1 Introduction↩︎

1.1 Background and Motivation↩︎

Throughout this paper, we work over \(\mathbb{C}\), and all varieties \(X\) are assumed to be irreducible, smooth, projective algebraic varieties of dimension \(n\).

The theory of Newton-Okounkov bodies, introduced by [1], [2], [3], and [4], provides a powerful tool for studying divisors on projective varieties through convex geometry. This construction associates to a divisor a convex body in Euclidean space that encodes asymptotic information about the corresponding linear series. A refinement of this theory is the infinitesimal Newton-Okounkov body, also developed by [3], which offers certain advantages such as independence from auxiliary flag choices in favorable situations. The extension from single-point to multipoint configurations has been pursued by [5] and [6], providing finer geometric invariants.

On the algebraic side, the notion of \(k\)-jet ampleness, systematically studied by [7], generalizes classical positivity conditions: \(0\)-jet ampleness corresponds to global generation, while \(1\)-jet ampleness corresponds to very ampleness. This concept is also referred to as generation of jets or high-order embeddings in the literature. A local variant is jet separation (see [8]), which requires that global sections separate jets at a given point up to a specified order, and can be viewed as a pointwise version of jet ampleness.

The connection between Newton-Okounkov bodies and positivity notions was established by [9], [10]. In particular, [10] provides a criterion for jet separation at a single point using infinitesimal Newton-Okounkov bodies: the inclusion of an appropriately sized inverted standard simplex guarantees the desired jet separation property.

A natural question is whether this criterion extends to the multipoint setting, where one requires simultaneous jet separation at a prescribed finite set of points. This is the main focus of the present paper. We note that [5] also addresses multipoint jet separation criteria, but based on a different construction of multipoint Newton-Okounkov bodies. Our approach instead builds on Trusiani’s framework [6], which is particularly well-suited for infinitesimal constructions.

The motivation for studying multipoint jet separation criteria via Okounkov bodies is closely related to classical questions about effective positivity of line bundles, for example Matsusaka’s Big Theorem.

In the toric setting, such positivity questions admit purely combinatorial answers in terms of polytope data: [12] proved Fujita’s very ampleness conjecture for toric varieties using combinatorial properties of associated polytopes, and [13] characterized \(k\)-jet ampleness via intersection numbers with invariant curves and lattice lengths of polytope edges. Our work contributes to this philosophy by providing combinatorial criteria for jet separation—a local positivity notion—using the convex geometry of multipoint Newton-Okounkov bodies that determine for which values of \(k\) a given line bundle \(L\) generates \(k\)-jets at all prescribed points.

A key distinction of our work is that we provide criteria for \(k\)-jet ampleness directly in terms of the line bundle \(L\) itself, rather than relying on the adjoint bundle \(K_X + L\). Since global generation and very ampleness correspond to \(0\)- and \(1\)-jet ampleness respectively, our multipoint criterion provides a computational framework for testing such positivity phenomena via convex geometry.

1.2 Main Results↩︎

Our main results extend the jet separation criterion of [10] in three directions: removing the canonical divisor, allowing multiple points, and combining both generalizations.

The above result addresses jet separation at a single point. We now extend this to the multipoint setting.

For a big divisor \(D\) on \(X\) and a 0-cycle \(Z = \{x_1, \ldots, x_N\}\) of distinct smooth points, let \(\pi: \overline{X} = \mathrm{Bl}_{Z}(X) \to X\) be the blow-up along \(Z\). For each point \(x_j\), let \(\tilde{Y}^{(j)}_{\bullet}\) be an infinitesimal flag over \(x_j\), and denote \({\tilde{Y}_{\bullet}}=({\tilde{Y}_\bullet^{(1)}},...,{\tilde{Y}_\bullet^{(N)}})\). Define \[V_{k,j} = \{ s \in H^0(X, kD) \setminus \{0\} : \nu_{\tilde{Y}^{(j)}_\bullet}(s) < \nu_{\tilde{Y}^{(i)}_\bullet}(s) \text{ for all } i \neq j \}.\] The multipoint infinitesimal Newton-Okounkov body at \(x_j\) is \[\Delta_{\tilde{Y}_\bullet,j}(D) = \overline{\bigcup_{k \geq 1} \frac{\nu_{\tilde{Y}^{(j)}_\bullet}(V_{k,j})}{k}} \subset \mathbb{R}^n .\]

With this notion, we obtain the following multipoint jet separation criterion involving the canonical divisor.

