Rational Complexity-One -Varieties are Well-Poised
Given an affine rational complexity-one -variety , we construct an explicit embedding of in affine space . We show that this embedding is well-poised, that is, every initial ideal of is a prime ideal, and we determine the tropicalization . We then study valuations of the coordinate ring of which respect the torus action, showing that for full rank valuations, the natural generators of form a Khovanskii basis. This allows us to determine Newton-Okounkov bodies of rational projective complexity-one -varieties, partially recovering (and generalizing) results of Petersen. We apply our results to describe all integral special fibers of -equivariant degenerations of rational projective complexity-one -varieties, generalizing a result of Süß and the first author.
Let be an irreducible affine variety over a trivially valued, algebraically closed field . There is a corresponding presentation of the coordinate ring of by the polynomial algebra , where :
Here is the prime ideal of polynomials in which vanish on . We are concerned with the tropical geometry of the variety . We consider the tropical variety associated to the very affine variety ([MS15, Definition 3.2.1] and §4.1). We assume that . Recall that to each point there is an associated initial ideal . The purpose of this paper is in part to introduce the following property for affine embeddings:
The embedding of into is said to be well-poised if is a prime ideal for all .
Well-poised embeddings can tell us much about the geometry of . For example, when is a homogeneous ideal, all of the Gröbner degenerations associated to the tropical points of a well-poised embedding are reduced, irreducible varieties. In particular, a generic tropical point defines a (possibly non-normal) toric degeneration of .
Given the connection with toric geometry, it is not surprising that affine toric varieties themselves always have well-poised embeddings. It can be arranged that the coordinate ring of an affine toric variety be presented by a prime binomial ideal [CLS11, Proposition 1.1.9]. With respect to this embedding, is a rational vector space of dimension , and every initial ideal is itself.
A more exotic example of a well-poised embedding is provided by the Plücker embedding of the Grassmannian variety of -planes:
or rather, the embedding of the affine cone of this variety in . The tropical variety of this embedding is known as the tropical Grassmannian, and is described in the seminal paper [SS04] of Speyer and Sturmfels. The toric degenerations defined by the initial ideals of the tropical Grassmannian are also interesting objects. They have appeared in the study of integrable systems [NNU12] [HMM11], and invariant theory [HMSV09]. The variety is an instructive special case. This variety is cut out by a single equation among the six Plücker variables . There are three ways to drop a monomial from this equation in order to make an irreducible binomial; these correspond to the three maximal cones of the tropical variety .
We can generalize the description of the case to a special type of hypersurface. Consider a polynomial ring and any partition of . This defines an irreducible polynomial . Any initial form extracted from this polynomial is likewise irreducible, so it follows that the embedded hypersurface cut out by is well-poised. This last example is also interesting because when , the variety admits an effective action of an algebraic torus of dimension one less than .
In general, a complexity- -variety is a normal, irreducible variety equipped with an effective action by an algebraic torus such that . By considering the action of a subtorus, any complexity- -variety can be viewed as a complexity- -variety for . In this sense, complexity-one -varieties are a natural generalization of normal toric varieties, namely -varieties of complexity-zero. Our first theorem generalizes the observation that affine toric varieties possess well-poised embeddings to a larger class of -varieties:
Theorem 1.2 (Theorem 4.9).
Every affine rational complexity-one -variety has an equivariant embedding which is well-poised.
To prove this theorem, for any affine rational complexity-one -variety , we construct an explicit family of embeddings, which we call semi-canonical embeddings. Such an embedding is canonically determined by the geometry of up to equivariant isomorphism and the action of .
The main idea behind the semi-canonical embedding is to embed in such a fashion such that the geometry of becomes as simple as possible. Under this embedding , the intersection of with is simply the product of the torus with , where is a certain line embedded in projective space meeting the dense torus . Furthermore, the ideal describing a semi-canonical embedding can be determined from the quasi-combinatorial data describing the geometry of . We describe this embedding in detail in §2 and §3, and show in §4.2 that every semi-canonical embedding is in fact well-poised.
