ R r onto ∆ r − ⊆ R r − .Explicitly, if ( ν , . . . , ν r ) are coordinates in R r − then (cid:3) t = ( ( ν , . . . , ν r ) ∈ ∆ r − ⊆ R r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s − r X i =2 ν i ! + r X i =2 s i ν i ≥ t ) and( ⋆ ) (cid:3) wt = ( ( ν , . . . , ν r ) ∈ ∆ r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r X i =2 σ w ( i − ν i + σ w ( r ) − r X i =2 ν i ! ≥ t ) . Corollary 4.8 . —
For any admissible flag Y • on P ( E ) and any big rationalclass ξ − tf on P ( E ) we have that vol R r (∆ Y • ( ξ − tf )) = Zb (cid:3) t ℓ X i =1 µ i β i − t ! β r − · · · β r ℓ − ℓ ( r − · · · ( r ℓ − dβ, for b (cid:3) t ⊆ R ℓ as in Notation 4.7. Remark 4.9 . —
Let F = π − ( q ) ∼ = P r − be any fiber of π : P ( E ) → C .Then, for any big R -divisor D on P ( E ) we have that F B + ( D ) . In fact,if D ∼ R A + E , with A ample R -divisor and E effective R -divisor, and if F ⊆ Supp( E ) , then we can write E = aF + E ′ with a > and F Supp( E ′ ) ,which implies that F B + ( D ) since A + aF is ample. EWTON-OKOUNKOV BODIES ON PROJECTIVE BUNDLES OVER CURVES As a direct consequence, we compute the restricted volume function on afiber F = π − ( q ) ∼ = P r − . Corollary 4.10 . —
Let F be a fiber of π : P ( E ) → C and let ξ − tf be a bigrational class. Then, vol P ( E ) | F ( ξ − tf ) = ( r − · Zb (cid:3) t β r − · · · β r ℓ − ℓ ( r − · · · ( r ℓ − dβ, for b (cid:3) t ⊆ R ℓ as in Notation 4.7. In particular, if ≤ τ ≤ µ ℓ − t then we havethat vol R r − (∆ Y • ( ξ − tf ) ν = τ ) = Zb (cid:3) t + τ β r − · · · β r ℓ − ℓ ( r − · · · ( r ℓ − dβ, where Y • is any admissible flag on P ( E ) with divisorial component Y = F .In particular, these volumes depend only on gr(HN • ( E )) , the graded vectorbundle associated to the Harder-Narasimhan filtration of E .Proof . — We consider v ( τ ) = vol P ( E ) ( ξ − tf + τ f ) = r ! · Zb (cid:3) t − τ ℓ X i =1 µ i β i − t + τ ! β r − · · · β r ℓ − ℓ ( r − · · · ( r ℓ − dβ. Since F B + ( ξ − tf ) by Remark 4.9, Theorem 3.1 and differentiation underthe integral sign givevol P ( E ) | F ( ξ − tf ) = 1 r · ddτ v ( τ ) (cid:12)(cid:12)(cid:12)(cid:12) τ =0 = ( r − · Zb (cid:3) t β r − · · · β r ℓ − ℓ ( r − · · · ( r ℓ − dβ. Lemma 4.11 . —
Following Notation 4.7 ( ⋆ ) , we have that vol R r (∆ Y • ( ξ − tf )) = Z (cid:3) t r X j =1 s j λ j − t dλ and vol R r − (∆ Y • ( ξ − tf ) | ν = τ ) = Z (cid:3) t + τ dλ = vol R r − ( (cid:3) t + τ ) . Moreover, vol R r − ( (cid:3) t + τ ) = vol R r − ( (cid:3) wt + τ ) for every w ∈ S r .Proof . — The first two equalities follow from Remark 4.6. For the last as-sertion consider (cid:3) t ⊆ R r and (cid:3) wt ⊆ R r as in Notation 4.7. Then, there is alinear transformation T : R r → R r , whose associated matrix in the canonicalbasis of R r is given by a permutation matrix, such that T ( (cid:3) t ) = (cid:3) wt and | det( T ) | = 1. PEDRO
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5. Newton-Okounkov bodies on projective bundles over curves
