Newton–Okounkov theory in an abstract setting

Abstract

We extend the theory of Newton–Okounkov bodies, originally developed by Boucksom–Chen, Kaveh–Khovanskii, and Lazarsfeld–Mustaţă for lattice semigroups, to the context of cancellative torsion–free semigroups.

Filtered Newton–Okounkov bodies

Let as before k be an algebraically closed field and K/k an extension with finite transcendence degree n, and let v be a k-valuation of K, of maximal rational rank n. Denote $\Gamma _v=v(K^\times )$ the value group of v. Let $\Gamma$ be a graded (abelian) group and R a graded k-subalgebra of the group algebra $K[\Gamma ]$. Denote $\sigma (R)$ the set of additive vector subspaces of R. It is partially ordered by inclusion.

Definition A.1

A filtration of R indexed by the ordered abelian group $\tilde{\Gamma }$ is an order-reversing map

$$\begin{aligned} \tilde{\Gamma }\overset{F}{\longrightarrow }&\,\sigma (R),\\ \tilde{\gamma } \longmapsto&\,F_{\tilde{\gamma }}. \end{aligned}$$

The filtrations F of interest for us are moreover

• (A.1.1) complete, i.e., $\bigcup _{\tilde{\gamma } \in \tilde{\Gamma }} F_{\tilde{\gamma }}=R$,

• (A.1.2) multiplicative, i.e., $F_{\tilde{\gamma }} \cdot F_{\tilde{\eta }} \subseteq F_{\tilde{\gamma }+\tilde{\eta }}$ for all $\tilde{\gamma },\tilde{\eta } \in \tilde{\Gamma }$.

• (A.1.3) homogeneous, i.e., $F_{\tilde{\gamma }}=\bigoplus _{\gamma \in \Gamma } F_{\tilde{\gamma }}(\gamma )$, where $F_{\tilde{\gamma }}(\gamma )=F_{\tilde{\gamma }}\cap R(\gamma )$ for every $\tilde{\gamma }\in \tilde{\Gamma }$.

Multiplicative filtrations on sections rings $R=R(X,L)$, where L is a line bundle on a smooth variety X, arise naturally in various ways. One immediate example is to consider the order of vanishing along a smooth subvariety, but especially important are arithmetic filtrations. Donaldson’s test configurations Donaldson (2002) are another source of multiplicative filtrations Ross and Thomas (2006, 2007), Székelyhidi (2015) and Nyström (2012).

Such filtrations give rise to “refinements” of the Newton–Okounkov body of R.

Definition A.2

The Rees algebra of the filtration F, indexed by $\tilde{\Gamma }$, is defined as the following graded subalgebra of $R[\tilde{\Gamma }]$:

$$\begin{aligned} \mathrm {Rees}(F) = \bigoplus _{\tilde{\gamma } \in \tilde{\Gamma }} F_{\tilde{\gamma }} u^{\tilde{\gamma }} \subset R[\tilde{\Gamma }] \subset K[\Gamma \times \tilde{\Gamma }].\end{aligned}$$

Every subsemigroup of $\tilde{\Gamma }$ determines a subalgebra of the Rees algebra; especially relevant are bounded Rees algebras. Assume an embedding $\iota :\tilde{\Gamma }\hookrightarrow \mathbb R^r$ is given, and for every $\tilde{\gamma }\in \tilde{\Gamma }$, denote $\tilde{\gamma }_i$ the i-th component of $\iota (\tilde{\gamma })$. Given $\mathbf {b}=(b_1\dots ,b_r)\in \mathbb R^r$, we define the Rees algebra of F bounded by $\mathbf {b}$ as

$$\begin{aligned} \mathrm {Rees}_{\mathbf {b}}( F) = \bigoplus _{\begin{array}{c} \tilde{\gamma }\in \Gamma _{{F}}\\ \tilde{\gamma }_i \ge b_i\deg (\gamma ) \end{array}} F_{\tilde{\gamma }}(\gamma ) u^{\tilde{\gamma }} \subset \mathrm {Rees}(F) \subset K[\Gamma \times \tilde{\Gamma }]. \end{aligned}$$

The semigroup generated by the support of a bounded Rees algebra is linearly bounded, and its Newton–Okounkov body encodes additional information to that of $\Delta (R)$. In particular, for real filtrations ($\tilde{\Gamma }\subset \mathbb R$) we recover the filtered Newton–Okounkov body introduced by Boucksom–Chen, motivated by the case of arithmetic filtrations and the study of arithmetic volumes (see Boucksom and Chen (2011) and also Boucksom et al. (2015)). Let us now recall their construction. With notation as above, assume $\tilde{\Gamma }=\mathbb R$, and for each $t\in \mathbb R$ set

$$\begin{aligned} V_t(d) {\mathop {=}\limits ^{\text {def}}}F_{td}(d) \ . \end{aligned}$$

It is immediate that $V_t(\bullet )$ forms a graded subalgebra of R(X, L), corresponding to a graded linear series, and the Newton–Okounkov bodies $\Delta _v(V_t(\bullet ))$ are a non-increasing collection of compact convex subsets of $\Delta _v(L)$.

Definition A.3

With notation as above, the the filtered Newton–Okounkov body of the filtration F is

$$\begin{aligned} \widehat{\Delta }(L, F) {\mathop {=}\limits ^{\text {def}}}\{(\alpha ,t)\in \Delta _v(L)\times \mathbb R\,|\, 0\le t, \text { and }\alpha \in \Delta _v(V_t(\bullet ))\}. \end{aligned}$$

It is a convex body, built from its ”horizontal” slices, which are, at each level t, the Newton–Okounkov body of the graded linear series $V_t(\bullet )$.

This fits in our setting, as the filtered Newton–Okounkov body turns out to be (up to exchanging first and last coordinates) the Newton–Okounkov body of a suitable bounded Rees algebra.

Proposition A.4

((Küronya et al. 2020, Proposition 2.26)) Let X be an n-dimensional projective variety, $v:K(X)^\times \rightarrow \Gamma$ a valuation of maximal rational rank, L a big line bundle on X, and F a homogeneous, linearly bounded, and complete multiplicative filtration on $R=R(X,L)$ indexed by a subgroup of $\mathbb R$. Let $\mathrm {Rees}_0( F)$ Rees algebra bounded by $b=0$ and let $\chi :\mathbb R^{n}\times \mathbb R\rightarrow \mathbb R\times \mathbb R^n$ be the isomorphism that exchanges both factors. Then $\chi (\widehat{\Delta }(L,{F}))=\Delta _v(\mathrm {Rees}_0({F}))$.