Newton-Okounkov bodies on projective bundles over curves

TL;DR

This paper computes Newton-Okounkov bodies for projective bundles over curves, using Wolfe's estimates, and characterizes semi-stable vector bundles through these geometric objects.

Contribution

It provides explicit computations of Newton-Okounkov bodies on these varieties and links them to the stability of vector bundles, offering new geometric insights.

Findings

Explicit Newton-Okounkov bodies for projective bundles over curves

Characterization of semi-stability via Newton-Okounkov bodies

Application of Wolfe's estimates to volume computations

Abstract

In this article, we study Newton-Okounkov bodies on projective vector bundles over curves. Inspired by Wolfe's estimates used to compute the volume function on these varieties, we compute all Newton-Okounkov bodies with respect to linear flags. Moreover, we characterize semi-stable vector bundles over curves via Newton-Okounkov bodies.

Equations

$${\mathrm{HN}}_{\bullet }(E):0=E_{\ell }\subseteq E_{\ell -1}\subseteq \cdots \subseteq E_1\subseteq E_0=E$$

$${\mathrm{vol}}_{\mathbb{P}(E)}(\xi -tf)=r!\cdot {\int }_{{\hat{\Delta }}_{r-1}}\max \left\{\sum _{j=1}^r{\sigma }_{r+1-j}{\lambda }_j-t,0\right\}d\lambda ,$$

$${\square }_t=\left\{({\nu }_2,\dots ,{\nu }_r)\in {\Delta }_{r-1}|\sum _{i=2}^r{\sigma }_{i-1}{\nu }_i+{\sigma }_r\left(1-\sum _{i=2}^r{\nu }_i\right)\ge t\right\}.$$

$${\Delta }_{Y_{\bullet }}(\xi -tf)=\left\{({\nu }_1,\dots ,{\nu }_r)\in {\mathbb{R}}_{\ge 0}^r|0\le {\nu }_1\le {\mu }_{\ell }-t,({\nu }_2,\dots ,{\nu }_r)\in {\square }_{t+{\nu }_1}\right\}.$$

$$\mathbf{B}(D)=\bigcap _{m>0}\mathrm{Bs}(mD)$$

$${\mathbf{B}}_{+}(D)=\bigcap _A\mathbf{B}(D-A),$$

$${\mathbf{B}}_{-}(D)=\bigcup _A\mathbf{B}(D+A),$$

$${\mathbb{F}}_r=\coprod _{w\in {\mathfrak{S}}_r}{\Omega }_w.$$

$$Y_i=\{x_1=\dots =x_i=0\}\subseteq {\mathbb{P}}^{r-1}$$

$$Y_i^w=\{x_{w(1)}=\dots =x_{w(i)}=0\}\subseteq {\mathbb{P}}^{r-1}.$$

$$Y_{\bullet }:X=Y_0\supseteq Y_1\supseteq Y_2\supseteq \cdots \supseteq Y_{n-1}\supseteq Y_n=\{p\}$$

$${\nu }_{Y_{\bullet }}:H^0(X,{\mathcal{O}}_X(D))\setminus \{0\}\to {\mathbb{Z}}^n,s\mapsto ({\nu }_1(s),\dots ,{\nu }_n(s)).$$

$${\Gamma }_{Y_{\bullet }}(D)=\left\{({\nu }_{Y_{\bullet }}(s),m)\in {\mathbb{N}}^n\times \mathbb{N}|0\ne s\in H^0(X,{\mathcal{O}}_X(mD)\right\}.$$

$${\Delta }_{Y_{\bullet }}(D)=\mathrm{cone}({\Gamma }_{Y_{\bullet }}(D))\cap ({\mathbb{R}}^n\times \{1\}),$$

$${\mathrm{vol}}_{{\mathbb{R}}^n}({\Delta }_{Y_{\bullet }}(D))=\frac{1}{n!}\cdot {\mathrm{vol}}_X(D),$$

$${\Delta }_{Y_{\bullet }}(X)\subseteq {\mathbb{R}}^n\times N^1(X{)}_{\mathbb{R}}$$

$$A_m=H^0(X|F,{\mathcal{O}}_X(mD))=\text{Im}\left(H^0(X,{\mathcal{O}}_X(mD))\stackrel{\mathrm{rest}}{\to }{H}^0(F,{\mathcal{O}}_F(mD))\right)$$

$${\Delta }_{X|F}(D)\subseteq {\mathbb{R}}^d,$$

$${\mathrm{vol}}_{{\mathbb{R}}^d}({\Delta }_{X|F}(D))=\frac{1}{d!}{\mathrm{vol}}_{X|F}(D),$$

$${\mathrm{vol}}_{X|F}(D)=\underset{m\to \infty }{\lim }\frac{{\dim }_k{A}_m}{m^d/d!}$$

$$Y_{\bullet }:X=Y_0\supseteq Y_1\supseteq Y_2\supseteq \cdots \supseteq Y_{n-1}\supseteq Y_n=\{p\}$$

$${\Delta }_{Y_{\bullet }}(\eta {)}_{{\nu }_1=t}$$

$${\Delta }_{Y_{\bullet }}(\eta {)}_{{\nu }_1\ge t}$$

$${\tau }_F(\eta )=\sup \{s>0|\eta -s\cdot f\in \mathrm{Big}(X)\},$$

$$\frac{d}{dt}\left({\mathrm{vol}}_X(\eta +tf)\right){|}_{t=0}=n\cdot {\mathrm{vol}}_{X|F}(\eta ).$$

$${\Delta }_{Y_{\bullet }}(X,D{)}_{{\nu }_1=t}={\Delta }_{Y_{\bullet }|Y_1}(Y_1,(D-t{Y}_1){|}_{Y_1}),$$

$$\{(t,{\nu }_2,\dots ,{\nu }_n)\in {\mathbb{R}}^n|0\le t<{\tau }_{Y_1}(D),({\nu }_2,\dots ,{\nu }_n)\in {\Delta }_{Y_{\bullet }|Y_1}(Y_1,(D-t{Y}_1){|}_{Y_1})\}.$$

$$D=P(D)+N(D)$$

$${\Delta }_{Y_{\bullet }}(D)=\left\{(t,y)\in {\mathbb{R}}^2|\nu \le t\le {\tau }_C(D),\alpha (t)\le t\le \beta (t)\right\}$$

$$\mu (E)=\frac{d}{r}$$

$${\mathrm{HN}}_{\bullet }(E):0=E_{\ell }\subseteq E_{\ell -1}\subseteq \cdots \subseteq E_1\subseteq E_0=E$$

Full text

1. 1 Introduction

2. 2 Preliminaries

3. 2.1 Numerical classes and positivity

4. 2.2 Flag varieties and Schubert cells

5. 3 Newton-Okounkov bodies and Semi-stability

6. 3.1 Newton-Okounkov bodies

7. 3.2 Semi-stability and Harder-Narasimhan filtrations

8. 4 Divisors on projective bundles over curves

9. 5 Newton-Okounkov bodies on projective bundles over curves

10. 5.1 Ruled surfaces

11. 5.2 Some reductions

12. 5.3 A toric computation

13. 5.4 Proofs of main results

1. Introduction

Let $C$ be a smooth projective curve over an algebraically closed field of characteristic zero and let $E$ be vector bundle over $C$ of rank $r\geq 2$. It is well-known since Hartshorne’s work [12] that numerical information coming from semi-stability properties of $E$ can be translated into positivity conditions. Namely, the shape of the nef and pseudo-effective cone of the projective vector bundle $\mathbb{P}(E)$ were determined by Hartshorne [12] (cf. [17, §6.4.B]) and Nakayama [24, Chapter IV] (cf. [22]) in terms of the Harder-Narasimhan filtration of $E$. More precisely, if we denote by $\xi$ the class of the tautological line bundle $\mathcal{O}_{\mathbb{P}(E)}(1)$ and by $f$ the class of a fiber of $\pi:\mathbb{P}(E)\to C$ then we have that for $t\in\mathbb{R}$ the class $\xi-tf$ is nef (resp. pseudo-effective) if and only if $t\leq\mu_{\min}(E)$ (resp. $t\leq\mu_{\max}(E)$), where $\mu_{\min}(E)$ (resp. $\mu_{\max}(E)$) is the minimal (resp. maximal) slope of $E$. In particular, the nef and pseudo-effective cone of $\mathbb{P}(E)$ coincide if and only if $E$ is semi-stable (cf. [23, Theo. 3.1] and [11]).

Indirectly, the pseudo-effective cone can be also deduced from the work of Wolfe [28] and Chen [7], who explicitly computed the volume function on $\mathbb{P}(E)$. In fact, they showed that for every $t\in\mathbb{R}$ the volume of the class $\xi-tf$ on $\mathbb{P}(E)$ can be expressed in terms of numerical information coming from the Harder-Narasimhan filtration of $E$,

[TABLE]

with successive semi-stable quotients $Q_{i}=E_{i-1}/E_{i}$ of rank $r_{i}$ and slope $\mu_{i}$. More precisely, if we consider $\sigma_{1}\geq\ldots\geq\sigma_{r}$ to be the ordered slopes of $E$ counted with multiplicities equal to the rank of the corresponding semi-stable quotient111In other words, $\mu_{\max}(E)=\sigma_{1}\geq\ldots\geq\sigma_{r}=\mu_{\min}(E)$ can be viewed as the components of the vector $\boldsymbol{\sigma}=(\sigma_{1},\ldots,\sigma_{r})=(\underbrace{\mu_{\ell},\ldots,\mu_{\ell}}_{r_{\ell}\text{ times}},\underbrace{\mu_{\ell-1},\ldots,\mu_{\ell-1}}_{r_{\ell-1}\text{ times}},\ldots,\underbrace{\mu_{1},\ldots,\mu_{1}}_{r_{1}\text{ times}})\in\mathbb{Q}^{r}$., their results can be summarized as follows.

