Toric completions and bounded functions on real algebraic varieties

Toric completions and bounded functions on real algebraic varieties

Daniel Plaumann and Claus Scheiderer

Abstract

Given a semi-algebraic setS, we study compactifications ofS that arise from embeddings into complete toric varieties. This makes it possible to describe the asymptotic growth of polynomial functions on S in terms of combinatorial data. We extend our earlier work in Plaumann and Scheiderer [‘The ring of bounded polynomials on a semi-algebraic set’,Trans. Amer. Math. Soc.

364 (2012) 4663–4682] to compute the ring of bounded functions in this setting, and discuss applications to positive polynomials and the moment problem. Complete results are obtained in special cases, like sets defined by binomial inequalities. We also show that the wild behaviour of certain examples constructed by Krug [‘Geometric interpretations of a counterexample to Hilbert’s 14th problem, and rings of bounded polynomials on semialgebraic sets’, Preprint, 2011, arXiv:1105.2029] and Mondal-Netzer [‘How fast do polynomials grow on semialgebraic sets?’, J. Algebra413 (2014) 320–344] cannot occur in a toric setting.

Introduction

The simplest measure for the asymptotic growth of a real polynomial innvariables onRⁿis its total degree. However, when we pass fromRⁿ to an unbounded semi-algebraic subsetS⊆Rⁿ, the total degree of a polynomial may not reflect the growth of the restrictionf|S any more.

The degree of a polynomial can be understood as its pole order along the hyperplane at infinity whenRⁿ is embedded into projective space in the usual way. How this relates to the growth off|Sdepends on how the closure ofSinPⁿ(R) meets the hyperplane at infinity. Unless this intersection is empty (which would mean thatSis bounded inRⁿ) or of maximal dimension, the total degree alone will usually not suffice to understand the growth of polynomials onS.

We may however hope to improve control of the growth by suitable blow-ups at infinity.

To make this idea more precise, we consider the following setup. Suppose thatV is an affine real variety and S is the closure of an open semi-algebraic subset of V(R). An open dense embedding of V into a complete variety X is called compatible with S if the geometry of S at infinity is regular in the following sense: ifZ is any hypersurface at infinity, that is, any irreducible component of the complement of V in X, the closure S of S in X(R) meets Z either in a Zariski-dense subset ofZ or not at all. Under this condition, the pole orders of a regular function f on V along the hypersurfaces at infinity intersectingS accurately reflects the qualitative growth off onS.

Compatible completions were introduced by the authors in [11], as well as in the dissertation of the first author, motivated by earlier work of Powers and Scheiderer [12]. AnS-compatible completion ofV yields in particular a description of

BV(S) ={f ∈R[V] :∃λ∈R|f|λon S},

2010Mathematics Subject Classification14P99 (primary), 14C20, 14M25, 14P10 (secondary).

The first author was supported by DFG grant PL 549/3-1 and the Zukunftskolleg of the University of Konstanz, the second author by DFG grant SCHE 281/10-1.

Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-0-374717

the ring of regular functions onV that are bounded onS. IfV →X is such anS-compatible completion andY is the union of those irreducible components ofXV that are disjoint from S, then B_V(S) is naturally identified with OX(XY), the ring of regular functions of the varietyXY.

The main goal of this paper is to improve on the results in [11] and make them more explicit in the controlled setting of toric varieties. Specifically, we study the following questions:

(1) One of the main results of [11] is the existence of regular completions in the case dimV 2. The higher-dimensional case remains open and hinges on the existence of a certain type of embedded resolution of singularities. In the toric setting, we introduce a stronger, purely combinatorial compatibility condition in the spirit of toric geometry (Section2). We show that this condition can be satisfied ifS is defined by binomial inequalities (Corollary2.16) or ifS is what we call a tentacle (Corollary2.20), generalizing a concept introduced by Netzer [10].

Since the compatible completions in dimension 2 constructed in [11] are built from an embedded resolution of singularities, they are typically quite hard to compute explicitly. In contrast, our results in the toric setting only require the usual arithmetic of semigroups derived from rational polyhedral cones.

(2) The transcendence degree of the ring of regular functionsO(XY) of the complement of a divisor Y in a complete variety X is called the Iitaka dimension of Y. It is a natural generalization of the Kodaira dimension studied extensively in complex algebraic geometry.

Thus, in the case of a compatible completion, whenB_V(S) is identified with O(XY), the Iitaka dimension measures in how many independent directions the setS is bounded.

In dimension 2, the Iitaka dimension is strongly related to the signature of the intersection matrix A_Y of the divisor Y. However, the correspondence is not perfect if A_Y is singular.

Specifically, ifA_Y is negative semidefinite, but not definite, Iitaka’s criterion (Proposition3.4) does not give anything. In the toric setting, on the other hand, we show that the signature of AY is sufficient to determine the Iitaka dimension (Proposition3.13). It seems plausible that this has been observed before, but we were unable to find any trace in the literature. We exploit the result in an application to positive polynomials explained below.

(3) The existence of anS-compatible completionX ofV yields a good description of the ring of bounded functionsBV(S). However, it does not imply that BV(S) is a finitely generated R-algebra. This was discussed in [11] and much further explored by Krug [5]. When a toric S-compatible completion exists,BV(S) is always finitely generated (Proposition2.10).