Since the geometry of is especially simple, so is that of : modulo lineality space, is just the tropical line . In particular, we are able to describe a tropical basis for , see Corollary 4.6.
When an embedding of is well-poised, work in [GRW16, Section 10] implies the existence of a section to the tropicalization map from the Berkovich skeleton of to its tropicalization. Our results here imply the existence of such a section over the tropicalization of any semi-canonical embedding. The presence of a large torus action gives this result a similar flavor to work of Draisma and Postinghel in [DP16].
In the second half of the paper, we study the relationship between semi-canonical embeddings, higher rank valuations, and the theory of Newton-Okounkov bodies. For any full rank valuation of the coordinate ring , its image is a semigroup . A set of generators of is a Khovanskii basis if their images generate , see §5.1. Our second main result shows the following:
Theorem 1.3 (Theorem 5.4).
Let be an affine rational complexity-one -variety, and a full rank valuation which is homogeneous with respect to the grading induced by the -action. Then has a semi-canonical embedding for which the corresponding generators of form a Khovanskii basis.
Our second main result has a number of consequences. We are able to give explicit descriptions of the value semigroups for homogeneous valuations (Corollary 5.5). It turns out that all such valuations can be constructed as weight valuations using the machinery of Kaveh and the second author, see §5.3, Theorem 5.8, and [KM, §4].
Newton-Okounkov bodies of complexity -varieties have been studied by Petersen in [Pet], where he gives a construction of Newton-Okounkov polytopes for any projective, complexity-one -variety with respect to valuations obtained from -invariant flags. As a corollary of our results mentioned above, we recover Petersen’s description in the case when is rational, and show that in fact, the Newton-Okounkov body with respect to any homogeneous valuation is of the form described by Petersen, see §5.5. Petersen also shows that the global Newton-Okounkov bodies for complexity-one -varieties are polyhedral; we easily recover this result again in the case of rational -varieties, see §5.6.
As a further application of our second main theorem, we determine all integral special fibers of -equivariant degenerations of rational projective complexity-one -varieties, see Theorem 6.1. In [IS17], Süß and the first author described all normal irreducible special fibers of such degenerations. Our theorem generalizes this result to allow for non-normal special fibers.
There is a large amount of literature devoted to studying the algebra of coordinate rings and ideals of toric varieties, for example, quadratic generation and Gröbner bases (e.g. [PRS98]), Koszulness (e.g. [BGT97]), and free resolutions (e.g. [Pee11]). We hope that the introduction of semi-canonical presentations for the coordinate rings of affine rational complexity-one -varieties will lead to a similar study of the algebra of such -varieties.
We now describe the structure of the remainer of this paper. In §2, we recall basics about -varieties, construct our semi-canonical embeddings for affine rational complexity-one -varieties, relate this to previous constructions, and discuss the projective case. We then precisely describe the ideal of a semi-canonical embedding in §3. In §4 we recall basics of Gröbner theory and tropical geometry, and then proceed to analyze the initial ideals arising for semi-canonical embeddings, proving our Theorem 1.2. In §5, we move on to a discussion of valuations on coordinate rings of rational complexity-one -varieties and their connections to semi-canonical embeddings. Finally, we apply our results in §6 to classify integral special fibers of -equivariant degenerations of rational projective complexity-one -varieties.
2. Semi-Canonical Embeddings
Let be an algebraic torus, with character lattice and co-character lattice . Recall that a -variety is a normal variety equipped with an effective action . The complexity of is . Such varieties may be described in terms of a ‘quotient’ variety of dimension equal to the complexity, equipped with some combinatorial data [AH06, AHS08, AIP12].