Let E be a vector bundle on a smooth projective curve C , of rank r ≥ d . In this section we study the geometry of Newton-Okounkov bodiesof rational big classes in N ( P ( E )) in terms of the numerical information ofthe Harder-Narasimhan filtration of E . In particular, Theorem A will bea consequence of Lemma 5.3, Theorem 5.8 and Corollary 5.9. We followNotation 3.6. Definition 5.1 (Linear flag). —
A complete flag of subvarieties Y • on theprojective vector bundle P ( E ) π −→ C is called a linear flag centered at p ∈ P ( E ) ,over the point q ∈ C (or simply a linear flag ) if Y = P ( E ), Y = π − ( q ) ∼ = P r − and Y i ∼ = P r − i is a linear subspace of Y i − for i = 2 , . . . , r , with Y r = { p } . Let us begin with the following example that illus-trates the general case. Namely, that the shape of Newton-Okounkov bodieson P ( E ) will depend on the semi-stability of E . Example 5.2 . —
Suppose that rank( E ) = 2 and let η = a ( ξ − µ ℓ f ) + bf ∈ N ( P ( E )) Q be a big class. In other words, a, b ∈ Q > . Let Y • : P ( E ) ⊇ F = π − ( q ) ⊇ { p } be the linear flag centered at p ∈ P ( E ), over q ∈ C . The Newton-Okounkovbody of η can be computed by applying [LM09, Theo. 6.4] (see Example 3.3). Semi-stable case. If E is semi-stable then Corollary 4.3 implies that every bigclass is ample. In particular, for every η ∈ Big( X ) Q and every Q -divisor D η with numerical class η , we have that N ( D η ) = 0 and P ( D η ) = D η . It followsthat, with the notation as in Example 3.3, ν = 0, τ F ( η ) = b , α ( t ) = 0 for every t ∈ [0 , b ] and β ( t ) = a for every t ∈ [0 , b ]. The Newton-Okounkov bodies aregiven by rectangles in this case. ty (0 , , a ) ( b, Figure 1.
Newton-Okounkov body ∆ Y • ( η ) for E semi-stable EWTON-OKOUNKOV BODIES ON PROJECTIVE BUNDLES OVER CURVES Unstable case. If E is unstable we consider its Harder-Narasimhan filtration0 → E → E → Q → , and we note that in this case E = Q ℓ . The quotient E → Q → s : C → P ( E ) with [ s ( C )] = ξ − µ ℓ f in N ( P ( E )). The curve s ( C )is the only irreducible curve on P ( E ) with negative self-intersection. On theother hand, if D η is any Q -divisor with numerical class η then, either η isinside the nef cone of P ( E ) and thus P ( D η ) = D η and N ( D η ) = 0, or η is bigand not nef in which case we compute that[ P ( D η )] = bµ ℓ − µ ( ξ − µ f ) and [ N ( D η )] = (cid:18) a ( µ ℓ − µ ) − bµ ℓ − µ (cid:19) ( ξ − µ ℓ f )in N ( P ( E )) Q . We notice that N ( D η ) = (cid:16) a ( µ ℓ − µ ) − bµ ℓ − µ (cid:17) · s ( C ) as Q -divisor, byminimality of the negative part and by the fact that the negative part is uniquein its numerical equivalence class. See [Băd01, Lemm. 14.10, Cor. 14.13] fordetails.Let t ∗ = b − a ( µ ℓ − µ ). We have that η is big and nef if and only if t ∗ ≥ η − tf is big and nef for 0 ≤ t ≤ t ∗ . For t ∗ ≤ t ≤ b ,the same computation above enables us to find the Zariski decomposition of D η − tF , which is big and not nef.We notice that, with the notation as in Example 3.3, ν = 0. On the otherhand, the functions α ( t ) and β ( t ) will depend on whether or not we have { p } = s ( C ) ∩ F . A straightforward computation shows that the Newton-Okounkov bodies are given by the following finite polygons in R . ty (0 ,