\theoname** \the\smf@thm.**

Let $C$ and $E$ as above. Then,

[TABLE]

where ${\widehat{\Delta}}_{r-1}\subseteq\mathbb{R}^{r}$ is the standard $(r-1)$-simplex with coordinates $\lambda_{1},\ldots,\lambda_{r}$ and $d\lambda$ is the standard induced Lebesgue measure for which $\widehat{\Delta}_{r-1}$ has volume $\frac{1}{(r-1)!}$.

Following the idea that numerical information encoded by the Harder-Narasimhan filtration of $E$ should be related to asymptotic numerical invariants of $\mathbb{P}(E)$, we study the geometry of Newton-Okounkov bodies on $\mathbb{P}(E)$. These compact convex bodies were introduced by Okounkov in his original article [25] and they were studied later on by Kaveh and Khovanskii [14] and Lazarsfeld and Mustaţă [19], who associated to any big divisor $D$ on a normal projective variety $X$ of dimension $r$, and any complete flag of subvarieties $Y_{\bullet}$ on $X$ satisfying suitable conditions, a convex body $\Delta_{Y_{\bullet}}(D)\subseteq\mathbb{R}^{r}$ depending only on the numerical equivalence class of $D$. Moreover, there exists a global Newton-Okounkov body $\Delta_{Y_{\bullet}}(X)\subseteq\mathbb{R}^{r}\times\operatorname{N}^{1}(X)_{\mathbb{R}}$ such that the slice of $\Delta_{Y_{\bullet}}(X)$ over any big rational class $\eta\in\operatorname{N}^{1}(X)_{\mathbb{Q}}$ is given by $\Delta_{Y_{\bullet}}(\eta)\subseteq\mathbb{R}^{r}\times\{\eta\}$.

Newton-Okounkov bodies of big divisors on geometrically ruled surfaces with respect to linear flags (see Definition 5.1) can be computed via Zariski decomposition (see Example 5.2). In higher dimension, we will use methods similar to those used by Wolfe to compute the volume function in [28]. More precisely, we will first reduce ourselves to the case of the nef and big class $\xi-\mu_{\min}(E)f$ (see Lemma 5.3). Afterwards, we will need to understand the Harder-Narasimhan filtration of the symmetric products $S^{m}E$ for $m\geq 1$ and then to consider suitable refinements of these filtrations.

Let $Y_{\bullet}$ be a complete linear flag which is compatible with the Harder-Narasimhan filtration of $E$ (see Definition 5.6). With the notation of Theorem 1 above, define for each real number $t\in\mathbb{R}$ the following polytope inside the full dimensional standard simplex $\Delta_{r-1}$ in $\mathbb{R}^{r-1}$ (see Notation 4.7)

[TABLE]

Then, we prove the following result (see Corollary 5.9).

Theorem A**.**

Let $C$ be a smooth projective curve and let $E$ be a vector bundle over $C$ of rank $r\geq 2$. Then, for every real number $t<\mu_{\max}(E)=\mu_{\ell}$ we have that

[TABLE]

In particular, the global Newton-Okounkov body $\Delta_{Y_{\bullet}}(\mathbb{P}(E))$ is a rational polyhedral cone and it depends only on $\operatorname{gr}(\operatorname{HN}_{\bullet}(E))$, the graded vector bundle associated to the Harder-Narasimhan filtration of $E$.

Moreover, we obtain the following characterization of semi-stability in terms of Newton-Okounkov bodies.

Proposition B**.**

Let $C$ be a smooth projective curve and let $E$ be a vector bundle over $C$ of rank $r\geq 2$. The following conditions are then equivalent:

• (1)

$E$* is semi-stable.*

• (2)

For every big rational class $\eta=a(\xi-\mu_{\ell}f)+bf$ on $\mathbb{P}(E)$ and every linear flag $Y_{\bullet}$ on $\mathbb{P}(E)$ we have that $\Delta_{Y_{\bullet}}(\eta)=[0,b]\times a\Delta_{r-1}\subseteq\mathbb{R}^{r}$.

Here, $a\Delta_{r-1}=\{(\nu_{2},\ldots,\nu_{r})\in\mathbb{R}_{\geq 0}^{r-1}\;|\;\sum_{i=2}^{r}\nu_{i}\leq a\}$ is the full dimensional standard $(r-1)$-simplex with side length $a$.

Outline of the article

First of all, we establish some notation and recall some basic facts in $\S 2$. Secondly, we recall in $\S 3$ definitions and well-known results about Newton-Okounkov bodies and semi-stability of vector bundles over curves. Next, $\S 4$ is devoted to the different cones of divisors on $\mathbb{P}(E)$ as well as results concerning their volume and restricted volume. Finally, we prove both Theorem A and Proposition B in $\S 5$.

Acknowledgements

I would like to express my gratitude to my thesis supervisors, Stéphane Druel and Catriona Maclean, for their advice, helpful discussions and encouragement throughout the preparation of this article. I also thank Bruno Laurent, Laurent Manivel and Bonala Narasimha Chary for fruitful discussions. Finally, I would like to thank the anonymous referee for a very helpful and detailed report.

2. Preliminaries

Throughout this article all varieties will be assumed to be reduced and irreducible schemes of finite type over a fixed algebraically closed field of characteristic zero222In positive characteristic one should take into account iterates of the absolute Frobenius as in [4, 2]. It is worth mentioning that in positive characteristic the semistability of a vector bundle over a curve does not imply the semi-stability of its symmetric powers. $k$.

2.1. Numerical classes and positivity

We denote by $\operatorname{N}^{1}(X)$ the group of numerical equivalence classes of Cartier divisors on $X$, and we define $\operatorname{N}^{1}(X)_{\Bbbk}=\operatorname{N}^{1}(X)\otimes_{\mathbb{Z}}\Bbbk$ for $\Bbbk=\mathbb{Q}$ or $\mathbb{R}$. All the $\mathbb{R}$-divisors that we consider are $\mathbb{R}$-Cartier. Dually, we denote by $\operatorname{N}_{1}(X)$ the group of numerical equivalence classes of 1-cycles on $X$. Inside $\operatorname{N}_{1}(X)_{\mathbb{R}}=\operatorname{N}_{1}(X)\otimes_{\mathbb{Z}}\mathbb{R}$ we distinguish the Mori cone $\overline{\operatorname{NE}}(X)\subseteq\operatorname{N}_{1}(X)_{\mathbb{R}}$, which is the closed and convex cone generated by numerical classes of 1-cycles with non-negative real coefficients.

Let $E$ be a locally free sheaf on a variety $X$. We follow Grothendieck’s convention and we define the projectivization $\mathbb{P}_{X}(E)=\mathbb{P}(E)$ of $E$ to be ${\mathbf{Proj}_{\mathcal{O}_{X}}\oplus_{m\geq 0}S^{m}E}$, where $S^{m}E$ denotes de $m$th symmetric power of $E$. This variety is endowed with a natural projection $\pi:\mathbb{P}(E)\to X$ and a tautological line bundle $\mathcal{O}_{\mathbb{P}(E)}(1)$.

Let $X$ be a normal projective variety. Following [17], we say that a numerical class $\eta\in\operatorname{N}^{1}(X)_{\mathbb{R}}$ is big if there exists an effective $\mathbb{R}$-divisor $E$ such that $\eta-E$ is ample. We denote by $\operatorname{Big}(X)\subseteq\operatorname{N}^{1}(X)_{\mathbb{R}}$ the open convex cone of big numerical classes. A numerical class $\eta\in\operatorname{N}^{1}(X)_{\mathbb{R}}$ is called pseudo-effective if it can be written as the limit of classes of effective $\mathbb{R}$-divisors. The pseudo-effective cone is the closure of the big cone: $\overline{\operatorname{Big}}(X)=\operatorname{Psef}(X)$ (see [17, Theo. 2.2.26], for instance). Moreover, a numerical class $\eta\in\operatorname{N}^{1}(X)_{\mathbb{R}}$ is nef if $\eta\cdot[C]\geq 0$ for every $[C]\in\overline{\operatorname{NE}}(X)$, and is ample if it is the numerical class of an $\mathbb{R}$-divisor that can be written as a finite sum of ample Cartier divisors with positive real coefficients. The cone $\operatorname{Nef}(X)\subseteq\operatorname{N}^{1}(X)_{\mathbb{R}}$ of nef classes is closed convex, and its interior $\operatorname{Amp}(X)$ is the cone of ample classes, by Kleiman’s ampleness criterion. A line bundle $L$ is big (resp. pseudo-effective, ample, nef) if and only if its numerical class $c_{1}(L)\in\operatorname{N}^{1}(X)_{\mathbb{R}}$ is big (resp. pseudo-effective, ample, nef).

Let us recall that the stable base locus of a $\mathbb{Q}$-divisor $D$ on $X$ is the closed set

[TABLE]

where $\operatorname{Bs}(mD)$ is the base locus of the complete linear series $|mD|$. Following [8], we define the augmented base locus of $D$ to be the Zariski closed set

[TABLE]

where the intersection runs over all ample $\mathbb{Q}$-divisors $A$. Similarly, the restricted base locus of $D$ is defined by

[TABLE]

where the union runs over all ample $\mathbb{Q}$-divisors $A$. The restricted base locus $\mathbf{B}_{-}(D)$ consist of at most a countable union of subvarieties whose Zariski closure is contained in $\mathbf{B}_{+}(D)$ (see [8, Rema. 1.13] and [18]). By [8, Prop. 1.4, Exam. 1.8, Prop. 1.15, Exam. 1.16], both $\mathbf{B}_{-}(D)$ and $\mathbf{B}_{+}(D)$ depend only on the numerical class of $D$, there is an inclusion $\mathbf{B}_{-}(D)\subseteq\mathbf{B}_{+}(D)$, and for any rational number $c>0$ we have ${\mathbf{B}_{-}(cD)=\mathbf{B}_{-}(D)}$ and ${\mathbf{B}_{+}(cD)=\mathbf{B}_{+}(D)}$. Moreover by [8, Exam. 1.7, Exam. 1.18] we have that ${\mathbf{B}_{+}(D)=\emptyset}$ if and only if $D$ is ample, and that $\mathbf{B}_{-}(D)=\emptyset$ if and only if $D$ is nef.