Finally, we present the canonical-free version for the multipoint case, which parallels the single-point result.

We conclude with an explicit computation based on an example originally due to [14], who considered a double covering of a product of two elliptic curves to exhibit a sequence of ample divisors whose multiples required larger and larger multiples to become very ample. We extend this phenomenon to k-jet ampleness and:

1. Computing explicitly the threshold \(k\) such that \((m+k)D_n\) is \(m\)-jet ample on the double cover, where \(D_n = f^* A_n\). This provides concrete numerical bounds for higher-order jet separation and illustrates how the required multiples grow with the jet order.

2. Calculating the Newton-Okounkov body \(\Delta(D_n)\) with respect to an admissible flag. We emphasize that our computation uses a standard flag construction; the infinitesimal flag approach, while theoretically more refined, presents significant computational challenges in this setting.

This calculation demonstrates the effectiveness of our Newton-Okounkov body approach and provides concrete numerical bounds for jet separation in a non-trivial geometric setting.

Acknowledgements↩︎

I gratefully acknowledge the financial support from the China Scholarship Council (CSC) under the grant number [202307300052].

I would like to express my sincere gratitude to my advisor, Dr. Thomas Eckl, for proposing the research problem and providing invaluable guidance throughout this work. I am deeply grateful to my co-advisor, Dr. Zhixian Zhu, for his theoretical support on toric varieties and meticulous verification of technical details in this paper.

This research was conducted under a joint supervision program.

2 Definitions and Preliminaries↩︎

2.1 Newton-Okounkov Bodies↩︎

Remark 3. The construction of the NO-body does not require the choice of a flag. A flag is merely one means to obtain a valuation \(\nu: K(X)^*\rightarrow \mathbb{Z}^n\) of rank \(n\), from which we define the NO-body \(\Delta(D,\nu)\) via the associated value semigroup. Such valuations can be constructed in many ways. This valuation-theoretic perspective, developed systematically in [4] and further extended in [15], provides a more intrinsic framework that encompasses the flag construction as a special case.

A notable example is the jet-valuation \(\nu_{jet}\), which arises from applying an invertible linear transformation to the valuation vector associated with an infinitesimal flag. This transformation, known as jet separation, establishes a natural correspondence between NO-bodies with respect to \(\nu_{jet}\) and those with respect to the infinitesimal flag. More precisely, if the jet valuation is given by \[\nu_{jet} = (\nu_1, \nu_2, \ldots, \nu_k),\] then the infinitesimal flag valuation is defined by the transformation \[\nu_{inf} = (\nu_1+\nu_2+...+\nu_k, \nu_2, \nu_3 , \ldots, \nu_k).\] This linear change of coordinates induces a corresponding transformation between the associated NO-bodies, as illustrated in Figure 1.

The construction of multipoint Newton-Okounkov bodies can be found in [6].

An explicit construction for toric varieties is given in [6].

Let \(D_j = Y_{j,1}\) be prime divisors and \(\mathbb{D} := \sum_{i=1}^N D_i\). Define \[\mu(L; \mathbb{D}) := \sup\{t \geq 0 : L - t\mathbb{D} \text{ is big}\}, \quad \mu(L; D_j) := \sup\{t \geq 0 : \Delta_j(L - t\mathbb{D})^{\circ} \neq \emptyset\}.\]

2.2 Jet Separation and Jet Ampleness↩︎

3 Main Theorems↩︎

The following theorem adopts the proof strategy from [10].

4 Example: Double Cover of Product of Elliptic Curves↩︎

We use \(\ell\geq 2\) for the Kollár–Lazarsfeld parameter in this example. This avoids confusion with the dimension variable in 9. In the application below, the variety is the surface \(Y\), so \(d:=\dim Y=2\); very ampleness corresponds to the jet order \(k=1\). We write \(s\) for the positive integer denoted by \(m\) in the theorem.