0) ( t ∗ , , a ) ( b, ty (0 ,
0) ( t ∗ , , a ) ( b, Figure 2. ∆ Y • ( η ) for E unstable and η big and nef(a) if { p } 6 = s ( C ) ∩ F (b) if { p } = s ( C ) ∩ F PEDRO
MONTERO ty (0 , , bµ ℓ − µ ) ( b, ty (0 , , − t ∗ µ ℓ − µ )(0 , a ) ( b, Figure 3. ∆ Y • ( η ) for E unstable and η big and not nef(a) if { p } 6 = s ( C ) ∩ F (b) if { p } = s ( C ) ∩ F We notice that Figure 3 provides examples of big and not nef divisors classes η such that the origin ~ ∈ ∆ Y • ( η ) for almost every linear flag Y • except for one.This shows in particular that condition (2) in the characterization of nefnessgiven in [KL15, Cor. 2.2] has to be checked for all linear flags. We first observe that Proposition B in § Proof of Proposition B . — (1) ⇒ (2). Let η = a ( ξ − µ ℓ f )+ bf be a big rationalclass on P ( E ) and Y • : X = Y ⊇ Y ⊇ Y ⊇ · · · ⊇ Y r − ⊇ Y r = { p } be a linearflag centered at p ∈ P ( E ), over the point q ∈ C . Since E is semi-stable wehave that Big( P ( E )) = Amp( P ( E )) by Corollary 4.3. Equivalently, we havethat B + ( η ) = ∅ for every big rational class η .We notice that τ F ( η ) = sup { s > | η − sf ∈ Big( P ( E )) } = b . Theimplication follows from Proposition 3.2 and [Bou12, Cor. 4.11], by notingthat if D η is a Q -divisor with numerical class η then∆ Y • ( η ) ν = t = ∆ Y • | F ( F, ( D η − tF ) | F ) = a ∆ Y • | F ( F, H ) = a ∆ r − ⊆ R r − , where H ⊆ F ∼ = P r − is an hyperplane section.(2) ⇒ (1). We notice that if for all linear flags Y • the Newton-Okounkov bodyof η = a ( ξ − µ ℓ f ) + bf is given by ∆ Y • ( η ) = [0 , b ] × a ∆ r − ⊆ R r then η is a bigand nef class, by [KL15, Cor. 2.2]. We can therefore compute the volume of η via the top self-intersection vol P ( E ) ( η ) = η r = ra r − ( b − a ( µ ℓ − µ ( E ))). Onthe other hand, we have thatvol R n (∆ Y • ( η )) = vol R r ([0 , b ] × a ∆ r − ) = a r − ( r − b, EWTON-OKOUNKOV BODIES ON PROJECTIVE BUNDLES OVER CURVES and hence [LM09, Theo. A] leads to µ ℓ = µ ( E ), implying the result.We will first reduce our problem to computing the Newton-Okounkov bodyof the big and nef divisor class ξ − µ f . Lemma 5.3 . —
Let E be a unstable vector bundle on a smooth projectivecurve C , with rank( E ) = r ≥ . Then for every big class η = a ( ξ − µ ℓ f ) + bf and every linear flag Y • we have that (1) ∆ Y • ( η ) = ([0 , t ∗ ] × a ∆ r − ) ∪ ( a ∆ Y • ( ξ − µ f ) + t ∗ ~e ) if η is big and nef; (2) ∆ Y • ( η ) = a ∆ Y • ( ξ − µ f ) ν ≥− t ∗ + t ∗ ~e if η is big and not nef.Here t ∗ = b − a ( µ ℓ − µ ) and a ∆ r − ⊆ R r − is the ( r − -simplex with sidelength a .Proof . — If η is an ample rational class then ∆ Y • ( η ) ν = t = a ∆ r − for 0 ≤ t ≤ t ∗ , by Proposition 3.2 and [LM09, Exam. 1.1]. On the other hand, we havethat ∆ Y • ( η ) ν ≥ t ∗ = a ∆ Y • ( ξ − µ f ) + t ∗ ~e by Theorem 3.1.If η is big and not nef then t ∗ <
0. Theorem 3.1 implies therefore that a ∆ Y • ( ξ − µ f ) ν ≥− t ∗ = ∆ Y • ( η ) − t ∗ ~e , which leads to (2). We will need the following result concerningthe Newton-Okounkov bodies of some toric graded algebras.