2.2. Flag varieties and Schubert cells

Let us denote by $\mathbb{F}_{r}$ the full flag variety parametrizing all complete linear flags on $\mathbb{P}^{r-1}$. Recall that if we fix a reference complete linear flag $Y_{\bullet}$ on $\mathbb{P}^{r-1}$, then there is a decomposition of $\mathbb{F}_{r}$ into Schubert cells

[TABLE]

More explicitly, if we consider homogeneous coordinates $[x_{1}:\ldots:x_{r}]$ on $\mathbb{P}^{r-1}$, we assume that for every $i=1,\ldots,r-1$ we have

[TABLE]

and we regard the permutation group $\mathfrak{S}_{r}$ as a subgroup of $\operatorname{PGL}_{r}(k)$ via its natural action on the standard basis points $e_{1},\ldots,e_{r}\in\mathbb{P}^{r-1}$ then we have that $\Omega_{w}$ is the orbit $B\cdot Y_{\bullet}^{w}$, where $B$ denotes the (Borel) subgroup of $\operatorname{PGL}_{r}(k)$ that fixes the flag $Y_{\bullet}$ and $Y_{\bullet}^{w}$ is the complete linear flag such that for every $i=1,\ldots,r-1$ we have

[TABLE]

We refer the reader to [5, $\S 1.2$] and [21, $\S 3.6$] for further details.

3. Newton-Okounkov bodies and Semi-stability

In this section, we review the construction of Newton-Okounkov bodies and semi-stability of vector bundles over smooth projective curves.

3.1. Newton-Okounkov bodies

Let $X$ be a smooth projective variety of dimension $n$ and let $L$ be a big line bundle on $X$. A full flag of closed subvarieties of $X$ centered at the point $p\in X$

[TABLE]

is an admissible flag if $\operatorname{codim}_{X}(Y_{i})=i$, and each $Y_{i}$ is smooth at the point $p$. In particular, $Y_{i+1}$ defines a Cartier divisor on $Y_{i}$ in a neighborhood of the point $p$. Following the work of Okounkov [25, 26], Kaveh and Khovanskii [14] and Lazarsfeld and Mustaţă [19] independently associated to $L$ and $Y_{\bullet}$ a convex body $\Delta_{Y_{\bullet}}(X,L)\subseteq\mathbb{R}^{n}$ encoding the asymptotic properties of the complete linear series $|L^{\otimes m}|$. We will follow the presentation of [19] and we refer the interested reader to the survey [3] for a comparison of both points of view.

Let $D$ be any divisor on $X$ and let $s=s_{1}\in H^{0}(X,\mathcal{O}_{X}(D))$ be a non-zero section. We shall compute successive vanishing orders of global sections in the following manner: let $D_{1}=D+\operatorname{div}(s_{1})$ be the effective divisor in the linear series $|D|$ defined by $s_{1}$ and set $\nu_{1}(s)=\operatorname{ord}_{Y_{1}}(D_{1})$ the coefficient of $Y_{1}$ in $D_{1}$. Then $D_{1}-\nu_{1}(s)Y_{1}$ is an effective divisor in the linear series $|D-\nu_{1}(s)Y_{1}|$, and does not contain $Y_{1}$ in its support, so we can define $D_{2}=(D_{1}-\nu_{1}(s)Y_{1})|_{Y_{1}}$ and set $\nu_{2}(s)=\operatorname{ord}_{Y_{2}}(D_{2})$. We proceed inductively in order to get $\nu_{Y_{\bullet}}(s)=(\nu_{1}(s),\ldots,\nu_{n}(s))\in\mathbb{N}^{d}$. This construction leads to a valuation-like function

[TABLE]

We then define the graded semigroup of $D$ to be the sub-semigroup of $\mathbb{N}^{n}\times\mathbb{N}$ defined by

[TABLE]

Finally, we define the Newton-Okounkov body of $D$ with respect to $Y_{\bullet}$ to be

[TABLE]

where $\operatorname{cone}(\Gamma_{Y_{\bullet}}(D))$ denotes the closed convex cone in $\mathbb{R}^{n}\times\mathbb{R}$ spanned by $\Gamma_{Y_{\bullet}}(D)$.

These convex sets $\Delta_{Y_{\bullet}}(D)$ are compact and they have non-empty interior whenever $D$ is big. Moreover, by [19, Theo. A], we have the following identity

[TABLE]

where $\operatorname{vol}_{X}(D)=\lim_{m\to\infty}\frac{h^{0}(X,\mathcal{O}_{X}(mD))}{m^{n}/n!}$. In particular, if $D$ is big and nef, then $\operatorname{vol}_{\mathbb{R}^{n}}(\Delta_{Y_{\bullet}}(D))=\frac{1}{n!}D^{n}$, by the Asymptotic Riemann-Roch theorem.

The Newton-Okounkov bodies of big divisors depend only on numerical classes: if $D\equiv_{\text{num}}D^{\prime}$ are big divisors then $\Delta_{Y_{\bullet}}(D)=\Delta_{Y_{\bullet}}(D^{\prime})$ for every admissible flag $Y_{\bullet}$ on $X$, by [19, Prop. 4.1] (see [13, Theo. A] for the converse). This fact, along with the identity $\Delta_{Y_{\bullet}}(pD)=p\cdot\Delta_{Y_{\bullet}}(D)$ for every positive integer $p$, enables us to define an Newton-Okounkov body $\Delta_{Y_{\bullet}}(\eta)\subseteq\mathbb{R}^{n}$ for every big rational class $\eta\in\operatorname{Big}(X)\cap\operatorname{N}^{1}(X)_{\mathbb{Q}}$. Moreover, by [19, Theo. B], there exists a global Newton-Okounkov body: a closed convex cone

[TABLE]

such that for each big rational class $\eta\in\operatorname{Big}(X)_{\mathbb{Q}}=\operatorname{Big}(X)\cap\operatorname{N}^{1}(X)_{\mathbb{Q}}$ the fiber of the second projection over $\eta$ is $\Delta_{Y_{\bullet}}(\eta)$. This enables us to define Newton-Okounkov bodies for big real classes by continuity.

The above construction works for graded linear series $A_{\bullet}$ associated to a big divisor $D$ on $X$. A graded linear series is a collection of subspaces $A_{m}\subseteq\operatorname{H}^{0}(X,\mathcal{O}_{X}(mD))$ such that $A_{\bullet}=\oplus_{m\geq 0}A_{m}$ is a graded subalgebra of the section ring ${R(D)=\oplus_{m\geq 0}\operatorname{H}^{0}(X,\mathcal{O}_{X}(mD))}$. The construction enables us to attach to any graded linear series $A_{\bullet}$ a closed and convex set $\Delta_{Y_{\bullet}}(A_{\bullet})\subseteq\mathbb{R}^{n}$. This set $\Delta_{Y_{\bullet}}(A_{\bullet})$ will be compact and will compute the volume of the linear series under some mild conditions listed in [19, $\S$ 2.3]. We will be specially interested on restricted complete linear series of a big divisor $D$, namely graded linear series of the form

[TABLE]

where $F\subseteq X$ is an irreducible subvariety of dimension $d\geq 1$. Under the hypothesis that $F\not\subseteq\mathbf{B}_{+}(D)$, the conditions listed in [19, $\S$ 2.3] are satisfied by [19, Lemm. 2.16]. Therefore, the Newton-Okounkov body associated to $A_{\bullet}$ above, the restricted Newton-Okounkov body (with respect to a fixed admissible flag)

[TABLE]

is compact and

[TABLE]

where

[TABLE]

is the restricted volume on $F$ of the divisor $D$. In particular, if $D$ is big and nef, then ${\operatorname{vol}_{X|F}(D)=(D^{d}\cdot F)}$, by [9, Cor. 2.17]. Restricted Newton-Okounkov bodies depend only on numerical classes (see [19, Rema. 4.25]), so it is meaningful to consider $\Delta_{X|F}(\eta)$ for every big rational class $\eta$ such that $F\not\subseteq\mathbf{B}_{+}(\eta)$.

As before, there exists a global Newton-Okounkov body $\Delta_{Y_{\bullet}}(X|F)$ that enables us to define, by continuity, $\Delta_{X|F}(\eta)$ for any big real numerical class $\eta$ such that $F\not\subseteq\mathbf{B}_{+}(\eta)$. See [19, Exam. 4.24] for details.

Restricted Newton-Okounkov bodies can be used to describe slices of Newton-Okounkov bodies.

Theorem 3.1** ([19, Theo. 4.26, Cor. 4.27]).**

Let $X$ be a normal projective variety of dimension $n$, and let $F\subseteq X$ be an irreducible and reduced Cartier divisor on $X$. Fix an admissible flag

[TABLE]

with divisorial component $Y_{1}=F$. Let $\eta\in\operatorname{Big}(X)_{\mathbb{Q}}$ be a rational big class, and consider the Newton-Okounkov body $\Delta_{Y_{\bullet}}(\eta)\subseteq\mathbb{R}^{n}$. Write $\operatorname{pr}_{1}:\Delta_{Y_{\bullet}}(\eta)\to\mathbb{R}$ for the projection onto the first coordinate, and set

[TABLE]

Assume that $F\not\subseteq\mathbf{B}_{+}(\eta)$ and let

[TABLE]

where $f\in\operatorname{N}^{1}(X)$ is the numerical class of $F$. Then, for any $t\in\mathbb{R}$ with $0\leq t<\tau_{F}(\eta)$ we have

1. (1)

$\Delta_{Y_{\bullet}}(\eta)_{\nu_{1}\geq t}=\Delta_{Y_{\bullet}}(\eta-tf)+t\cdot\vec{e}_{1}$, where $\vec{e}_{1}=(1,0,\ldots,0)\in\mathbb{N}^{n}$ is the first standard unit vector. 333In fact, this statement remains true even if we do not assume that $E\not\subseteq\mathbf{B}_{+}(\eta)$. See [15, Prop. 1.6]. 2. (2)

$\Delta_{Y_{\bullet}}(\eta)_{\nu_{1}=t}=\Delta_{X|F}(\eta-tf)$. 3. (3)

The function $t\mapsto\operatorname{vol}_{X}(\eta+tf)$ is differentiable at $t=0$, and

[TABLE]

We will need the following observation by Küronya, Lozovanu and Maclean.