Beyond bounded polynomials, an S-compatible completion also provides control over the asymptotic growth of arbitrary polynomials, as indicated in the beginning. Let Y be the union of all irreducible components ofXV that intersect the closure ofSinX. In Section4, we study the linear subspaces

L_X,m(S) ={f ∈R[V] : all poles off alongY have order at most m},

which consist of functions of bounded growth onS. Assume that B_V(S) =R. In analogy with the caseS=Rⁿ, one might expect that the spaces L_X,m(S) (m∈N) are finite-dimensional. If so, the filtrationL_X,0(S)⊆L_X,1(S)⊆L_X,2(S)⊆ · · · ofR[V] behaves much like the filtration of the polynomial ring by total degree. The properties of filtrations obtained in this way and further generalizations have also been studied in complex algebraic geometry (see [8]).

For us, this question is particularly relevant in the context of positive polynomials and the moment problem, as it concerns possible degree cancellations in sums of positive polynomials, as explained in Section 5. However, a subtle example due to Mondal and Netzer [9] (see Example 4.4) implies that the LX,m(S) may have infinite dimension. This construction is complemented by Theorem 5.5, which combines with the results of [13] to show that ifS is basic open of dimension at least 2 and admits anS-compatible toric completion, but no non- constant bounded function, then the spacesL_X,m(S) are finite-dimensional and, consequently, the moment problem forS is not finitely solvable. This comprises the results of Netzer [10] for

tentacles and of Powers and Scheiderer [12]. It is also related to a theorem of Vinzant [14], which constructs a certain kind of toric compatible completion under an algebraic assumption on the description ofS and the ideal ofV, as explained in the last section of [14].

1. Compatible completions of semi-algebraic sets We briefly summarize some of the definitions and results in [11].

Definition 1.1. LetV be a normal affineR-variety and letS be a semi-algebraic subset of V(R). By a completion of V, we mean an open dense embedding V →X into a normal complete R-variety. The completionX is said to be compatible with S (or S-compatible) if, for every irreducible componentZ ofXV, the following condition holds: The setZ(R)∩S is either empty or Zariski-dense inZ.

Here, when taking the closure S of a semi-algebraic subset S of X(R), we refer to the Euclidean topology on X(R), rather than the Zariski topology. Note that every irreducible component ofXV is a divisor onX, that is, has codimension 1 [3, p. 66].

Theorem 1.2([11, Theorem 3.8]). LetV be a normal affineR-variety,letS⊆V(R)be a semi-algebraic subset and assume that the completionV →X ofV is compatible withS. Let Y denote the union of those irreducible componentsZofXV for whichS∩Z(R) =∅,and putU =XY. Then the inclusionV ⊆U induces an isomorphism ofR-algebras

OX(U)∼=B_V(S).

A semi-algebraic set is calledregularif its closure coincides with the closure of its interior. It is calledregular at infinityif it is the union of a regular and a relatively compact semi-algebraic set. One of the main results of [11] is the existence of compatible completions for 2-dimensional semi-algebraic sets regular at infinity.

Theorem 1.3([11, Theorem 4.5]). LetV be a normal quasi-projective surface overR, and letSbe a semi-algebraic subset ofV(R)that is regular at infinity. ThenV has anS-compatible projective completion. If V is non-singular, then the completion can be chosen to be non- singular as well.

1.4. The proof of Theorem1.3is essentially constructive and relies on embedded resolution of singularities. We summarise the procedure for our present purposes. LetV →Xbe any open dense embedding ofV into a normal projective surface. LetC_∂ be the Zariski closure of the boundary of S in X(R) and letC_∞=XV. Put C=C_∂∪C_∞, a reduced curve in X. We write

∂_X^∞S=S∩C_∞(R)

for the set of boundary points of S at infinity in X. A sufficient condition for X to be an S-compatible completion of V is that C has only normal crossings in ∂_X^∞S. Explicitly, this means the following. IfP∈∂_X^∞S, then

(1) P is a non-singular point of all irreducible components ofC that contain it.

(2) P is contained in exactly one componentC₀ ofC∂ and one componentC₁ ofC_∞, and C₀(R) and C₁(R) have independent tangents in P. (Equivalently, the local equations forC₀ andC₁ generate the maximal ideal of the local ringOX,P.)

By blowing-up any points in∂^∞_XSin which conditions (1) or (2) are violated and proceeding inductively, we can produce a completionX and a corresponding curveC=C_∂∪C_∞, defined

as before, such that all points of ∂^∞

XS are normal crossings of C, which is therefore an S-compatible completion of V. Note that blowing-up increases the number of irreducible components in C_∞, since the exceptional divisor is added. In the resulting S-compatible completion X, the divisorY of Theorem1.2 consists of those irreducible components of C_∞ that are disjoint fromS.

Explicit computation of the ring of bounded polynomials following the above procedure is possible, but can quickly turn into a cumbersome task. We give the following simple example as an illustration. A much more interesting example will be discussed in Section4.

Example 1.5. Let S={(x, y)∈R²:−1x1} be a strip in the affine plane V =A²_R and consider the embedding V →P²_R into the projective plane given by (u, v) →(u:v: 1).

Then C_∞=P²V is the line at infinity and C_∂ is the Zariski closure of the two lines V(x−1) andV(x+ 1) inV. The set∂S_P^∞2 is the pointP = (0 : 1 : 0), which is also the unique intersection point of C_∂ and C_∞. In local coordinates r=x/y and s= 1/y of P²_R centred around P, we have C_∞=V(s) and C_∂ =V((r−s)(r+s)). Since all three components of C=C_∞∪C_∂ pass throughP,C does not have normal crossings inP. Indeed, the completion ofS is notS-compatible, sinceS∩C_∞(R) ={P} is not Zariski-dense inC_∞.