Lemma 5.4 . —
Fix A ∈ R , an integer r ≥ , σ = ( σ , . . . , σ r ) ∈ R r andhomogeneous coordinates [ x : . . . : x r ] on P r − . Put B = k and consider forevery integer m ≥ the vector subspace B m ⊆ H ( P r − , O P r − ( m )) generatedby monomials x α = x α · · · x α r r of total degree | α | = m such that P ri =1 σ i α i > A .Suppose that A ≥ and σ ≥ σ ≥ · · · ≥ σ r . Then B • = ⊕ m ≥ B m is a graded subalgebra of the coordinate ring k [ x , . . . , x r ] . Let us denote by V • the flag of linear subspaces V • : V = P r − ⊇ V ⊇ · · · ⊇ V r − , where V i = { x = . . . = x i = 0 } for i = 1 , . . . , r − and consider theSchubert cell decomposition of the full flag variety parametrizing completelinear flag on P r − F r = a w ∈ S r Ω w , with respect to V • . Fix a permutation w ∈ S r and let Y • ∈ Ω w . Then ∆ Y • ( B • ) is given by the projection of the polytope ( ( ν , . . . , ν r ) ∈ R r ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r X i =1 ν i = 1 and r X i =1 σ w ( i ) ν i ≥ ) onto the hyperplane { ν r = 0 } . PEDRO
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Proof . — Let us suppose that x α ∈ B m and x α ′ ∈ B m ′ , then x α + α ′ belongsto b m + m ′ since we have P ri =1 σ i ( α i + α ′ i ) > A ≥ A as long as A ≥
0. Thisproves (1).In order to prove (2) we will first show that every automorphism ϕ of P r − that fixes the flag V • induces an automorphism of the graded algebra B • . Forthis, we remark that automorphisms ϕ of P r − fixing the flag V • correspondto lower triangular matrices in PGL r ( k ) and hence, given ϕ = ( ϕ i,j ) ≤ i,j ≤ r such a matrix, the image of the monomial x α ∈ B m via the induced action on k [ x , . . . , x r ] is given by ϕ ( x α ) = r Y i =1 ( ϕ i, x + . . . + ϕ i,i x i ) α i . The above product can be written as a linear combination of monomials ofthe form x α ′ = x k + ... + k r where k i = ( k i, , . . . , k i,i , , . . . , ∈ N r is such that | k i | = α i . Let us prove that all these monomials belong to B m as well. In fact,we have that α ′ i = P rj = i k j,i and hence P ri =1 σ i α ′ i = P ri =1 P rj = i σ i k j,i = P ri =1 P ij =1 σ j k i,j ≥ P ri =1 P ij =1 σ i k i,j since σ ≥ · · · ≥ σ r = P ri =1 σ i | k i | = P ri =1 σ i α i > A. It follows that ϕ induces an automorphism of the graded algebra B • .In order to compute Newton-Okounkov bodies with respect to linear flagson a given Schubert cell, we note that [Bri05, Prop. 1.2.1] implies that given apermutation w ∈ S r and a linear flag Y • ∈ Ω w there exists an automorphism ϕ of P r − that fixes the reference flag V • and such that the image of the flag Y w • via the induced action of ϕ on F r is Y • , where Y w • is the linear flag suchthat for every i = 1 , . . . , r − Y wi = { x w (1) = . . . = x w ( i ) = 0 } ⊆ P r − . It follows from the previous paragraph that ϕ induces an automorphism of thegraded algebra B • and then for every m ≥ P ∈ B m we have that ν Y • ( ϕ ( P )) = ν ϕ ( Y w • ) ( ϕ ( P )) = ν Y w • ( P ) . In particular, we have { ν Y • ( P ) } P ∈ B m = { ν Y w • ( P ) } P ∈ B m ⊆ N r − and conse-quently ∆ Y • ( B • ) = ∆ Y w • ( B • ). Therefore, we can suppose that Y • = Y w • ∈ Ω w in order to prove (2).Since B m is generated by monomials we have that∆ Y w • ( B • ) = [ m ≥ (cid:26) ν Y w • ( x α ) m (cid:12)(cid:12)(cid:12)(cid:12) x α ∈ B m (cid:27) . EWTON-OKOUNKOV BODIES ON PROJECTIVE BUNDLES OVER CURVES We compute for every monomial x α = x α · · · x α r r ∈ B m that ν Y w • ( x α ) = ( α w (1) , α w (2) , . . . , α w ( r − ) ∈ N r − and hence for every m ≥ (cid:26) ν Y w • ( x α ) m (cid:12)(cid:12)(cid:12)(cid:12) x α ∈ B m (cid:27) is given by the set ofpoints of the form (cid:16) α w (1) m , . . . , α w ( r − m (cid:17) ∈ Q r − ≥ where α = ( α , . . . , α r ) ∈ N r issuch that | α | = α + . . . + α r = m and P ri =1 σ i α i > A . Equivalently, is givenby the set of points (cid:16) α w (1) m , . . . , α w ( r − m (cid:17) ∈ Q r − ≥ such that ◦ α w (1) m + . . . + α w ( r − m ≤ ◦ P ri =1 σ i α i m = P ri =1 σ w ( i ) α w ( i ) m = P r − i =1 σ w ( i ) α w ( i ) m + σ w ( r ) (cid:16) − P r − i =1 α w ( i ) m (cid:17) > Am . We conclude therefore that∆ Y w • ( B • ) = ( ( ν , . . . , ν r − ) ∈ ∆ r − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r − X i =1 σ w ( i ) ν i + σ w ( r ) − r − X i =1 ν i ! ≥ ) , from which (2) follows. Remark 5.5 . —