Proposition 3.2** ([16, Prop. 3.1]).**

Let $X$ be a normal projective variety together with an admissible flag $Y_{\bullet}$. Suppose that $D$ is a big $\mathbb{Q}$-divisor such that $Y_{1}\not\subseteq\mathbf{B}_{+}(D)$ and that $D-tY_{1}$ is ample for some $0\leq t<\tau_{Y_{1}}(D)$, where $\tau_{Y_{1}}(D)=\sup\{s>0\;|\;D-sY_{1}\text{ is big}\}$. Then

[TABLE]

where $Y_{\bullet}|Y_{1}:Y_{1}\supseteq Y_{2}\supseteq\cdots\supseteq Y_{n}$ is the induced admissible flag on $Y_{1}$.

In particular, if $\operatorname{Psef}(X)=\operatorname{Nef}(X)$ then we have that the Newton-Okounkov body $\Delta_{Y_{\bullet}}(D)$ is the closure in $\mathbb{R}^{n}$ of the following set

[TABLE]

Let us finish this section with the case of Newton-Okounkov bodies on surfaces (see [19, §6.2] for details). We will use this description in Example 5.2 in order to illustrate the shape of Newton-Okounkov bodies on ruled surfaces.

Example 3.3** (Surfaces).**

Let $S$ be a smooth projective surface together with a flag $Y_{\bullet}:Y_{0}=S\supseteq Y_{1}=C\supseteq Y_{2}=\{p\}$, where $C\subseteq S$ is a smooth curve and $p\in C$.

Let $D$ be a big $\mathbb{Q}$-divisor on $S$. Any such divisor admits a Zariski decomposition, that is we can uniquely write $D$ as a sum

[TABLE]

of $\mathbb{Q}$-divisors, with $P(D)$ nef and $N(D)$ either zero or effective with negative definite intersection matrix. Moreover, $P(D)\cdot\Gamma=0$ for every irreducible component $\Gamma$ of $N(D)$ and for all $m\geq 0$ there is an isomorphism $\operatorname{H}^{0}(S,\mathcal{O}_{S}(\lfloor mP(D)\rfloor))\cong\operatorname{H}^{0}(S,\mathcal{O}_{S}(\lfloor mD\rfloor))$. In this decomposition $P(D)$ is called the positive part and $N(D)$ the negative part. See [17, §2.3.E] and references therein for proofs and applications.

With the above notation, we have that

[TABLE]

where

1. (1)

$\nu\in\mathbb{Q}$ the coefficient of $C$ in $N(D)$, 2. (2)

$\tau_{C}(D)=\sup\{t>0\;|\;D-tC\text{ is big}\}$, 3. (3)

$\alpha(t)=\operatorname{ord}_{p}(N_{t}\cdot C)$, 4. (4)

$\beta(t)=\operatorname{ord}_{p}(N_{t}\cdot C)+P_{t}\cdot C$,

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 [16, Theo. B].

3.2. Semi-stability and Harder-Narasimhan filtrations

Throughout this section, $C$ is a smooth projective curve and $E$ is a locally free sheaf on $C$ of rank $r>0$ and degree $d=\deg(E)=\deg(c_{1}(E))$. Given such a bundle we call the rational number

[TABLE]

the slope of $E$.

Definition 3.4** (Semi-stability).**

Let $E$ be a vector bundle on $C$ of slope $\mu$. We say that $E$ is semi-stable if for every non-zero sub-bundle $S\subseteq E$, we have $\mu(S)\leq\mu$. Equivalently, $E$ is semi-stable if for every locally-free quotient $E\twoheadrightarrow Q$ of non-zero rank, we have $\mu\leq\mu(Q)$. A non semi-stable vector bundle will be called unstable.

Following [20, Prop. 5.4.2], there is a canonical filtration of $E$ with semi-stable quotients.

Proposition 3.5**.**

Let $E$ be a vector bundle on $C$. Then $E$ has an increasing filtration by sub-bundles

[TABLE]

where each of the quotients $E_{i-1}/E_{i}$ satisfies the following conditions:

1. (1)

Each quotient $E_{i-1}/E_{i}$ is a semi-stable vector bundle; 2. (2)

$\mu(E_{i-1}/E_{i})<\mu(E_{i}/E_{i+1})$* for $i=1,\ldots,\ell-1$.*

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 $Q_{i}=E_{i-1}/E_{i}$ the semi-stable quotients of the Harder-Narasimhan filtration of $E$, each one of rank $r_{i}=\operatorname{rank}(Q_{i})$, degree $d_{i}=\deg(c_{1}(Q_{i}))$ and slope $\mu_{i}=\mu(Q_{i})=d_{i}/r_{i}$. With this notation, $\mu_{1}$ and $\mu_{\ell}$ correspond to the minimal and maximal slopes, $\mu_{\min}(E)$ and $\mu_{\max}(E)$, respectively.

From a cohomological point of view, semi-stable vector bundles can be seen as the good higher-rank analogue of line bundles. For instance, we have the following classical properties (see [27] or [6, Lemm. 1.12, Lemm. 2.5]).

Lemma 3.7**.**

Let $E$ and $F$ be vector bundles on $C$ and $m\in\mathbb{N}$. Then,

1. (1)

$\mu_{\max}(E\otimes F)=\mu_{\max}(E)+\mu_{\max}(F)$. 2. (2)

$\mu_{\min}(E\otimes F)=\mu_{\min}(E)+\mu_{\min}(F)$. 3. (3)

$\mu_{\max}(S^{m}E)=m\mu_{\max}(E)$. 4. (4)

$\mu_{\min}(S^{m}E)=m\mu_{\min}(E)$. 5. (5)

If $\mu_{\max}(E)<0$, then $\dim_{k}\operatorname{H}^{0}(C,E)=0$. 6. (6)

If $\mu_{\min}(E)>2g-2$, then $\dim_{k}\operatorname{H}^{1}(C,E)=0$.

In particular, if $E$ and $F$ are semi-stable then $S^{m}E$ and $E\otimes F$ are semi-stable.

If $E_{1},\ldots,E_{\ell}$ are vector bundles on $C$ and $m_{1},\ldots,m_{\ell}$ be non-negative integers. By the splitting principle [10, Rema. 3.2.3] we can prove the following formula:

[TABLE]

Moreover, we have that for every $m\geq 1$ the Harder-Narasimhan filtration of the symmetric product $S^{m}E$ can be computed in terms of the one for $E$ (see [7, Prop. 3.4] and [28, Prop. 5.10], for instance).

Proposition 3.8**.**

Let $E$ be a vector bundle on $C$ with Harder-Narasimhan filtration

[TABLE]

and semi-stable quotients $Q_{i}=E_{i-1}/E_{i}$ with slopes $\mu_{i}=\mu(Q_{i})$, for $i=1,\ldots,\ell$. For every positive integer $m\geq 1$, let us consider the vector bundle $S^{m}E$ with Harder-Narasimhan filtration

[TABLE]

and semi-stable quotients $W_{j-1}/W_{j}$ with slopes $\nu_{j}=\mu(W_{j-1}/W_{j})$, for ${j=1,\ldots,M}$. Then, for every $j=1,\ldots,M$ we have that

[TABLE]

and

[TABLE]

where the sums are taken over all partitions $\mathbf{m}=(m_{1},\ldots,m_{\ell})\in\mathbb{N}^{\ell}$ of $m$, and $S^{m_{1}}E_{0}\otimes\cdots\otimes S^{m_{\ell}}E_{\ell-1}$ denotes the image of the composite

[TABLE]

In particular, there is a refinement $F_{\bullet}$ of $\operatorname{HN}_{\bullet}(S^{m}E)$ of length $L=L(m)$ and whose respective successive quotients are of the form

[TABLE]

for some partition $\mathbf{m}(i)=(m_{1},\ldots,m_{\ell})\in\mathbb{N}^{\ell}$ of $m$, and such that for every $i\in\{1,\ldots L\}$ we have $\mu(Q_{\mathbf{m}(i)})\leq\mu(Q_{\mathbf{m}(i+1)})$. Moreover, given any partition $\mathbf{m}\in\mathbb{N}^{\ell}$ of $m$, there is one and only one $i\in\{1,\ldots,L\}$ such that $\mathbf{m}(i)=\mathbf{m}$.

4. Divisors on projective bundles over curves

Let $E$ be a vector bundle on a smooth projective curve $C$, of rank $r\geq 2$ and degree $d$. In this section we study divisors on the projective bundle $\pi:\mathbb{P}(E)\to C$ of one-dimensional quotients. Let us recall that in this case the Néron-Severi group of $\mathbb{P}(E)$ is of the form

[TABLE]

where $f$ is the numerical class of a fiber of $\pi$ and $\xi=\xi_{E}$ is the numerical class of a divisor representing the tautological line bundle $\mathcal{O}_{\mathbb{P}(E)}(1)$. Moreover, if $[\text{pt}]$ denotes the class of a point in the ring $\operatorname{N}^{*}(\mathbb{P}(E))$ then we have the following relations:

[TABLE]

The cone of nef divisors can be described via Hartshorne’s characterization of ample vector bundles over curves [12, Theo. 2.4] (cf. [11, Lemm. 2.1]).