Let X be the blow-up of P²_R in P. It is given in local coordinates by the quadratic transformation r=r₁, s=r₁s₁. In the new coordinates r₁, s₁, the exceptional divisor is E=V(r₁). The strict transforms of the components of C in X are C_∞=V(s₁) and C∂=V((s₁−1)(s₁+ 1)). Now C_∞=XV has the two components C_∞ and E. Since C∂

meets E in the points (0,1) and (0,−1), but does not meet C_∞, we see that X is an S- compatible completion ofV andY =C_∞is the component ofC_∞that is disjoint fromS_X(R) . To computeO(XY), writef ∈R[x, y] asf =

i,jaijxⁱy^j =

i,jaijr₁^−js^−i−j₁ , so thatf lies inO(XY) if and only ifj= 0. Thus B(S) =O(XY) =R[x].

In dimensions3, it is not even guaranteed that the ringB(S) is finitely generated (see [11, Section 5]).

2. Toric completions

LetV be an affine toric variety. By atoric completionof V, we mean an open embedding of V into a complete toric variety X which is compatible with the torus actions. Let S⊆V(R) be a semi-algebraic subset. We are going to work out conditions onS ensuring that V has a toric completionV ⊆X that is compatible withS. The existence of such a completion allows us to make the ring of bounded polynomial functions onS completely explicit. It also prevents several pathologies that can occur in more general cases.

We start by reviewing some general notions on toric varieties. An excellent reference is the book of Coxet al.[2].

2.1. Let T be an n-dimensional split R-torus and let T(R)∼= (R^∗)ⁿ be the group of R-points. All toric varieties will be T-varieties. Let M =Hom(T,Gm) (respectively, N =Hom(Gm, T)), the group of characters (respectively, of co-characters) of T. Both are free abelian groups of rank n, each being the natural dual of the other. We write both groups additively and denote the character corresponding to α∈M by x^α, the co-character corresponding tov∈N byλv. The pairing between M andN will be denoted byα, v.

2.2. Let M_R=M ⊗R, N_R=N⊗R. By a cone σ⊆N_R we always mean a finitely generated rational convex cone. Let σ^∗⊆M_R denote the dual cone of σ, let H_σ =M ∩σ^∗,

and write R[Hσ] for the semigroup algebra of Hσ. Then Uσ=SpecR[Hσ] is an affine toric variety that contains a unique closedT-orbit, denotedOσ.

Assume that the coneσ⊆N_Ris pointed. Then the denseT-orbitU₀inU_σis isomorphic toT, and we may use any fixedξ₀∈U₀(R) to equivariantly identifyU₀withT. Letv∈N∩relint(σ).

For anyξ∈U₀(R), the limit

Lv(ξ) := lim

s→0(λv(s)·ξ)

exists inU_σ(R) and lies inO_σ(R). Clearly, the map L_v:U₀(R)→O_σ(R) is equivariant under theT(R)-action. In particular,L_v is an open map.

2.3. Fixing v∈N, we consider the v-grading of R[T], which is the grading that makes the character x^α homogeneous of degree α, v for every α∈M. We say that f ∈R[T] is v-homogeneous iffis homogeneous in thev-grading. For 0=f ∈R[T], let inv(f)∈R[T] denote the leading component off in the v-grading, that is, the non-zero v-homogeneous component of f of smallest v-degree. Two vectors v, v ∈N satisfy inv(f) = inv(f) if and only if v and v lie in the relative interior of the same cone of the normal fan of the Newton polytope of f. (Note that since we define the leading form inv(f) to be the homogeneous component of smallestv-degree, we are using inward, rather than outward, normal cones here.)

2.4. A fan is a finite non-empty set Σ of closed pointed rational cones in N_R, which is closed under taking faces and such that the intersection of any two elements of Σ is a face of both. The union of all cones in Σ is called the support of Σ, denoted by|Σ|; if |Σ|=N_R, then Σ is called complete. The fan Σ gives rise to a toric varietyX_Σ, obtained by glueing the affine toric varietiesU_σ,σ∈Σ. The varietyX_Σis complete if and only if the fan Σ is complete.

In general, the ring of global regular functions O(XΣ) is the semigroup algebra R[H], where H =M ∩ |Σ|^∗. By Dickson’s lemma, this is a finitely generatedR-algebra.

LetUσ be an affine toric variety and letS⊆Uσ(R) be a semi-algebraic set. We are going to study conditions under which there exists a toric completion ofUσ that is compatible withS, and which therefore allows the explicit computation of the ringBUσ(S) of polynomials bounded onS. We first propose an abstract framework; see Proposition2.10. After this, we will exhibit concrete situations to which the abstract framework applies.

We will always assume that the semi-algebraic setS is open and contained in the dense torus orbit inU_σ.

2.5. LetS⊆T(R) be an open semi-algebraic subset. Givenv∈N, put S(v) :={ξ∈T(R) :∀0< s1 λv(s)ξ∈S}.

It is easily seen that (S₁∪S₂)(v) =S₁(v)∪S₂(v) and (S₁∩S₂)(v) =S₁(v)∩S₂(v) hold for all v∈N and all open semi-algebraic setsS₁, S₂⊆T(R). Further let

K(S) :={v∈N :S(v)=∅}, K₀(S) :={v∈N : int(S(v))=∅}.