We briefly survey this correspondence for affine -varieties. Let be any normal variety, and fix a pointed, polyhedral cone in . A polyhedral divisor on with tailcone is a formal finite sum
where the are distinct prime divisors on , and the coefficients are either polyhedra in with tailcone , or the empty set.. Recall that the tailcone of a polyhedron is the set of all such that .
The polyhedral divisor induces a piecewise linear convex map
and if . We use this to construct a -scheme
Here an -coefficient in a divisor means that we allow poles with arbitrary order along the corresponding prime divisor. If one imposes certain positivity conditions on , is actually a -variety, and every affine -variety can be constructed in this fashion [AH06, Theorems 3.1 and 3.4]. Note that in [AH06], these theorems are only stated when the characteristic of is zero. However, the proofs of these theorems should go through essentially verbatim in positive characteristic as well. We are only interested in the complexity-one case, which is dealt with explicitly in [Lan15].
In this paper, we are primarily concerned with rational, complexity-one, affine -varieties. In this situation, we can take the quotient to be , and the positivity condition on is exactly that [AH06, Example 2.12]. In the following, we will show how to embed such a variety equivariantly in a particular toric variety. We will call this embedding a semi-canonical embedding of . The ambient toric variety is uniquely determined by up to equivariant isomorphism.
2.2. Affine Embeddings
Suppose that we are given a rational, complexity-one affine -variety , which we have described as in §2.1 in terms of a polyhedral divisor
on . As before, the have common pointed tailcone (or equal ), and . We will show how to embed into an affine toric variety.
From the above data, we construct a pointed polyhedral cone in :
where is the standard basis of , and . Note that is pointed, hence is full-dimensional.
For any line intersecting the torus , let be the restriction to of . We construct the following objects:
The functions are characters on the torus , and keep track of the multidegree of homogeneous elements of ; the functions are characters on the bigger torus and keep track of the multidegree of homogeneous elements of .
Our complexity-one -variety is equivariantly isomorphic to for an appropriate choice of . Indeed, choose any linear embedding of into such that the divisor on pulls back to the point . Taking to be the image of under this embedding, the pullback to of is exactly . It follows by construction that is equivariantly isomorphic to .
Concretely, such an embedding can be constructed as follows. Representing each point by a choice of homogeneous coordinates , we can embed into via
Note that different choices representing the points correspond to acting on by .
For any line , we will show that comes with a natural closed embedding in the toric variety . Let be the ideal of in . We define to be the ideal generated by
There is a natural exact sequence
for every , and hence an embedding of in with ideal .
Fix any rational affine complexity-one -variety . A semi-canonical embedding of is an affine embedding induced by the embedding , composed with an equivariant affine embedding of , where is any line in as in Remark 2.1.
Proof of Theorem 2.2.
It is straightforward to check that
We thus let the map be the map induced by restriction of sections. For degree , the degree piece is just
This is clearly surjective.
It remains to check that the kernel of this map is the degree part of . Note that if and only if
Now, if we take such an that also vanishes on , then clearly it restricts to zero on , so is contained in the kernel. On the other hand, sections
restricting to on are rational functions which vanish on . Furthermore, since is supported outside the torus, they are regular on . Hence, for such , as desired. ∎
Example 2.4 ( singularity).
We consider to be a normal surface singularity of type . This can be described by the polyhedral divisor
on , where
The cone is generated by the columns of
and the dual cone is generated by the columns of
A Hilbert basis for is given by the columns of
This means that the toric variety is cut out by a single binomial: . This corresponds to the linear relation between the above Hilbert basis elements.
Now, we can choose so that the ideal is generated by . To calculate , we check: for which are the columns of
in ? In this case, generates the ideal. Rewriting this in terms of variables corresponding to Hilbert basis elements, this becomes . Hence, is embedded in with ideal .
If all points are distinct, the construction of coincides with that of , where is the polyhedral divisor
on . In particular, such is normal.