Lemma 4.1**.**

$\operatorname{Nef}(\mathbb{P}(E))=\langle\xi-\mu_{\min}f,f\rangle.$**

The cone of pseudo-effective divisors was obtained by Nakayama in [24, Cor. IV.3.8], and it was indirectly computed by Wolfe [28] and Chen [7] who independently obtained the volume function $\operatorname{vol}_{\mathbb{P}(E)}$ on $\operatorname{N}^{1}(\mathbb{P}(E))_{\mathbb{R}}$. A more general result on the cone of effective cycles of arbitrary codimension can be found in [11, Theo. 1.1].

Lemma 4.2**.**

$\operatorname{Psef}(\mathbb{P}(E))=\langle\xi-\mu_{\max}f,f\rangle.$**

In particular, we recover a result of Miyaoka [23, Theo. 3.1] on semi-stable vector bundles over curves that was generalized by Fulger in [11, Prop. 1.5].

Corollary 4.3**.**

A vector bundle $E$ on a smooth projective curve $C$ is semi-stable if and only if $\operatorname{Nef}(\mathbb{P}(E))=\operatorname{Psef}(\mathbb{P}(E))$.

We finish this section by recalling Wolfe’s computation of the volume function on $\operatorname{N}^{1}(\mathbb{P}(E))$. See also [7, Theo. 1.2].

Notation 4.4**.**

Let $d\geq 1$ be an integer. We define the standard $d$-simplex $\widehat{\Delta}_{d}$ with $d+1$ vertices in $\mathbb{R}^{d+1}$ to be

[TABLE]

By projecting $\widehat{\Delta}_{d}$ onto the hyperplane $x_{1}=0$, we can identify $\widehat{\Delta}_{d}$ with the full dimensional standard $d$-simplex (or just $d$-simplex) in $\mathbb{R}^{d}$ given by

[TABLE]

Via the previous identification, we will denote by $\lambda$ the Lebesgue measure on $\widehat{\Delta}_{d}$ induced by the standard Lebesgue measure on $\Delta_{d}\subseteq\mathbb{R}^{d}$. In particular, we will have $\lambda(\widehat{\Delta}_{d})=\frac{1}{d!}$.

Given a positive real number $a>0$, we define the $d$-simplex with side length $a$ by a\Delta_{d}=\big{\{}(x_{2},\ldots,x_{d+1})\in\mathbb{R}^{d}\;\big{|}\;\sum_{i=2}^{d+1}x_{i}\leq a\text{ and }x_{i}\geq 0\text{ for all }i\big{\}}. Similar for $a\widehat{\Delta}_{d}\subseteq\mathbb{R}^{d+1}$, the standard $d$-simplex with side length $a$.

Theorem 4.5** ([28, Theo. 5.14]).**

Let $E$ be a vector bundle with Harder-Narasimhan filtration of length $\ell$ and semi-stable quotients $Q_{i}$ of ranks $r_{i}$ and slopes $\mu_{i}$. Then, for any $t\in\mathbb{R}$

[TABLE]

where $\widehat{\Delta}_{\ell-1}\subseteq\mathbb{R}^{\ell}$ is the standard $(\ell-1)$-simplex with coordinates $\beta_{1},\ldots,\beta_{\ell}$, and $\beta$ be the standard induced Lebesgue measure.

Remark 4.6**.**

Alternatively, Chen computed in [7, Theo. 1.2] a similar volume formula, but slightly simplified by integrating in $\mathbb{R}^{r}$ instead of $\mathbb{R}^{\ell}$ (cf. [7, Prop. 3.5]). More precisely, with the same notation as above

[TABLE]

where $\widehat{\Delta}_{r-1}\subseteq\mathbb{R}^{r}$ is the standard $(r-1)$-simplex with coordinates $\lambda_{1},\ldots,\lambda_{r}$, $d\lambda$ is the standard induced Lebesgue measure444Unlike Chen, we do not normalize the measure in order to have $\lambda(\Delta_{r-1})=1$., and ${\mathbf{s}=(s_{1},\ldots,s_{r})}$ is a vector in $\mathbb{R}^{r}$ such that the value $\mu_{i}$ appears exactly $r_{i}$ times in the coordinates of $\mathbf{s}$ as in Notation 4.7 below.

Notation 4.7**.**

Fix $\ell\geq 1$ and $r\geq 1$ two integers, $(r_{1},\ldots,r_{\ell})\in\mathbb{N}^{\ell}$ a partition of $r$ and $t\in\mathbb{R}$. We define for $(\mu_{1},\ldots,\mu_{\ell})\in\mathbb{Q}^{\ell}$ the following polytopes:

[TABLE]

and

[TABLE]

where

[TABLE]

Similarly, for every permutation $w\in\mathfrak{S}_{r}$ we define

[TABLE]

where

[TABLE]

By abuse of notation, we will also denote by $\square_{t}$ and $\square_{t}^{w}$ the full dimensional polytopes in $\mathbb{R}^{r-1}$ obtained via the projection of $\widehat{\Delta}_{r-1}\subseteq\mathbb{R}^{r}$ onto $\Delta_{r-1}\subseteq\mathbb{R}^{r-1}$. Explicitly, if $(\nu_{2},\ldots,\nu_{r})$ are coordinates in $\mathbb{R}^{r-1}$ then

[TABLE]

and

[TABLE]

Corollary 4.8**.**

For any admissible flag $Y_{\bullet}$ on $\mathbb{P}(E)$ and any big rational class $\xi-tf$ on $\mathbb{P}(E)$ we have that

[TABLE]

for $\widehat{\square}_{t}\subseteq\mathbb{R}^{\ell}$ as in Notation 4.7.

Remark 4.9**.**

Let $F=\pi^{-1}(q)\cong\mathbb{P}^{r-1}$ be any fiber of $\pi:\mathbb{P}(E)\to C$. Then, for any big $\mathbb{R}$-divisor $D$ on $\mathbb{P}(E)$ we have that $F\not\subseteq\mathbf{B}_{+}(D)$. In fact, if $D\sim_{\mathbb{R}}A+E$, with $A$ ample $\mathbb{R}$-divisor and $E$ effective $\mathbb{R}$-divisor, and if $F\subseteq\operatorname{Supp}(E)$, then we can write $E=aF+E^{\prime}$ with $a>0$ and $F\not\subseteq\operatorname{Supp}(E^{\prime})$, which implies that $F\not\subseteq\mathbf{B}_{+}(D)$ since $A+aF$ is ample.

As a direct consequence, we compute the restricted volume function on a fiber $F=\pi^{-1}(q)\cong\mathbb{P}^{r-1}$.

Corollary 4.10**.**

Let $F$ be a fiber of $\pi:\mathbb{P}(E)\to C$ and let $\xi-tf$ be a big rational class. Then,

[TABLE]

for $\widehat{\square}_{t}\subseteq\mathbb{R}^{\ell}$ as in Notation 4.7. In particular, if $0\leq\tau\leq\mu_{\ell}-t$ then we have that

[TABLE]

where $Y_{\bullet}$ is any admissible flag on $\mathbb{P}(E)$ with divisorial component $Y_{1}=F$. In particular, these volumes depend only on $\operatorname{gr}(\operatorname{HN}_{\bullet}(E))$, the graded vector bundle associated to the Harder-Narasimhan filtration of $E$.

Démonstration.

We consider

[TABLE]

Since $F\not\subseteq\mathbf{B}_{+}(\xi-tf)$ by Remark 4.9, Theorem 3.1 and differentiation under the integral sign give

[TABLE]

∎

Lemma 4.11**.**

Following Notation 4.7 ($\star$ ‣ 4.7), we have that

[TABLE]

and

[TABLE]

Moreover, $\operatorname{vol}_{\mathbb{R}^{r-1}}(\square_{t+\tau})=\operatorname{vol}_{\mathbb{R}^{r-1}}(\square_{t+\tau}^{w})$ for every $w\in\mathfrak{S}_{r}$.

Démonstration.

The first two equalities follow from Remark 4.6. For the last assertion consider $\square_{t}\subseteq\mathbb{R}^{r}$ and $\square_{t}^{w}\subseteq\mathbb{R}^{r}$ as in Notation 4.7. Then, there is a linear transformation $T:\mathbb{R}^{r}\to\mathbb{R}^{r}$, whose associated matrix in the canonical basis of $\mathbb{R}^{r}$ is given by a permutation matrix, such that $T(\square_{t})=\square_{t}^{w}$ and $|\det(T)|=1$. ∎

5. Newton-Okounkov bodies on projective bundles over curves

Let $E$ be a vector bundle on a smooth projective curve $C$, of rank $r\geq 2$ and degree $d$. In this section we study the geometry of Newton-Okounkov bodies of rational big classes in $\operatorname{N}^{1}(\mathbb{P}(E))$ in terms of the numerical information of the Harder-Narasimhan filtration of $E$. In particular, Theorem A will be a consequence of Lemma 5.3, Theorem 5.8 and Corollary 5.9. We follow Notation 3.6.

Definition 5.1** (Linear flag).**

A complete flag of subvarieties $Y_{\bullet}$ on the projective vector bundle $\mathbb{P}(E)\xrightarrow{\pi}C$ is called a linear flag centered at $p\in\mathbb{P}(E)$, over the point $q\in C$ (or simply a linear flag) if $Y_{0}=\mathbb{P}(E)$, $Y_{1}=\pi^{-1}(q)\cong\mathbb{P}^{r-1}$ and $Y_{i}\cong\mathbb{P}^{r-i}$ is a linear subspace of $Y_{i-1}$ for $i=2,\ldots,r$, with $Y_{r}=\{p\}$.

5.1. Ruled surfaces

Let us begin with the following example that illustrates the general case. Namely, that the shape of Newton-Okounkov bodies on $\mathbb{P}(E)$ will depend on the semi-stability of $E$.