ThenK(S₁∪S₂) =K(S₁)∪K(S₂) andK₀(S₁∪S₂) =K₀(S₁)∪K₀(S₂) hold.

Lemma 2.6. Given any open semi-algebraic setS⊆T(R),there exists a fanΣinN_Rsuch that

K(S) =N∩

σ∈E

relint(σ), K₀(S) =N∩

σ∈E0

relint(σ) hold for suitable subsetsE₀, E ofΣ. Any such fanΣis said to be adapted toS.

Proof. We may assume that S={ξ∈T(R) :fi(ξ)>0 (i= 1, . . . , r)} is basic open, with f₁, . . . , fr∈R[T]. Givenf ∈R[T] andv∈N, letfv,d ∈R[T] be thev-homogeneous component off of degreed. Thus

f(λv(s)ξ) =

d∈Z

fv,d(ξ)·s^d

for s∈R and ξ∈T(R). So ξ∈S(v) holds if and only if, for every i= 1, . . . , r, there exists di∈Z with (fi)v,di(ξ)>0 and with (fi)v,d(ξ) = 0 for all d< di. Let Λ(f, v) denote the sequence of non-zero v-homogeneous components of f, ordered by increasing degree. Then, ifv, v∈N satisfy Λ(f_i, v) = Λ(f_i, v) fori= 1, . . . , r, it follows thatS(v) =S(v). It is clear that there is a fan Σ such that any two vectors v, v in the relative interior of the same cone of Σ satisfy this condition. Such Σ satisfies the condition of the lemma.

Remark 2.7. IfSis a subset of the positive orthant inRⁿ, the setK(S) is closely related to the tropicalization ofS constructed by Alessandrini [1].

Lemma 2.8. Let S⊆T(R) be an open semi-algebraic set and let ρ⊆N_R be a pointed cone:

(a) IfK(S)∩relint(ρ)=∅,thenS∩O_ρ(R)=∅.

(b) IfK₀(S)∩relint(ρ)=∅,then S∩O_ρ(R)is Zariski-dense inO_ρ.

Here we fix an equivariant identification T =U₀. The closures are taken inside the affine toric varietyUρ and with respect to the Euclidean topology. Recall that Uρ contains U₀=T (respectively,O_ρ) as an open dense (respectively, as a closed)T-orbit.

Proof. Given v∈N∩relint(ρ), the map Lv :T(R)→Oρ(R) (see 2.2) is open and maps S(v) into S∩Oρ(R). The hypothesis v∈K(S) (respectively, v∈K₀(S)) means S(v)=∅ (respectively, int(S(v))=∅). This proves the lemma.

2.9. Now let Σ be a complete fan in N_R and let X_Σ be the associated complete toric variety. We write Σ(d) for the set ofd-dimensional cones in Σ. Forτ ∈Σ letYτ be the Zariski closure ofOτ inX_Σ. In particular,Yτ is a prime Weil divisor onX_Σwhenτ∈Σ(1). We fix a coneσ∈Σ and considerX_Σas a toric completion of the affine toric varietyUσ.

Let S⊆T(R) =U₀(R) be an open semi-algebraic set. We will require the following toric compatibility assumption:

(TC) For anyτ∈Σ(1) withτσ, eitherS∩Yτ(R) is empty orK₀(S)∩relint(τ) is non-empty.

(The two cases are mutually exclusive by Lemma2.8.) We define the subfan F_S of Σ by F_S :=

ρ∈Σ : Every 1−dimensionalfaceτ ofρsatisfies τ ⊆σorK₀(S)∩relint(τ)=∅

.

Proposition 2.10. With the above notation,assume that the toric compatibility condition (T C)holds. Then the toric varietyX_Σis anS-compatible completion ofUσ. In particular,let BUσ(S) be the subring of R[Uσ] consisting of the regular functions that are bounded on S.

Then

B_Uσ(S) =O(XFS) =R[H]

withH =M ∩ |FS|^∗. In particular, theR-algebraBUσ(S)is finitely generated.

Figure 1.The semi-algebraic set from Example 2.11(2) and its corresponding fan.

Proof. WriteX=X_Σ, a normal and complete toric variety containingU_σas an open affine toric subvariety. The irreducible components ofXU_σare theY_τ, whereτ ∈Σ(1) andτσ.

Given suchτ withS∩Yτ(R)=∅, we know thatS∩Yτ(R) is Zariski-dense inYτ, by condition (T C) and Lemma 2.8(b). So the completion X of Uσ is compatible with the semi-algebraic set S⊆Uσ(R), in the sense of Definition 1.1. By Theorem 1.2, we therefore have BUσ(S) = O(XY), where Y is the union of those irreducible components Yτ of XUσ for which S∩Yτ(R) =∅. By condition (T C), the latter meansτ ∈Σ(1),τσand relint(τ)∩K₀(S) =

∅. SoXY is precisely the toric variety associated to the subfanFS of Σ defined above.

Example 2.11. Letn= 2. We compatibly identify M =Z², N =Z² and T(R) = (R^∗)². We denote by (e₁, e₂) the standard basis ofN_R=R²and by (e^∗₁, e^∗₂) the dual basis ofM_R. Let σ= cone(e₁, e₂) be the positive quadrant in N_R, so thatR[Uσ] =R[x₁, x₂] andUσ=A². Let Σ₀be the standard fan ofP²with ray generatorse₁,e₂, and−(e₁+e₂). We use homogeneous coordinates (u₀:u₁:u₂) onP² withx_i=_u^ui

0 (i= 1,2).