On the other hand, if some points coincide, may no longer be normal. We say that the collection of polyhedra (with tailcone ) is admissible for if for every either
for some ; or
for every , is non-integral for at most one with .
If intersects the boundary of in exactly two points, then is a (potentially non-normal) toric variety. Indeed, after reordering suppose that coincide, as do . After identifying these points respectively with , it follows that is the semigroup algebra for the semigroup
It is a priori not obvious that this is a finitely generated semigroup. However, by our embedding in , we see not only that the semigroup is finitely generated, but are even given generators for it, namely, the restriction of generators for the semigroup of .
2.3. Similar Constructions
Our embedding of is related to two other constructions. The first is Altmann’s construction of so-called toric deformations [Alt95]. The setup is as follows: given a pointed polyhedral cone and an admissible collection of polyhedra with tailcone (see Remark 2.5), Altmann constructs an -parameter deformation of the affine toric variety whose cone of one-parameter subgroups is generated by and in . This is done by embedding as a relative complete intersection in the affine toric variety whose cone of one-parameter subgroups is generated by and in . The toric deformation of arises by perturbing the defining equations of in . It is straightforward to verify that Altmann’s embedding coincides with our embedding of in when we take as input , as above, and , .
The second related construction is the embedding of a rational complexity-one -variety in a toric variety obtained via their Cox rings. Hausen and Süß describe this embedding in [HS10, Corollary 5.2] for the case of complete A2 (e.g. projective) varieties, but it also makes sense for affine varieties with no non-constant invariant functions. We briefly recall this embedding and show that it coincides with our embedding of in .
Fix a rational complexity-one affine -variety described by the polyhedral divisor
on as in §2.1. The condition that have no non-constant invariant functions corresponds to the requirement that all coefficients of are non-empty. Represent each point by a choice of homogeneous coordinates . The Cox ring of has a presentation of the following form, see [HS10, Theorem 1.3, Corollary 4.9]:
where is the set of rays of such that , and is the set of vertices of . The ideal is generated by the minors of
and denotes the smallest integer such that is in . The -grading on is described by the exact sequence
where the map sends
Here, and are the corresponding basis elements, , and by abuse of notation we use to denote both a ray and its primitive lattice generator.
Taking the quotient of by the quasitorus , we recover . On the other hand, the quotient of the spectrum of the -graded polynomial ring by this quasitorus is an affine toric variety , see [CLS11, Chapter 5] for details. Since , we also obtain a closed embedding .
The embedding obtained from the Cox ring of is the same as the embedding described above.
We begin by considering the Cox ring of , which is a polynomial ring whose free generators correspond to rays of . By construction of , these rays are in bijection with elements of and elements of . Hence, the Cox ring of is naturally isomorphic to [CLS11, §5.1]. In fact, this isomorphism preserves the grading, since the grading on the Cox ring of is also induced by the map above. It follows that and are naturally isomorphic, since both arise as the same quotient of .
It remains to check that this isomorphism of and maps to . Since is integral, we only need to check this on the open torus. For this, it suffices to show that the image of in agrees with . But the ideal of in is generated by the minors of
for homogeneous coordinates on , and the inclusion of is induced by the map sending to . The claim then follows. ∎
2.4. Projective -Varieties
We are also interested in studying embeddings of projective -varieties. Let be a projective -variety, and any ample line bundle. Then the ring
is a finitely generated normal domain, and . Choosing homogeneous generators for as a -algebra of degrees leads to a presentation
and hence an embedding of in the weighted projective space .
On the other hand, is an affine -variety. If we assume that was rational and of complexity one, then so is , and we have a semi-canonical presentation of via the semi-canonical embedding of §2.2. Thus, after choosing the line bundle , we have a semi-canonical embedding of the polarized pair in some weighted projective space.
Example 2.8 (The projectivized cotangent bundle on ).