Example 5.2**.**

Suppose that $\operatorname{rank}(E)=2$ and let $\eta=a(\xi-\mu_{\ell}f)+bf\in\operatorname{N}^{1}(\mathbb{P}(E))_{\mathbb{Q}}$ be a big class. In other words, $a,b\in\mathbb{Q}_{>0}$. Let

[TABLE]

be the linear flag centered at $p\in\mathbb{P}(E)$, over $q\in C$. The Newton-Okounkov body of $\eta$ can be computed by applying [19, Theo. 6.4] (see Example 3.3).

Semi-stable case. If $E$ is semi-stable then Corollary 4.3 implies that every big class is ample. In particular, for every $\eta\in\operatorname{Big}(X)_{\mathbb{Q}}$ and every $\mathbb{Q}$-divisor $D_{\eta}$ with numerical class $\eta$, we have that $N(D_{\eta})=0$ and $P(D_{\eta})=D_{\eta}$. It follows that, with the notation as in Example 3.3, $\nu=0$, $\tau_{F}(\eta)=b$, $\alpha(t)=0$ for every $t\in[0,b]$ and $\beta(t)=a$ for every $t\in[0,b]$. The Newton-Okounkov bodies are given by rectangles in this case.

Unstable case. If $E$ is unstable we consider its Harder-Narasimhan filtration

[TABLE]

and we note that in this case $E_{1}=Q_{\ell}$. The quotient $E\to Q_{1}\to 0$ corresponds to a section $s:C\to\mathbb{P}(E)$ with $[s(C)]=\xi-\mu_{\ell}f$ in $\operatorname{N}^{1}(\mathbb{P}(E))$. The curve $s(C)$ is the only irreducible curve on $\mathbb{P}(E)$ with negative self-intersection. On the other hand, if $D_{\eta}$ is any $\mathbb{Q}$-divisor with numerical class $\eta$ then, either $\eta$ is inside the nef cone of $\mathbb{P}(E)$ and thus $P(D_{\eta})=D_{\eta}$ and $N(D_{\eta})=0$, or $\eta$ is big and not nef in which case we compute that

[TABLE]

in $\operatorname{N}^{1}(\mathbb{P}(E))_{\mathbb{Q}}$. We notice that $N(D_{\eta})=\left(\frac{a(\mu_{\ell}-\mu_{1})-b}{\mu_{\ell}-\mu_{1}}\right)\cdot s(C)$ as $\mathbb{Q}$-divisor, by minimality of the negative part and by the fact that the negative part is unique in its numerical equivalence class. See [1, Lemm. 14.10, Cor. 14.13] for details.

Let $t^{*}=b-a(\mu_{\ell}-\mu_{1})$. We have that $\eta$ is big and nef if and only if $t^{*}\geq 0$, in which case the class $\eta-tf$ is big and nef for $0\leq t\leq t^{*}$. For $t^{*}\leq t\leq b$, the same computation above enables us to find the Zariski decomposition of $D_{\eta}-tF$, which is big and not nef.

We notice that, with the notation as in Example 3.3, $\nu=0$. On the other hand, the functions $\alpha(t)$ and $\beta(t)$ will depend on whether or not we have $\{p\}=s(C)\cap F$. A straightforward computation shows that the Newton-Okounkov bodies are given by the following finite polygons in $\mathbb{R}^{2}$.

We notice that Figure 3 provides examples of big and not nef divisors classes $\eta$ such that the origin $\vec{0}\in\Delta_{Y_{\bullet}}(\eta)$ for almost every linear flag $Y_{\bullet}$ except for one. This shows in particular that condition (2) in the characterization of nefness given in [15, Cor. 2.2] has to be checked for all linear flags.

5.2. Some reductions

We first observe that Proposition B in $\S 1$ states that the shape of Newton-Okounkov bodies on projective semi-stable vector bundles will be similar as in Example 5.2 above. Moreover, this provides a characterization of semi-stability in terms of Newton-Okounkov bodies.

Proof of Proposition B.

$(1)\Rightarrow(2)$. Let $\eta=a(\xi-\mu_{\ell}f)+bf$ be a big rational class on $\mathbb{P}(E)$ and $Y_{\bullet}:X=Y_{0}\supseteq Y_{1}\supseteq Y_{2}\supseteq\cdots\supseteq Y_{r-1}\supseteq Y_{r}=\{p\}$ be a linear flag centered at $p\in\mathbb{P}(E)$, over the point $q\in C$. Since $E$ is semi-stable we have that $\operatorname{Big}(\mathbb{P}(E))=\operatorname{Amp}(\mathbb{P}(E))$ by Corollary 4.3. Equivalently, we have that $\mathbf{B}_{+}(\eta)=\emptyset$ for every big rational class $\eta$.

We notice that $\tau_{F}(\eta)=\sup\{s>0\;|\;\eta-sf\in\operatorname{Big}(\mathbb{P}(E))\}=b$. The implication follows from Proposition 3.2 and [3, Cor. 4.11], by noting that if $D_{\eta}$ is a $\mathbb{Q}$-divisor with numerical class $\eta$ then

[TABLE]

where $H\subseteq F\cong\mathbb{P}^{r-1}$ is an hyperplane section.

$(2)\Rightarrow(1)$. We notice that if for all linear flags $Y_{\bullet}$ the Newton-Okounkov body of $\eta=a(\xi-\mu_{\ell}f)+bf$ is given by $\Delta_{Y_{\bullet}}(\eta)=[0,b]\times a\Delta_{r-1}\subseteq\mathbb{R}^{r}$ then $\eta$ is a big and nef class, by [15, Cor. 2.2]. We can therefore compute the volume of $\eta$ via the top self-intersection $\operatorname{vol}_{\mathbb{P}(E)}(\eta)=\eta^{r}=ra^{r-1}\left(b-a(\mu_{\ell}-\mu(E))\right)$. On the other hand, we have that

[TABLE]

and hence [19, Theo. A] leads to $\mu_{\ell}=\mu(E)$, implying the result. ∎

We will first reduce our problem to computing the Newton-Okounkov body of the big and nef divisor class $\xi-\mu_{1}f$.

Lemma 5.3**.**

Let $E$ be a unstable vector bundle on a smooth projective curve $C$, with $\operatorname{rank}(E)=r\geq 2$. Then for every big class $\eta=a(\xi-\mu_{\ell}f)+bf$ and every linear flag $Y_{\bullet}$ we have that

• (1)

$\Delta_{Y_{\bullet}}(\eta)=\left([0,t^{*}]\times a\Delta_{r-1}\right)\cup\left(a\Delta_{Y_{\bullet}}(\xi-\mu_{1}f)+t^{*}\vec{e}_{1}\right)$* if $\eta$ is big and nef;*

• (2)

$\Delta_{Y_{\bullet}}(\eta)=a\Delta_{Y_{\bullet}}(\xi-\mu_{1}f)_{\nu_{1}\geq-t^{*}}+t^{*}\vec{e}_{1}$* if $\eta$ is big and not nef.*

Here $t^{*}=b-a(\mu_{\ell}-\mu_{1})$ and $a\Delta_{r-1}\subseteq\mathbb{R}^{r-1}$ is the $(r-1)$-simplex with side length $a$.

Démonstration.

If $\eta$ is an ample rational class then ${\Delta_{Y_{\bullet}}(\eta)_{\nu_{1}=t}=a\Delta_{r-1}}$ for $0\leq t\leq t^{*}$, by Proposition 3.2 and [19, Exam. 1.1]. On the other hand, we have that ${\Delta_{Y_{\bullet}}(\eta)_{\nu_{1}\geq t^{*}}=a\Delta_{Y_{\bullet}}(\xi-\mu_{1}f)+t^{*}\vec{e}_{1}}$ by Theorem 3.1.

If $\eta$ is big and not nef then $t^{*}<0$. Theorem 3.1 implies therefore that ${a\Delta_{Y_{\bullet}}(\xi-\mu_{1}f)_{\nu_{1}\geq-t^{*}}=\Delta_{Y_{\bullet}}(\eta)-t^{*}\vec{e}_{1}}$, which leads to (2). ∎

5.3. A toric computation

We will need the following result concerning the Newton-Okounkov bodies of some toric graded algebras.

Lemma 5.4**.**

Fix $A\in\mathbb{R}$, an integer $r\geq 2$, $\boldsymbol{\sigma}=(\sigma_{1},\ldots,\sigma_{r})\in\mathbb{R}^{r}$ and homogeneous coordinates $[x_{1}:\ldots:x_{r}]$ on $\mathbb{P}^{r-1}$. Put $B_{0}=k$ and consider for every integer $m\geq 1$ the vector subspace $B_{m}\subseteq\operatorname{H}^{0}(\mathbb{P}^{r-1},\mathcal{O}_{\mathbb{P}^{r-1}}(m))$ generated by monomials ${x^{\alpha}=x_{1}^{\alpha_{1}}\cdots x_{r}^{\alpha_{r}}}$ of total degree $|\alpha|=m$ such that $\sum_{i=1}^{r}\sigma_{i}\alpha_{i}>A$. Suppose that $A\geq 0$ and $\sigma_{1}\geq\sigma_{2}\geq\cdots\geq\sigma_{r}$. Then

1. (1)

$B_{\bullet}=\oplus_{m\geq 0}B_{m}$* is a graded subalgebra of the coordinate ring $k[x_{1},\ldots,x_{r}]$.* 2. (2)

Let us denote by $V_{\bullet}$ the flag of linear subspaces

[TABLE]

where $V_{i}=\{x_{1}=\ldots=x_{i}=0\}$ for $i=1,\ldots,r-1$ and consider the Schubert cell decomposition of the full flag variety parametrizing complete linear flag on $\mathbb{P}^{r-1}$

[TABLE]

with respect to $V_{\bullet}$. Fix a permutation $w\in\mathfrak{S}_{r}$ and let $Y_{\bullet}\in\Omega_{w}$. Then $\Delta_{Y_{\bullet}}(B_{\bullet})$ is given by the projection of the polytope

[TABLE]

onto the hyperplane $\{\nu_{r}=0\}$.

Démonstration.