(1) Consider the set

S:={(ξ1, ξ₂)∈(R^∗)²:−1< ξ₁<1}.

It is easily seen that K(S) =K₀(S) ={(v1, v₂)∈N :v₁0}. Let Σ be the refinement of Σ0

generated by the additional ray generator−e2. Then Σ is adapted toS; cf. Lemma2.6. The toric varietyX_Σis the blow-up ofP²in the point (0:0:1), which is exactly the compatible completion of the stripS⊆R² we considered in Example1.5. By definition,|FS|={(v1, v₂)∈N_R:v₁ 0}, so thatM∩ |FS|^∗={(k,0)∈M :k0}, whenceO(XFS) =R[x₁]. It is not hard to check that condition (T C) is met in this example, so that Proposition2.10yieldsB_A²(S) =R[x₁].

(2) Letk1 and let

S:={(ξ₁, ξ₂)∈R²:ξ₁^kξ₂<1, ξ₁>0, ξ₂>0}

(see Figure1 fork= 2). Here we find thatK(S) =K₀(S) is the half-spacekv₁+v₂0 inN. We again refine Σ₀ by adding ray generators ±(e₁−ke₂) to Σ₀, and obtain the fan Σ shown on the right of Figure1.

By construction, Σ is adapted to S, and |FS|={v∈N_R:kv₁+v₂0}. We check that condition (T C) is satisfied. This amounts to showing, forτ = cone(−e₁−e₂), thatYτ(R)∩S =

∅. Indeed, letρ= cone(−e₁−e₂,−e₁+ke₂). Thenρ^∗is generated bye^∗₂−e^∗₁and−(ke^∗₁+e^∗₂), so that R[Uρ] =R[H], where H is the saturated semigroup generated by y₁=x^−k₁ x⁻¹₂ , y₂=x⁻¹₁ , and y₃=x⁻¹₁ x₂, so that R[Uρ]∼=R[y₁, y₂, y₃]/(y₁y₃−y₂^k+1). Under this identifi- cation, we find y₁= 0 on Y_τ∩U_ρ while y₁>1 on S∩U_ρ(R). So S∩(U_ρ∩Y_τ)(R) =∅.

Essentially the same computation applies toρ = cone(−e1−e₂, e₁−ke₂). Hence we conclude B_A2(S) =O(XFS) =R[x^k₁x₂]. This will be discussed in general below (cf. Corollary2.16).

(3) Let

S:={(ξ1, ξ₂)∈R²:ξ₁(ξ₁−ξ₂) + 1>0, ξ₂(ξ₂−ξ₁) + 1>0, ξ₁>0, ξ₂>0}

Figure 2.The semi-algebraic set from Example 2.11(3) and its corresponding fan.

(see Figure2). In this example, we have K₀(S) ={v∈N :v₁+v₂0}, whileK(S) consists of K₀(S) and the half-line τ generated by −(e1+e₂). Let Σ be the complete fan with ray generatorse₁,e₂,±(e1−e₂) and−(e1+e₂) (see Figure 2). Then Σ is adapted toS and |FS| is the half-planev₁+v₂0. But condition (T C) is not satisfied. If it were, we could conclude B_A²(S) =R[x1x₂], which is clearly not true sincex₁x₂ is unbounded on S. Indeed, it follows from Lemma2.8thatYτ(R)∩S=∅. (It is not hard to check directly that it is a single point).

(4) The key property for Proposition2.10to apply is condition (T C) from2.9. For a general open semi-algebraic set, this condition cannot be satisfied by any choice of a fan Σ inN_R, as is demonstrated by the following simple example: for the open setS={ξ∈(R^∗₊)²:ξ₁, ξ₂>1 and 1< ξ₁−ξ₂<2} in the 2-dimensional torus, we haveK(S) =Z₊(−e₁−e₂) andK₀(S) ={0}.

So, at least with respect toσ={0}, condition (T C) cannot hold for any complete fan Σ.

Remark 2.12. Example (4) can still be saved by making a linear change of coordinates.

However, it is clear that more complicated examples of open sets may be constructed for which no linear coordinate change allows to apply condition (T C). There is also an indirect way to see this. Whenever condition (T C) applies, we see from Proposition2.10that the ringB_Uσ(S) of bounded polynomials onU_σ is finitely generated as anR-algebra. On the other hand, it is known that there exist open semi-algebraic subsetsSof (R^∗)ⁿforn3 for which theR-algebra B_Aⁿ(S) fails to be finitely generated (see [5]).

Condition (T C) can be rather cumbersome to check, as the above examples show. We therefore seek favourable situations in which this condition can be guaranteed, and therefore allows a purely combinatorial computation of the ring of bounded functions. We discuss two classes of sets where this approach is successful, namely, binomially defined sets and the so-called ‘tentacles’ considered by Netzer [10].

2.13. LetQ:= (R^∗₊)ⁿ⊆T(R), and let

S={ξ∈Q:a_iξ^αi < b_iξ^βi (i= 1, . . . , r)}

be a non-empty basic open set inQdefined by binomial inequalities, where 0=a_i, b_i∈R, and α_i, β_i∈M =Zⁿ(i= 1, . . . , r). An easy argument shows that the inequalities can be rewritten witha_i= 1 andβ_i= 0 for alli. For the following discussion we will therefore assume

S={ξ∈Q:ξ^γi < c_i (i= 1, . . . , r)}, whereγ_i∈M andc_i>0 (i= 1, . . . , r).