The assumption σ ≥ σ ≥ · · · ≥ σ r in Lemma 5.4 canalways be fulfilled by considering an automorphism of P r − permuting thechosen homogeneous coordinates. Let us note that given q ∈ C , the Harder-Narasimhan filtrationHN • ( E ) : 0 = E ℓ ⊆ E ℓ − ⊆ · · · ⊆ E ⊆ E = E induces a (not necessarily complete) flag on P ( E ) in the following way: let Y = P ( E ) and for every i = 2 , . . . , ℓ the exact sequence0 → Q i → E/E i → E/E i − → P (( E/E i − ) | q ) ֒ → P (( E/E i ) | q ). We obtain therefore thefollowing (possibly partial) flag of linear subvarieties P (( E/E ) | q ) ⊆ P (( E/E ) | q ) ⊆ · · · ⊆ P (( E/E ℓ − ) | q ) ⊆ P ( E | q ) = π − ( q ) ⊆ P ( E )with codim P ( E ) P (( E/E i ) | q ) = rank( E i ) + 1. We also note that this flagis complete if and only if all the semi-stable quotients Q i are line bundles.In general, it will be necessary to choose a complete linear flag on each P (( E i − /E i ) | q ) = P ( Q i | q ) in order to complete the flag above.We shall consider linear flags that are compatible with the Harder-Narasimhan filtration of E in the sense that they complete the previousflag. PEDRO
MONTERO
Definition 5.6 (Compatible linear flag). —
A linear flag Y • on P ( E ) over q ∈ C is said to be compatible with the Harder-Narasimhan filtration of E if Y rank E i +1 = P (( E/E i ) | q ) ∼ = P r − rank E i − ⊆ P ( E )for every i = 1 , . . . , ℓ .We will adopt the following convention. Convention 5.7 . —
Let V • be a fixed linear flag on P ( E ) over q ∈ C andconsider the corresponding Schubert cell decomposition F r = a w ∈ S r Ω w , where F r is the full flag variety parametrizing complete flags of linear subspacesof V = π − ( q ) ∼ = P r − . We say that a linear flag Y • on P ( E ) over q ∈ C belongs to a Schubert cell Ω w if the induced linear flag Y • | Y belongs to Ω w .We can prove now the main reduction step. Namely, compute the Newton-Okounkov body of the big and nef class ξ − µ f with respect to any linearflag. Theorem 5.8 . —
Let C be a smooth projective curve and let E be a vectorbundle over C of rank r ≥ . Fix a linear flag Y HN • on P ( E ) over q ∈ C whichis compatible with the Harder-Narasimhan filtration of E and let F r = a w ∈ S r Ω w be the corresponding Schubert cell decomposition of the full flag variety F r parametrizing linear flags on π − ( q ) ∼ = P r − . Then, for every linear flag Y • on P ( E ) over q ∈ C that belongs to Ω w we have that ∆ Y • ( ξ − µ f ) = n ( ν , . . . , ν r ) ∈ R r ≥ | ≤ ν ≤ µ ℓ − µ , ( ν , . . . , ν r ) ∈ (cid:3) wµ + ν o , where (cid:3) wµ + ν ⊆ R r − is the full dimensional polytope defined in Notation4.7 ( ⋆ ) , with ( µ , . . . , µ ℓ ) and ( r , . . . , r ℓ ) given by the Harder-Narasimhanfiltration of E as in Notation 3.6.Proof . — We follow Notation 3.6. We first note that if E is semi-stable then µ = µ ℓ = µ ( E ) and hence the Theorem follows from Proposition B. Let ussuppose from now on that E is unstable.For the reader’s convenience, the proof is subdivided into several steps. Wefirst observe that it is enough to compute rational slices of ∆ Y • ( ξ − µ f ). Inorder to do so, we consider the restricted algebra A • whose Newton-Okounkovbody computes the desired slice. After performing a suitable Veronese embed-ding A n • ⊆ A • , we define a graded subalgebra B n • ⊆ A n • which turns out to EWTON-OKOUNKOV BODIES ON PROJECTIVE BUNDLES OVER CURVES be a toric graded algebra as in Lemma 5.4. A comparison of volumes leads tothe result. Step 1. Reduction to rational slices.