Let us suppose that $x^{\alpha}\in B_{m}$ and $x^{\alpha^{\prime}}\in B_{m^{\prime}}$, then $x^{\alpha+\alpha^{\prime}}$ belongs to $b_{m+m^{\prime}}$ since we have $\sum_{i=1}^{r}\sigma_{i}(\alpha_{i}+\alpha_{i}^{\prime})>2A\geq A$ as long as $A\geq 0$. This proves (1).

In order to prove (2) we will first show that every automorphism $\varphi$ of $\mathbb{P}^{r-1}$ that fixes the flag $V_{\bullet}$ induces an automorphism of the graded algebra $B_{\bullet}$. For this, we remark that automorphisms $\varphi$ of $\mathbb{P}^{r-1}$ fixing the flag $V_{\bullet}$ correspond to lower triangular matrices in $\operatorname{PGL}_{r}(k)$ and hence, given $\varphi=(\varphi_{i,j})_{1\leq i,j\leq r}$ such a matrix, the image of the monomial $x^{\alpha}\in B_{m}$ via the induced action on $k[x_{1},\ldots,x_{r}]$ is given by

[TABLE]

The above product can be written as a linear combination of monomials of the form $x^{\alpha^{\prime}}=x^{k_{1}+\ldots+k_{r}}$ where $k_{i}=(k_{i,1},\ldots,k_{i,i},0,\ldots,0)\in\mathbb{N}^{r}$ is such that $|k_{i}|=\alpha_{i}$. Let us prove that all these monomials belong to $B_{m}$ as well. In fact, we have that $\alpha_{i}^{\prime}=\sum_{j=i}^{r}k_{j,i}$ and hence

[TABLE]

It follows that $\varphi$ induces an automorphism of the graded algebra $B_{\bullet}$.

In order to compute Newton-Okounkov bodies with respect to linear flags on a given Schubert cell, we note that [5, Prop. 1.2.1] implies that given a permutation $w\in\mathfrak{S}_{r}$ and a linear flag $Y_{\bullet}\in\Omega_{w}$ there exists an automorphism $\varphi$ of $\mathbb{P}^{r-1}$ that fixes the reference flag $V_{\bullet}$ and such that the image of the flag $Y_{\bullet}^{w}$ via the induced action of $\varphi$ on $\mathbb{F}_{r}$ is $Y_{\bullet}$, where $Y_{\bullet}^{w}$ is the linear flag such that for every $i=1,\ldots,r-1$ we have

[TABLE]

It follows from the previous paragraph that $\varphi$ induces an automorphism of the graded algebra $B_{\bullet}$ and then for every $m\geq 1$ and every $P\in B_{m}$ we have that

[TABLE]

In particular, we have $\{\nu_{Y_{\bullet}}(P)\}_{P\in B_{m}}=\{\nu_{Y_{\bullet}^{w}}(P)\}_{P\in B_{m}}\subseteq\mathbb{N}^{r-1}$ and consequently ${\Delta_{Y_{\bullet}}(B_{\bullet})=\Delta_{Y_{\bullet}^{w}}(B_{\bullet})}$. Therefore, we can suppose that $Y_{\bullet}=Y_{\bullet}^{w}\in\Omega_{w}$ in order to prove (2).

Since $B_{m}$ is generated by monomials we have that

[TABLE]

We compute for every monomial $x^{\alpha}=x_{1}^{\alpha_{1}}\cdots x_{r}^{\alpha_{r}}\in B_{m}$ that

[TABLE]

and hence for every $m\geq 1$ the set $\left\{\left.\frac{\nu_{Y_{\bullet}^{w}}(x^{\alpha})}{m}\;\right|\;x^{\alpha}\in B_{m}\right\}$ is given by the set of points of the form $\left(\frac{\alpha_{w(1)}}{m},\ldots,\frac{\alpha_{w(r-1)}}{m}\right)\in\mathbb{Q}^{r-1}_{\geq 0}$ where $\alpha=(\alpha_{1},\ldots,\alpha_{r})\in\mathbb{N}^{r}$ is such that $|\alpha|=\alpha_{1}+\ldots+\alpha_{r}=m$ and $\sum_{i=1}^{r}\sigma_{i}\alpha_{i}>A$. Equivalently, is given by the set of points $\left(\frac{\alpha_{w(1)}}{m},\ldots,\frac{\alpha_{w(r-1)}}{m}\right)\in\mathbb{Q}^{r-1}_{\geq 0}$ such that

• $\circ$

$\frac{\alpha_{w(1)}}{m}+\ldots+\frac{\alpha_{w(r-1)}}{m}\leq 1$ and

• $\circ$

{\textstyle\begin{array}[]{ll}\sum_{i=1}^{r}\sigma_{i}\frac{\alpha_{i}}{m}&=\sum_{i=1}^{r}\sigma_{w(i)}\frac{\alpha_{w(i)}}{m}\\ &=\sum_{i=1}^{r-1}\sigma_{w(i)}\frac{\alpha_{w(i)}}{m}+\sigma_{w(r)}\left(1-\sum_{i=1}^{r-1}\frac{\alpha_{w(i)}}{m}\right)>\frac{A}{m}.\end{array}}

We conclude therefore that

[TABLE]

from which (2) follows. ∎

Remark 5.5**.**

The assumption $\sigma_{1}\geq\sigma_{2}\geq\cdots\geq\sigma_{r}$ in Lemma 5.4 can always be fulfilled by considering an automorphism of $\mathbb{P}^{r-1}$ permuting the chosen homogeneous coordinates.

5.4. Proofs of main results

Let us note that given $q\in C$, the Harder-Narasimhan filtration

[TABLE]

induces a (not necessarily complete) flag on $\mathbb{P}(E)$ in the following way: let $Y_{0}=\mathbb{P}(E)$ and for every $i=2,\ldots,\ell$ the exact sequence

[TABLE]

induces an inclusion $\mathbb{P}((E/E_{i-1})|_{q})\hookrightarrow\mathbb{P}((E/E_{i})|_{q})$. We obtain therefore the following (possibly partial) flag of linear subvarieties

[TABLE]

with $\operatorname{codim}_{\mathbb{P}(E)}\mathbb{P}((E/E_{i})|_{q})=\operatorname{rank}(E_{i})+1$. We also note that this flag is 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 $\mathbb{P}((E_{i-1}/E_{i})|_{q})=\mathbb{P}(Q_{i}|_{q})$ 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 previous flag.

Definition 5.6** (Compatible linear flag).**

A linear flag $Y_{\bullet}$ on $\mathbb{P}(E)$ over $q\in C$ is said to be compatible with the Harder-Narasimhan filtration of $E$ if

[TABLE]

for every $i=1,\ldots,\ell$.

We will adopt the following convention.

Convention 5.7**.**

Let $V_{\bullet}$ be a fixed linear flag on $\mathbb{P}(E)$ over $q\in C$ and consider the corresponding Schubert cell decomposition

[TABLE]

where $\mathbb{F}_{r}$ is the full flag variety parametrizing complete flags of linear subspaces of $V_{1}=\pi^{-1}(q)\cong\mathbb{P}^{r-1}$. We say that a linear flag $Y_{\bullet}$ on $\mathbb{P}(E)$ over $q\in C$ belongs to a Schubert cell $\Omega_{w}$ if the induced linear flag $Y_{\bullet}|Y_{1}$ belongs to $\Omega_{w}$.

We can prove now the main reduction step. Namely, compute the Newton-Okounkov body of the big and nef class $\xi-\mu_{1}f$ with respect to any linear flag.

Theorem 5.8**.**

Let $C$ be a smooth projective curve and let $E$ be a vector bundle over $C$ of rank $r\geq 2$. Fix a linear flag $Y_{\bullet}^{\operatorname{HN}}$ on $\mathbb{P}(E)$ over $q\in C$ which is compatible with the Harder-Narasimhan filtration of $E$ and let

[TABLE]

be the corresponding Schubert cell decomposition of the full flag variety $\mathbb{F}_{r}$ parametrizing linear flags on $\pi^{-1}(q)\cong\mathbb{P}^{r-1}$. Then, for every linear flag $Y_{\bullet}$ on $\mathbb{P}(E)$ over $q\in C$ that belongs to $\Omega_{w}$ we have that

[TABLE]

where $\square_{\mu_{1}+\nu_{1}}^{w}\subseteq\mathbb{R}^{r-1}$ is the full dimensional polytope defined in Notation 4.7 ($\star$ ‣ 4.7), with $(\mu_{1},\ldots,\mu_{\ell})$ and $(r_{1},\ldots,r_{\ell})$ given by the Harder-Narasimhan filtration of $E$ as in Notation 3.6.

Démonstration.

We follow Notation 3.6. We first note that if $E$ is semi-stable then $\mu_{1}=\mu_{\ell}=\mu(E)$ and hence the Theorem follows from Proposition B. Let us suppose from now on that $E$ is unstable.

For the reader’s convenience, the proof is subdivided into several steps. We first observe that it is enough to compute rational slices of $\Delta_{Y_{\bullet}}(\xi-\mu_{1}f)$. In order to do so, we consider the restricted algebra $A_{\bullet}$ whose Newton-Okounkov body computes the desired slice. After performing a suitable Veronese embedding $A_{n\bullet}\subseteq A_{\bullet}$, we define a graded subalgebra $B_{n\bullet}\subseteq A_{n\bullet}$ which turns out to be a toric graded algebra as in Lemma 5.4. A comparison of volumes leads to the result.

Step 1. Reduction to rational slices. It follows from Theorem 3.1 and Lemma 4.2 that the projection of the Newton-Okounkov body of $\xi-\mu_{1}f$ onto the first coordinate is given by

[TABLE]

By continuity of slices of Newton-Okounkov bodies (cf. [15, Lemm. 1.7]), it suffices to consider a fixed $t\in]0,\mu_{\ell}-\mu_{1}[\cap\mathbb{Q}$ and show that the slice of $\Delta_{Y_{\bullet}}(\xi-\mu_{1}f)$ at $t\in]0,\mu_{\ell}-\mu_{1}[\cap\mathbb{Q}$ is given by

[TABLE]

for linear flags $Y_{\bullet}$ on $\mathbb{P}(E)$ that belong to the Schubert cell $\Omega_{w}$ (see Convention 5.7).