4.1 Setup and ampleness↩︎

Let \(S=E\times E\), where \(E\) is an elliptic curve. Let \(F_1,F_2\) be the two fiber classes and let \(\Delta\) be the diagonal. The intersection numbers are \[F_1^2=F_2^2=\Delta^2=0, \qquad F_1\cdot F_2=F_1\cdot\Delta=F_2\cdot\Delta=1.\] Define \[A_\ell=\ell F_1+(\ell^2-\ell+1)F_2-(\ell-1)\Delta.\] Put \(H=F_1+F_2\). Since \[A_\ell^2=2, \qquad A_\ell\cdot H=\ell^2-2\ell+3>0,\] and on an abelian surface the ample cone is the component of the positive cone containing \(H\), the divisor \(A_\ell\) is ample [8].

Set \(R=F_1+F_2\), choose a smooth divisor \(B\in |2R|\), and let \[f:Y\to S\] be the double cover branched along \(B\). Define \(D_\ell=f^*A_\ell\). Since \(f\) is finite, \(D_\ell\) is ample. Equivalently, by Nakai–Moishezon, if \(\Gamma\subset Y\) is irreducible and \(C=f(\Gamma)\), then \[f_*\Gamma=eC \quad (e=1 \text{ or }2), \qquad (f^*L)\cdot\Gamma=L\cdot f_*\Gamma=e(L\cdot C).\] Together with \((f^*L)^2=2L^2\), this shows that \(f^*L\) is ample on \(Y\) if and only if \(L\) is ample on \(S\).

4.2 Computation of \(m(D_\ell)\)↩︎

The first condition in 9 is \[m(D_\ell)D_\ell-K_Y \quad \text{ample}.\] The double-cover formula gives \[K_Y=f^*(K_S+R)=f^*R,\] since \(K_S=0\). For an integer \(q\), set \(L_q=qA_\ell-R\). Then \[qD_\ell-K_Y=f^*L_q,\] so \(qD_\ell-K_Y\) is ample if and only if \(L_q\) is ample on \(S\).

Let \[N_\ell:=A_\ell\cdot R=\ell^2-2\ell+3.\] Using the abelian-surface criterion with the ample class \(A_\ell\), \(L_q\) is ample if and only if \[L_q\cdot A_\ell=2q-N_\ell>0, \qquad L_q^2=2q^2-2N_\ell q+2>0.\] Equivalently, \[q>\frac{N_\ell+\sqrt{N_\ell^2-4}}{2}.\] Since \(N_\ell\geq 3\), this real number lies strictly between \(N_\ell-1\) and \(N_\ell\). Hence the minimal integer is \[\boxed{m(D_\ell)=N_\ell=\ell^2-2\ell+3.}\]

4.3 The Newton–Okounkov condition and the resulting coefficient↩︎

The second condition in 9 is the inverted-simplex containment \[\Delta_{d+k+\varepsilon}^{-1} \subseteq \Delta_{\widetilde{Y}_\bullet}(s\pi_y^*D_\ell)\] for every infinitesimal flag over the point. Here \(d=2\) and \(k=1\), so the required size is \(3+\varepsilon\).

The relevant convex-body size is controlled by the infinitesimal Newton–Okounkov body. For any \(y\in Y\), put \(x=f(y)\). The projection formula above, together with the local multiplicity inequality for finite maps, gives \[\varepsilon(D_\ell;y)\geq \varepsilon(A_\ell;x).\] On the abelian surface \(S\), one has \[\varepsilon(A_\ell;x)=1:\] the translate of \(F_2\) through \(x\) gives the upper bound \(A_\ell\cdot F_2=1\), while every ample integral line bundle on an abelian variety has Seshadri constant at least \(1\) [8]. By the surface theorem of Küronya–Lozovanu [10], this gives \[\Delta_1^{-1}\subseteq \Delta_{\widetilde{Y}_\bullet}(\pi_y^*D_\ell), \qquad \Delta_s^{-1}\subseteq \Delta_{\widetilde{Y}_\bullet}(s\pi_y^*D_\ell)\] for every infinitesimal flag. Therefore, the preceding lower-bound estimate guarantees that the strict condition \[\Delta_{3+\varepsilon}^{-1} \subseteq \Delta_{\widetilde{Y}_\bullet}(s\pi_y^*D_\ell)\] holds for \(s=4\) and for some \(\varepsilon>0\). Combining this with the value of \(m(D_\ell)\), the coefficient in the conclusion of 9 is \[s+m(D_\ell)=4+(\ell^2-2\ell+3)=\ell^2-2\ell+7.\] Thus the theorem verifies \(1\)-jet separation for \[\boxed{(\ell^2-2\ell+7)D_\ell.}\]

References↩︎