We use the notation introduced in2.5. Let v∈N. If γi, v>0 for alli, thenS(v) =Q. If γi, v0 for alli, thenS⊆S(v). Ifγi, v<0 for somei, thenS(v) =∅. So we see that

K(S) =K₀(S) =C_S^∗∩N,

whereC_S := cone(γ₁, . . . , γ_r)⊆M_RandC_S^∗⊆N_R is the dual cone ofC_S.

The next lemma contains the reason why condition (T C) can be met.

Lemma 2.14. Let ρ⊆N_R be a pointed cone satisfying S∩O_ρ(R)=∅. Then C_S^∗∩ relint(ρ)=∅.

Proof. We may work in the toric affine variety U_ρ=SpecR[Hρ] with H_ρ=M∩ρ^∗. Any point ξ∈O_ρ(R) satisfies ξ^γ = 0 for all γ∈H_ρ(−Hρ). Let us write τ:=−C_S^∗, so that H_τ :=M∩τ^∗ is the saturation inside M of the semi-group generated by−γ1, . . . ,−γr. Any β∈H_τ can be written in the formβ =−_r

i=1b_iγ_i with rational numbers b_i0. Therefore, there exists c >0 with ξ^β> c for all ξ∈S. Hence we have ξ^β c >0 for any ξ∈S, which implies β /∈Hρ(−Hρ). Thus Hτ∩Hρ⊆ −Hρ, or equivalently, by dualizing, −ρ⊆ρ+τ.

Choose anyu∈relint(ρ). There existsv∈ρwith−u∈v+τ, that is, withu+v∈ −τ=C_S^∗. This proves the lemma sinceu+v∈relint(ρ).

Corollary 2.15. LetΣbe a complete fan inN_R which is adapted toS. Then condition (T C)from2.9is satisfied.

Proof. Adapted simply means here thatC_S^∗ is a union of cones from Σ. The claim is clear from Lemma 2.14: if τ∈Σ(1) satisfies S∩Y_τ(R)=∅, then S∩O_ρ(R)=∅ for some ρ∈Σ containing τ. By Lemma 2.14, this implies C_S^∗∩relint(ρ)=∅. By adaptedness, this implies τ⊆C_S^∗.

We conclude that anS-compatible toric completion exists wheneverSis defined by binomial inequalities.

Corollary 2.16. Let σ be a pointed cone in N_R, and let S={ξ∈Q:ξ^γi < ci

(i= 1, . . . , r)} as before, considered as a subset of Uσ(R). The ring of polynomials on Uσ

that are bounded onS is given by

BUσ(S) =R[H], whereH =M∩σ^∗∩C_S.

2.17. A polynomial functionf ∈R[Uσ] is therefore bounded onS if and only if, for every monomialm occurring in f, some power ofm is a product of x^γ1, . . . ,x^γr. It is obvious that suchf is bounded onS; the content of Corollary 2.16is that no otherf is bounded onS. In particular, we see thatB_Uσ(S) =Rif and only ifσ+C_S^∗=N_R.

2.18. For a second class of examples, letU be a non-empty open semi-algebraic subset of Q={ξ∈T(R) = (R^∗)ⁿ:ξi>0 (i= 1, . . . , n)}, and letv∈N. We consider the open set

S:=Sv(U) :={λv(s)ξ:ξ∈U, 0< s1}

inQ, which we may call av-tentacle, following Netzer [10]. Multiplyingvby a positive integer does not changeS, therefore we may assume thatv is a primitive element ofN.

Lemma 2.19. Assume that U is relatively compact inQ. LetS=S_v(U)be the associated v-tentacle as above.

(a) K(S) =K₀(S) =Z+v.

(b) If{0} =ρ⊆N_R is a pointed cone withS∩Oρ(R)=∅,thenv∈relint(ρ).

Proof. (a) We obviously have U ⊆S(v), and therefore v∈K₀(S). Conversely let u∈N with S(u)=∅. So there is ξ∈Q such that λ_u(s)ξ∈S for all sufficiently small real s >0.

Thus, for any small s >0 there exist 0< t1 and η∈U with λu(s)ξ=λv(t)η. Assume u /∈R₊v. Then there exists α∈M with α, u>0>α, v. Evaluating the character x^α, we gets^α,uξ^α=t^α,vη^αη^α. The right-hand side is positive and bounded away from zero, since x^αdoes not approach zero onU. On the other hand, the left-hand side tends to zero fors→0.

This contradiction proves the claim.

(b) The proof is similar to that of Lemma 2.14. Again we may work in the affine toric variety U_ρ. Let γ∈H_ρ(−Hρ). For any ξ∈U_ρ(R) we have ξ^γ = 0. Since U is relatively compact, there existsc >1 with c⁻¹ξ^γ c for all ξ∈U. We have x^γ(λ_v(s)ξ) =s^α,v·ξ^γ fors >0, and we conclude α, v>0. Thus α, v>0 holds for every γ∈H_ρ(−Hρ). This means M ∩(−R+v)^∗∩ρ^∗⊆ −ρ^∗, or −ρ⊆ρ−R+v after dualizing. As before, this implies v∈relint(ρ).

Similarly to Proposition2.16, we deduce the following corollary.