Let us fix from now on a permutation $w\in\mathfrak{S}_{r}$ and a linear flag $Y_{\bullet}$ on $\mathbb{P}(E)$, over $q\in C$, and let us denote by $Y_{\bullet}|F$ the induced flag on $F$. Suppose that $Y_{\bullet}|F$ belongs to the Schubert cell $\Omega_{w}$ with respect to the reference flag $Y_{\bullet}^{\operatorname{HN}}|F$.

Step 2. Restricted algebra $A_{\bullet}$. Consider a $\mathbb{Q}$-divisor $D$ such that $[D]=\xi-\mu_{1}f$ in $\operatorname{N}^{1}(\mathbb{P}(E))_{\mathbb{Q}}$. For every integer $m\geq 1$, let us define the subspace

[TABLE]

If follows from [19, Prop. 4.1, Rem. 4.25] that the restricted algebra $A_{\bullet}$ above computes the desired slice

[TABLE]

and that for every integer $n\geq 1$ we have

[TABLE]

Step 3. Veronese embedding $A_{n\bullet}\subseteq A_{\bullet}$. We will consider $A_{n\bullet}=\{a_{nm}\}_{m\geq 0}$ instead of $A_{\bullet}=\{a_{m}\}_{m\geq 0}$, for $n$ fixed and divisible enough such that $n\mu_{1}\in\mathbb{Z}$ and $nt\in\mathbb{Z}$. We also note that for $0<t<\mu_{\ell}-\mu_{1}$ we have that

[TABLE]

and

[TABLE]

Therefore, by considering $n$ above large enough we may also assume that ${\mu_{\max}(S^{nm}E\otimes\mathcal{O}_{C}(-nm(\mu_{1}+t)\cdot q))>2g-1}$ for every $m\geq 1$.

Let $[x_{1}:\cdots:x_{r}]$ be homogeneous coordinates on $F\cong\mathbb{P}^{r-1}$. Since $Y_{\bullet}$ is a linear flag on $\mathbb{P}(E)$, there is an isomorphism of graded algebras

[TABLE]

such that $A_{m}$ can be regarded as a subspace of $k[x_{1},\ldots,x_{r}]_{m}$, the $k$-vector space of homogeneous polynomials of degree $m$ in the variables $x_{1},\ldots,x_{r}$, for all $m\geq 0$. Via this identification, $A_{\bullet}$ can be seen as a graded subalgebra of $k[x_{1},\ldots,x_{r}]$. Moreover, the projection formula implies that we can identify $A_{m}$ with

[TABLE]

Step 4. Toric graded subalgebra $B_{n\bullet}\subseteq A_{n\bullet}$. We shall define a graded subalgebra $B_{n\bullet}\subseteq A_{n\bullet}$ for which we can explicitly compute that

[TABLE]

and we will prove that $\Delta_{Y_{\bullet}|F}(B_{n\bullet})=\Delta_{Y_{\bullet}|F}(A_{n\bullet})=n\Delta_{Y_{\bullet}|F}(A_{\bullet})$.

In order to construct $B_{n\bullet}$ let us note that Proposition 3.8 implies that for every $m\geq 1$ there is a filtration

[TABLE]

whose successive quotients have the form

[TABLE]

for some partition $\mathbf{m}(j)\in\mathbb{N}^{\ell}$ of $nm$, and $\mu(Q_{\mathbf{m}(j),\mu_{1}+t})\leq\mu(Q_{\mathbf{m}(j+1),\mu_{1}+t})$ for every $j\in\{1,\ldots,L\}$.

Let us define $J=J(m)\in\{1,\ldots,L\}$ to be the largest index such that ${\mu(Q_{\mathbf{m}(J),\mu_{1}+t})\leq 2g-1}$. We have that for every $j\in\{J,\ldots,L-1\}$, the short exact sequence

[TABLE]

gives an exact sequence in cohomology

[TABLE]

since we have that $h^{1}(C,F_{j+1}\otimes\mathcal{O}_{C}(-q))=0$, by Lemma 3.7. In particular, we get for every $j\in\{J,\ldots,L-1\}$ a surjection $\operatorname{H}^{0}(C,F_{j+1})\to\operatorname{H}^{0}(C,F_{j+1}|_{q})$. Therefore, let us consider the subspaces

[TABLE]

Let us choose homogeneous coordinates $[x_{1}:\ldots:x_{r}]$ on $F$ such that

[TABLE]

for $i=1,\ldots,r-1$. In particular, we have that

[TABLE]

for $i=1,\ldots,\ell$ and therefore the degree 1 part of the isomorphism $\phi$,

[TABLE]

is such that for every $i=0,\ldots,\ell-1$ the image of the subspace

[TABLE]

via $\phi_{1}$ coincide with the subspace generated by the variables $x_{1},\ldots,x_{\operatorname{rank}E_{i}}$. By taking symmetric powers it follows from Proposition 3.8 that for each $m\geq 1$ we have that ${B_{nm}\subseteq k[x_{1},\ldots,x_{r}]_{nm}}$, the image of ${\operatorname{H}^{0}(C,F_{J+1}|_{q})\subseteq\operatorname{H}^{0}(C,F_{0}|_{q})}$, corresponds to the subspace of homogeneous polynomials of degree $nm$ generated by polynomials of the form

[TABLE]

where $P_{i}$ is an homogeneous polynomial of degree $m_{i}\geq 0$ in the variables $\mathbf{x}_{i}=(x_{i,1},\ldots,x_{i,{r_{i}}})$, where $(x_{1},\ldots,x_{r})=(\mathbf{x}_{\ell},\ldots,\mathbf{x}_{1})$, and the $m_{i}$ are such that $m_{1}+\ldots+m_{\ell}=nm$ and

[TABLE]

In other words, $B_{nm}$ is the subspace generated by monomials $x^{\alpha}=x_{1}^{\alpha_{1}}\cdots x_{r}^{\alpha_{r}}$ of total degree $|\alpha|=nm$ such that $\sum_{i=1}^{r}(\sigma_{i}-\mu_{1}-t)\alpha_{i}>2g-1$, where $\boldsymbol{\sigma}=(\underbrace{\mu_{\ell},\ldots,\mu_{\ell}}_{r_{\ell}\text{ times}},\underbrace{\mu_{\ell-1},\ldots,\mu_{\ell-1}}_{r_{\ell-1}\text{ times}},\ldots,\underbrace{\mu_{1},\ldots,\mu_{1}}_{r_{1}\text{ times}})\in\mathbb{Q}^{r}$.

Step 5. Volume comparison and conclusion. It follows from Lemma 5.4 applied555We note that if $C\cong\mathbb{P}^{1}$ then all the slopes $\mu_{i}=\mu(Q_{i})$ are integer numbers and hence the inequality “$>2g-1$” becomes “$\geq 0$”. to the collection of subspaces $\{B_{nm}\}_{m\geq 1}$ that $B_{n\bullet}$ is a graded subalgebra of $k[x_{1},\ldots,x_{r}]$ whose Newton-Okounkov body, with respect to a linear flag $Y_{\bullet}$ that belongs to the Schubert cell $\Omega_{w}$ (see Convention 5.7) is given by

[TABLE]

where

[TABLE]

and $\boldsymbol{\sigma}=(\underbrace{\mu_{\ell},\ldots,\mu_{\ell}}_{r_{\ell}\text{ times}},\underbrace{\mu_{\ell-1},\ldots,\mu_{\ell-1}}_{r_{\ell-1}\text{ times}},\ldots,\underbrace{\mu_{1},\ldots,\mu_{1}}_{r_{1}\text{ times}})\in\mathbb{Q}^{r}$. Finally, we have that

[TABLE]

and hence the inclusion $\Delta_{Y_{\bullet}|F}(B_{n\bullet})\subseteq\Delta_{Y_{\bullet}|F}(A_{n\bullet})$ leads to the equality $\Delta_{Y_{\bullet}|F}(B_{n\bullet})=\Delta_{Y_{\bullet}|F}(A_{n\bullet})$, as the two convex bodies have equal volume. From this we conclude that

[TABLE]

∎

The following result (from which Theorem A is easily deduced) is an immediate consequence. We keep the same notation as in Theorem 5.8.

Corollary 5.9**.**

For every linear flag $Y_{\bullet}$ on $\mathbb{P}(E)$ that belongs to the Schubert cell $\Omega_{w}$ and every big rational class $\eta=a(\xi-\mu_{\ell}f)+bf$ we have that

[TABLE]

and hence the global Newton-Okounkov body of $\mathbb{P}(E)$ with respect to $Y_{\bullet}$ is given by

[TABLE]

In particular, the global Newton-Okounkov body $\Delta_{Y_{\bullet}}(\mathbb{P}(E))$ is a rational polyhedral cone and it depends only on $\operatorname{gr}(\operatorname{HN}_{\bullet}(E))$, the graded vector bundle associated to the Harder-Narasimhan filtration of $E$.

Démonstration.

We note that if $\eta=a(\xi-\mu_{\ell}f)+bf$ is a big rational class on $\mathbb{P}(E)$ and the induced linear flag $Y_{\bullet}$ belongs to the Schubert cell $\Omega_{w}$ with respect to a reference flag $Y_{\bullet}^{\operatorname{HN}}$, then Theorem 5.8 gives

[TABLE]

and hence

[TABLE]

We compute that for $t^{*}=b-a(\mu_{\ell}-\mu_{1})$ we have

[TABLE]

with $\mu_{1}+\frac{1}{a}(\nu_{1}-t^{*})=\mu_{\ell}-\frac{1}{a}(b-\nu_{1})$. The result follows from Lemma 5.3. ∎

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