It follows from Theorem 3.1 and Lemma4.2 that the projection of the Newton-Okounkov body of ξ − µ f onto the firstcoordinate is given by pr (∆ Y • ( ξ − µ f )) = [0 , µ ℓ − µ ] . By continuity of slices of Newton-Okounkov bodies (cf. [KL15, Lemm. 1.7]),it suffices to consider a fixed t ∈ ]0 , µ ℓ − µ [ ∩ Q and show that the slice of∆ Y • ( ξ − µ f ) at t ∈ ]0 , µ ℓ − µ [ ∩ Q is given by∆ Y • ( ξ − µ f ) | ν = t = (cid:3) wµ + t ⊆ R r − for linear flags Y • on P ( E ) that belong to the Schubert cell Ω w (see Convention5.7).Let us fix from now on a permutation w ∈ S r and a linear flag Y • on P ( E ),over q ∈ C , and let us denote by Y • | F the induced flag on F . Suppose that Y • | F belongs to the Schubert cell Ω w with respect to the reference flag Y HN • | F . Step 2. Restricted algebra A • . Consider a Q -divisor D such that [ D ] = ξ − µ f in N ( P ( E )) Q . For every integer m ≥
1, let us define the subspace A m = A m,t = H ( P ( E ) | F, O P ( E ) ( ⌊ m ( D − tF ) ⌋ ))= Im(H ( P ( E ) , O P ( E ) ( ⌊ m ( D − tF ) ⌋ )) rest −−→ H ( F, O F ( m ))) ⊆ H ( F, O F ( m )) . If follows from [LM09, Prop. 4.1, Rem. 4.25] that the restricted algebra A • above computes the desired slice∆ Y • | F ( A • ) = ∆ Y • ( ξ − µ f ) | ν = t , and that for every integer n ≥ Y • ( nD ) | ν = nt = ∆ Y • | F ( n ( D − tF )) = n ∆ Y • | F ( D − tF ) = n ∆ Y • ( D ) | ν = t . Step 3. Veronese embedding A n • ⊆ A • . We will consider A n • = { a nm } m ≥ instead of A • = { a m } m ≥ , for n fixed and divisible enough such that nµ ∈ Z and nt ∈ Z . We also note that for 0 < t < µ ℓ − µ we have that µ max ( S m E ⊗ O C ( − m ( µ + t ) · q )) = m ( µ ℓ − µ − t ) > µ min ( S m E ⊗ O C ( − m ( µ + t ) · q )) = − mt < . Therefore, by considering n above large enough we may also assume that µ max ( S nm E ⊗ O C ( − nm ( µ + t ) · q )) > g − m ≥ PEDRO
MONTERO
Let [ x : · · · : x r ] be homogeneous coordinates on F ∼ = P r − . Since Y • is alinear flag on P ( E ), there is an isomorphism of graded algebras φ : M m ≥ H ( F, O F ( m )) → k [ x , . . . , x r ]such that A m can be regarded as a subspace of k [ x , . . . , x r ] m , the k -vectorspace of homogeneous polynomials of degree m in the variables x , . . . , x r , forall m ≥
0. Via this identification, A • can be seen as a graded subalgebra of k [ x , . . . , x r ]. Moreover, the projection formula implies that we can identify A m withIm (cid:16) H ( C, S m E ⊗O C ( − m ( µ + t ) · q )) rest −−→ H ( C, ( S m E ⊗O C ( − m ( µ + t ) · q )) | q ) (cid:17) . Step 4. Toric graded subalgebra B n • ⊆ A n • . We shall define a graded subalge-bra B n • ⊆ A n • for which we can explicitly compute that∆ Y • | F ( B n • ) = n (cid:3) wµ + t , and we will prove that ∆ Y • | F ( B n • ) = ∆ Y • | F ( A n • ) = n ∆ Y • | F ( A • ).In order to construct B n • let us note that Proposition 3.8 implies that forevery m ≥ F • : 0 = F L ⊆ F L − ⊆ · · · ⊆ F ⊆ F = S nm E ⊗ O C ( − nm ( µ + t ) · q )whose successive quotients have the form F j − /F j ∼ = Q m ( j ) ,µ + t = S m Q ⊗ · · · ⊗ S m ℓ Q ℓ ⊗ O C ( − nm ( µ + t ) · q )for some partition m ( j ) ∈ N ℓ of nm , and µ ( Q m ( j ) ,µ + t ) ≤ µ ( Q m ( j +1) ,µ + t ) forevery j ∈ { , . . . , L } .Let us define J = J ( m ) ∈ { , . . . , L } to be the largest index such that µ ( Q m ( J ) ,µ + t ) ≤ g −
1. We have that for every j ∈ { J, . . . , L − } , the shortexact sequence 0 → F j +1 ⊗ O C ( − q ) → F j +1 → F j +1 | q → → H ( C, F j +1 ⊗ O C ( − q )) → H ( C, F j +1 ) → H ( C, F j +1 | q ) → , since we have that h ( C, F j +1 ⊗ O C ( − q )) = 0, by Lemma 3.7. In particular,we get for every j ∈ { J, . . . , L − } a surjection H ( C, F j +1 ) → H ( C, F j +1 | q ).Therefore, let us consider the subspaces B nm = Im(H ( C, F J +1 ) rest −−→ H ( C, F J +1 | q )) = H ( C, F J +1 | q ) ⊆ A nm . Let us choose homogeneous coordinates [ x : . . . : x r ] on F such that Y HN i +1 = { x = . . . = x i = 0 } ⊆ P ( E ) EWTON-OKOUNKOV BODIES ON PROJECTIVE BUNDLES OVER CURVES for i = 1 , . . . , r −
1. In particular, we have that Y HNrank E i +1 = P (( E/E i ) | q ) = { x = . . . = x rank E i = 0 } ⊆ P ( E )for i = 1 , . . . , ℓ and therefore the degree 1 part of the isomorphism φ , φ : H ( F, O F (1)) ∼ = H ( C, ( E ⊗ O C ( − ( µ + t ) · q )) | q ) → k [ x , . . . , x r ] , is such that for every i = 0 , . . . , ℓ − ( C, ( E i ⊗ O C ( − ( µ + t ) · q )) | q ) ⊆ H ( C, ( E ⊗ O C ( − ( µ + t ) · q )) | q )via φ coincide with the subspace generated by the variables x , . . . , x rank E i .By taking symmetric powers it follows from Proposition 3.8 that for each m ≥ B nm ⊆ k [ x , . . . , x r ] nm , the image of H ( C, F J +1 | q ) ⊆ H ( C, F | q ),corresponds to the subspace of homogeneous polynomials of degree nm gener-ated by polynomials of the form P ( x ) = P ( x ) · · · P ℓ ( x ℓ )where P i is an homogeneous polynomial of degree m i ≥ x i = ( x i, , . . . , x i,r i ), where ( x , . . . , x r ) = ( x ℓ , . . . , x ), and the m i are suchthat m + . . . + m ℓ = nm and µ m + . . . + µ ℓ m ℓ > nm ( µ + t ) + 2 g − . In other words, B nm is the subspace generated by monomials x α = x α · · · x α r r of total degree | α | = nm such that P ri =1 ( σ i − µ − t ) α i > g −
1, where σ = ( µ ℓ , . . . , µ ℓ | {z } r ℓ times , µ ℓ − , . . . , µ ℓ − | {z } r ℓ − times , . . . , µ , . . . , µ | {z } r times ) ∈ Q r . Step 5. Volume comparison and conclusion.