Corollary 2.20. Let U =∅ be an open and relatively compact subset of Q, and let S=S_v(U)be the associatedv-tentacle. Letσbe a pointed cone inN_R. The ring of polynomials onU_σ that are bounded onS isB_Uσ(S) =R[H], whereH =M∩σ^∗∩(R+v)^∗.

2.21. Thus a polynomial functionf ∈R[Uσ] is bounded onSif and only if every monomial x^αoccurring inf satisfiesα, v0. In particular, B_Uσ(S) =Ris equivalent toσ+R₊v=N_R.

3. Iitaka dimension on toric surfaces

LetX be a non-singular projective surface over a fieldk. We always assume thatXis absolutely irreducible. We first discuss how the intersection matrix AD of an effective divisor D on X relates to the Iitaka dimensionκ(D) ofD. Sinceκ(D) is the transcendence degree ofO(XD), these facts have implications for rings of bounded polynomials on 2-dimensional semi-algebraic sets, by Theorem1.3. In general, the intersection matrix does not uniquely determineκ(D).

However, whenX is a toric surface and the divisorD is toric, we show thatκ(D) can be read off fromADin a simple manner (Proposition3.13).

3.1. Given two divisors D, D on X, we denote by D . D the intersection number ofD and D. The intersection pairing is invariant under linear equivalence and therefore induces a bilinear pairing on the divisor class groupP ic(X). As usual, we write D²:=D . D for the self-intersection number ofD.

Definition 3.2. Let D be an effective (not necessarily reduced) divisor on X whose distinct irreducible components are C₁, . . . , Cr. We define the intersection matrix of D to be the symmetricr×r matrix with integer entriesCi. Cj (i, j = 1, . . . , r); cf. [4, 8.3]. It will be denoted byAD.

3.3. LetD be an effective divisor onX. Form1 let φm:X |mD|be the rational map associated with the complete linear series |mD|. The Iitaka dimension of D is defined to be

κ(X, D) := max

m1dimφm(X);

see [4, Section 10.1] or [6, 2.1.3]. It is well known that κ(D, X) is equal to the transcendence degree of O(XD), the ring of regular functions on the open subvariety XD:=X\ supp(D) of X (see [4, Proposition 10.1]). The Iitaka dimension of D is closely related to the intersection matrixA_D.

Proposition 3.4. (a)IfADis negative definite,thenκ(X, D) = 0,that is,O(XD) =k.

(b) IfD²>0, thenκ(X, D) = 2.

Proof. (a) is [4, Proposition 8.5]; (b) is Lemma 8.5. Assertion (a) is also a consequence of Proposition3.6.

Corollary 3.5. IfO(XD) has transcendence degree 1, then AD is negative semi- definite.

In particular,κ(X, D) is determined by the Sylvester signature ofA_D wheneverA_D is non- singular.

Proof. Let C₁, . . . , Cr be the irreducible components of D. The intersection matrix AD

has non-negative off-diagonal entries. Therefore, if AD has a positive eigenvalue, there exist integersmi0 with (

imiCi)²>0. By Proposition3.4(b), this impliestrdegO(XD) = 2.

Part (a) of3.4can be generalized as follows.

Proposition 3.6. Let D⊆X be an effective divisor whose intersection matrix AD is negative definite. Then, for any line bundle Lon X, the space H⁰(XD,O(L))is a finite- dimensionalk-vector space.

For the proof we need two lemmas.

Lemma 3.7. Let D be an effective divisor with irreducible components C₁, . . . , Cr, and assumeCi. D <0 fori= 1, . . . , s. Then, for any divisorE there exists an integern₀=n₀(E) such that

|E+nD|= (n−n₀)D+|E+n₀D|

holds for allnn₀. Proof. SayD=_r

i=1miCi, withmi1. Choose an integernsuch that the inequality n(Ci. D)<−Ci.(E+a₁C₁+· · ·+asCs) (3.1) holds for i= 1, . . . , r and every tuple (a₁, . . . , a_r) with 0a_jm_j (j= 1, . . . , r). Then we claim

|E+ (n+ 1)D|=|E+nD|+D.

Indeed, ifa₁, . . . , ar are integers with 0aj mj (j= 1, . . . , r), we show

E+nD+

j

a_jC_j

=|E+nD|+

j

a_jC_j by induction on

jaj. The assertion is trivial for

jaj = 0. If (a₁, . . . , as)= (0, . . . ,0) is a tuple with 0a_j m_j, and ifiis an index witha_i1, we have

C_i.

⎛

⎝E+nD+

j

a_jC_j

⎞

⎠<0

by (3.1). Any effective divisor linearly equivalent toE+nD+

jajCjmust therefore contain Ci, which implies

E+nD+

j

ajCj

=

−Ci+E+nD+

j

ajCj

+Ci, and so

E+nD+

j

ajCj

=|E+nD|+

j

ajCj

by the inductive hypothesis.

Lemma 3.8. Ifx₁, . . . , xr is a linear basis of R^r, there exist integersm₁, . . . , mr1such thatx=

imixi satisfiesx, xi>0for alli= 1, . . . , r.

Proof. LetK be the convex cone spanned byx₁, . . . , x_r, and letK^∗={y∈R^r:x, y0}

be the dual cone. Sincex₁, . . . , x_rare a basis, bothKandK^∗have non-empty interior. We have to show that the interiors intersect. Assuming int(K)∩int(K^∗) =∅, there exists 0=z∈R^r withx, z0 for allx∈K andy, z0 for ally∈K^∗. Hence z∈K^∗∩(−K^∗∗) = (K^∗)∩ (−K), which implies z, z0, whence z= 0, which is a contradiction. Now interior(K)∩ interior(K^∗) is a non-empty open cone, hence it contains integer points with respect to the basisx₁, . . . , xr.