We consider , the projectivization of the cotangent bundle on . This is a Fano threefold often called , and equal to number 2.32 in the list of Mori and Mukai [MM86]. It is also isomorphic to the variety of complete flags in . This variety is equipped with a natural two-torus action.
On , the anticanonical class is divisible by two. We take to be half the anticanonical bundle and consider the polarised pair . Using the data found in [Süs14], one determines that is encoded by the polyhedral divisor
on , where the vertices of are , , the vertices of are , , the vertices of are , , and all coefficients have tailcone generated by the columns of
The distinguished -action on corresponds to the co-character .
The dual cone is generated by the columns of
These elements also form a Hilbert basis for the semigroup . Note that all elements have degree one with respect to our -grading, so we obtain an embedding of the Fano threefold in .
Concretely, the corresponding toric ideal is generated by the minors of
and we recognize that is simply the (cone over the) Segre embedding of . The variety is further cut out by the additional equation .
In general, the above construction produces a polarized pair in a (potentially weighted) projective space. If we wish to embed in a standard projective space, we may then pass to the th Veronese subring for appropriate choice of . It is straightforward to check that the construction of §2.2 commutes with passing to Veronese subrings.
This means that the construction of §2.2 applied to will yield a semi-canonical embedding of the polarized pair in projective space.
One might also be interested in the total coordinate or Cox ring of a rational, complexity-one projective -variety . See [ADHL15] for details on Cox rings.
The variety is itself a complexity-one affine -variety, see [AP12] for a description of the corresponding polyhedral divisor. As long as has only log terminal singularities, will also be rational, see [ABHW, Corollary 5.11]. In these cases, we may apply §2.2 to produce a semi-canonical presentation for the ring . See also §2.3 for further connections with Cox rings.
3. The Ideal of
In this section, we study the ideal of in the embedding constructed in §2.2. In particular, we determine explicitly its generators. This will be important in the subsequent §4.2, in which we show that each inital ideal corresponding to a point of is prime.
We continue using notation as in §2.2. To begin with, we have the following:
Let be the projection. Then is the ideal of the closure of in .
The ideal of in is generated by the image of in . The ideal of the closure in consists of those such that is regular on , that is, . Clearly, is contained in .
On the other hand, is in if and only if each -graded piece is. But is itself of the form for some , hence is contained in .
Geometrically, the above lemma means that the intersection is just the product of with the torus .
We are now interested in finding generators for . We set for .
Fix a multidegree . For any Laurent monomial , is in if and only if
Let be any set of generators for which are linear in the . The other ingredient we need is the polytope consisting of those defined by the inequalities
A set of lattice generators for as a -module is any set such that every lattice point of is a sum of an element of with a lattice point of .
A finite set of lattice generators for may be computed by considering a Hilbert basis for the cone
Selecting all elements of with final coordinate equal to one, and projecting back to leads to such a generating set .
By construction, we know that is generated by certain elements of the form , where is a rational function on and is a character of . We now specify precisely what form these elements and take:
The ideal is generated by the functions , where and .
Fix a multidegree . It follows from Remark 3.2 that the degree piece of is
The definition of is such that if and only if
are all in . In particular, for any , for each , since is linear in the .
We now need to show that these elements generate all of . Consider any homogeneous element in , which (by the above) we may write as , with . The polynomial is in . Using that the are linear and generate , we may write
with all .
Hence, we have reduced to showing that we can generate any element of the form such that , that is, . Now, any can be written as , where and . Thus, we can generate as
noting that is a regular function. ∎
4. Tropicalization and Gröbner Theory
Here we review the elements of Gröbner theory and tropical geometry necessary for our results. We recommend the books [CLO15, Stu96, MS15] as references. We will be working in the polynomial ring and the corresponding ring of Laurent polynomials .
Let and . The initial form of with respect to is
For an ideal in or , the initial ideal is the ideal generated by the set .
Let be an affine variety intersecting the torus with ideal , and the ideal of . The tropicalization of is the set of those