It follows from Lemma 5.4applied (5) to the collection of subspaces { B nm } m ≥ that B n • is a gradedsubalgebra of k [ x , . . . , x r ] whose Newton-Okounkov body, with respect toa linear flag Y • that belongs to the Schubert cell Ω w (see Convention 5.7) isgiven by ∆ Y • | F ( B n • ) = n (cid:3) wµ + t , where (cid:3) wµ + t = n ( ν , . . . , ν r ) ∈ ∆ r − (cid:12)(cid:12)(cid:12) P ri =2 σ w ( i − ν i + σ w ( r ) (1 − P ri =2 ν i ) ≥ µ + t o
5. We note that if C ∼ = P then all the slopes µ i = µ ( Q i ) are integer numbers and hencethe inequality “ > g −
1” becomes “ ≥ PEDRO
MONTERO and σ = ( µ ℓ , . . . , µ ℓ | {z } r ℓ times , µ ℓ − , . . . , µ ℓ − | {z } r ℓ − times , . . . , µ , . . . , µ | {z } r times ) ∈ Q r . Finally, we havethatvol R r − (∆ Y • | F ( B n • )) = vol R r − ( n (cid:3) wµ + t )= vol R r − ( n ∆ Y • ( ξ − µ f ) | ν = t ) by Lemma 4.11= vol R r − (∆ Y • | F ( A n • ))and hence the inclusion ∆ Y • | F ( B n • ) ⊆ ∆ Y • | F ( A n • ) leads to the equality∆ Y • | F ( B n • ) = ∆ Y • | F ( A n • ), as the two convex bodies have equal volume. Fromthis we conclude that∆ Y • ( ξ − µ f ) | ν = t = (cid:3) wµ + t ⊆ R r − . The following result (from which Theorem A is easily deduced) is an imme-diate consequence. We keep the same notation as in Theorem 5.8.
Corollary 5.9 . —
For every linear flag Y • on P ( E ) that belongs to theSchubert cell Ω w and every big rational class η = a ( ξ − µ ℓ f ) + bf we havethat ∆ Y • ( η ) = (cid:26) ( ν , . . . , ν r ) ∈ R r ≥ | ≤ ν ≤ b, ( ν , . . . , ν r ) ∈ a (cid:3) wµ ℓ − a ( b − ν ) (cid:27) , and hence the global Newton-Okounkov body of P ( E ) with respect to Y • is givenby ∆ Y • ( P ( E )) = n (( a ( ξ − µ ℓ f ) + bf ) , ( ν , . . . , ν r )) ∈ N ( P ( E )) R × R r such that ≤ ν ≤ b and ( ν , . . . , ν r ) ∈ a (cid:3) wµ ℓ − a ( b − ν ) o . In particular, the global Newton-Okounkov body ∆ Y • ( P ( E )) is a rational poly-hedral cone and it depends only on gr(HN • ( E )) , the graded vector bundle as-sociated to the Harder-Narasimhan filtration of E .Proof . — We note that if η = a ( ξ − µ ℓ f ) + bf is a big rational class on P ( E )and the induced linear flag Y • belongs to the Schubert cell Ω w with respect toa reference flag Y HN • , then Theorem 5.8 gives∆ Y • ( ξ − µ f ) = n ( ν , . . . , ν r ) ∈ R r ≥ | ≤ ν ≤ µ ℓ − µ , ( ν , . . . , ν r ) ∈ (cid:3) wµ + ν o and hence a ∆ Y • ( ξ − µ f ) = (cid:26) ( ν , . . . , ν r ) ∈ R r ≥ | ≤ ν ≤ a ( µ ℓ − µ ) , ( ν , . . . , ν r ) ∈ a (cid:3) wµ + a ν (cid:27) . EWTON-OKOUNKOV BODIES ON PROJECTIVE BUNDLES OVER CURVES We compute that for t ∗ = b − a ( µ ℓ − µ ) we have a ∆ Y • ( ξ − µ f )+ t ∗ ~e = (cid:26) ( ν , . . . , ν r ) ∈ R r ≥ | ≤ ν ≤ b, ( ν , . . . , ν r ) ∈ a (cid:3) wµ + a ( ν − t ∗ ) (cid:27) with µ + a ( ν − t ∗ ) = µ ℓ − a ( b − ν ). The result follows from Lemma 5.3. References [Băd01] L. Bădescu.
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