Proof of Proposition 3.6. Let C₁, . . . , Cr be the irreducible components of D, let U :=XD and let E be a divisor on X such that L∼=OX(E). Every section in Γ(U, L) is a meromorphic section ofLonX, which means that

Γ(U, L) =

n1

Γ(X, E+nD)

(ascending union). Since A_D is negative definite, we find integers m₁, . . . , m_r1 such that D:=_r

i=1m_iC_i satisfies C_i. D <0 (i= 1, . . . , r), using Lemma 3.8. By Lemma 3.7, there exists n₀1 such that |E+nD|= (n−n₀)D+|E+n₀D| for all nn₀, which means Γ(X, E+nD) = Γ(X, E+n₀D). Hence Γ(U, L) = Γ(X, E+n₀D), and so this space has finite dimension.

Remark 3.9. The hypothesis thatA_Dis negative definite in Proposition3.6entailsO(X D) =k. One may wonder whether 3.6 remains true if only O(XD) =k is assumed. An example due to Mondal and Netzer [9] shows that this usually fails. We will revisit their construction in Example4.4. On the other hand, we will see in3.13that such problems do not occur in a toric setting.

3.10. Let Σ be the fan of a non-singular projective toric surface X. For ρ∈Σ(1) let Yρ=Oρ. Let ρ₀, . . . , ρ_m−1, ρm=ρ₀ be the elements of Σ(1), written in cyclic order, so that ρ_i−1 and ρi bound a cone from Σ(2) for i= 1, . . . , m. Let vi be the primitive generator of ρi. The divisor class group of X is generated by the Yi=Yρi, and the intersection form on X has the following description (see [2, Section 10.4]): given 1i < m, there is an integer b_i such thatb_iv_i=v_i−1+v_i+1. Then we haveY_i²=−bi,Y_i. Y_j= 1 ifj−i=±1, andY_i. Y_j = 0 otherwise. Similarly fori= 0.

Lemma 3.11. Let n1 and let ρ₀, . . . , ρ_n+1 be a sequence of pairwise different cones in Σ(1) such that ρ_i−1 and ρi bound a cone from Σ(2) for i= 1, . . . , n+ 1. Let l=ρ₀∪(−ρ₀) and letAbe the intersection matrix of the divisor_n

i=1Y_ρi onX. Then:

(a) det(A) = 0 ⇔ ρ_n+1=−ρ₀;

(b) A≺0 ⇔ (−1)ⁿdet(A)>0 ⇔ ρ₁andρ_n+1lie(strictly)on the same side of the linel;

(c) Ais indefinite⇔ (−1)ⁿdet(A)<0 ⇔ ρ₁andρ_n+1lie(strictly)on opposite sides ofl.

In case(a)we haveA0 andrk(A) =n−1. In case(c)the matrixAhas a unique positive eigenvalue.

The hypothesis indicates that ρ₀, . . . , ρ_n+1 are given in cyclic order and that there exists no further cone from Σ(1) in between them. Sinceρ_n+1=ρ₀, there exists at least one cone in Σ(2) that is not of the form cone(ρ_i−1, ρ_i) with 1in+ 1.

Proof. Let v_i ∈N be the primitive vector generating ρ_i for i= 0, . . . , n+ 1. Let b₁, . . . , b_n ∈Zbe defined byv_i+1+v_i−1=b_iv_i (i= 1, . . . , n). Then

A=

⎛

⎜⎜

⎜⎜

⎜⎝

−b₁ 1 1 −b₂ 1

. .. ... . ..

1 −bn−1 1 1 −bn

⎞

⎟⎟

⎟⎟

⎟⎠.

Letδ(b₁, . . . , b_i) be the upper lefti×iprincipal minor ofA(i= 1, . . . , n). Then v_i+1= (−1)ⁱδ(b₁, . . . , b_i)v₁+ (−1)ⁱδ(b₂, . . . , b_i)v₀

holds fori= 1, . . . , n. In particular,

v_n+1= (−1)ⁿdet(A)v₁+ (−1)ⁿδ(b₂, . . . , b_n)v₀.

Since v₀, v₁ are linearly independent and v₀=v_n+1, the lemma follows easily from these identities.

3.12. We keep the previous hypotheses. LetT be a subset of Σ(1), letT = Σ(1)T and let

U =X

τ∈T

Y_τ,

an open toric subvariety of X. Let C= cone(τ :τ ∈T)⊆N_R; thenO(U) =k[M ∩C^∗], the semi-group algebra ofM∩C^∗. The following list exhausts all possible cases:

(1) C=N_R. Then |T|3 andO(U) =k.

(2) C is a half-plane. Then|T|3 andO(U)∼=k[u] (polynomial ring in one variable).

(3) C is a line. Then |T|= 2 and O(U)∼=k[u, u⁻¹] (ring of Laurent polynomials in one variable).

(4) C{0} is contained in an open half-plane. Then we havetrdegO(U) = 2.

The following result shows that, for toric divisors on non-singular toric surfaces, the Iitaka dimension is characterized by the signature of the intersection matrix.

Proposition 3.13. Let X be a non-singular toric projective surface with fan Σ. Let T ⊆Σ(1)with T=∅,letU =X

τ∈TY_τ and letAbe the intersection matrix ofT.