Computing the GIT-fan

Computing the GIT-fan

Simon Keicher Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany keicher@mail.mathematik.uni-tuebingen.de
Abstract.

We present an algorithm to compute the GIT-fan of algebraic torus actions on affine varieties.

2000 Mathematics Subject Classification:
14Q99, 14L24

1. Introduction

Given an action of a connected reductive linear algebraic group  on an algebraic variety , Mumford constructed open -sets admitting a good quotient , see [20]. His construction depends on the choice of a -linearized ample line bundle on and, in general, one obtains several distinct quotients. This variation of GIT-quotients is described by a combinatorial structure, the so-called GIT-fan; see the work by Dolgachev/Hu [9] and Thaddeus [22] for ample bundles on a projective variety and [3] for the affine case.

In the present note, we provide an algorithm for computing the GIT-fan describing the quotients arising from the possible linearizations of the trivial bundle for the case that is affine and is a torus. Note that the torus case is essential for the general one: if a connected reductive group acts on , then the associated GIT-fan equals that of the action of the torus on the affine variety , where is a maximal connected semisimple subgroup, see [3]. Moreover, our setting also occurs in the context of Mori dream spaces: there the Neron-Severi torus acts on the total coordinate space and the GIT-fan of this action is precisely the Mori chamber decomposition of the effective cone, see [14].

Our algorithm is based on the construction of the GIT-fan provided in [6]. We assume that is given by concrete equations. The main computational steps are to determine the toric orbits of intersecting , see section 3, a suitable number of the so-called orbit cones of the -action on and the GIT-chamber of a given weight, see section 2. The GIT-fan is then obtained by traversing a spanning tree of its dual graph; this idea also shows up in the computation of Gröbner-fans, symmetric fans and tropical varieties as presented in [11, 15, 7]. We discuss some examples in section 4. At the moment, a Maple/convex [10] implementation of our algorithm is available [16].

The author would like to thank Jürgen Hausen for valuable discussions and comments and the referee for helpful suggestions.

2. Computing the GIT-fan

Throughout the whole note, is an algebraically closed field of characteristic zero. In this section, we first recall the necessary concepts from [6] and thereby fix our notation. Then we present and prove our algorithms for the GIT-fan. Aspects of efficiency of the algorithms are discussed at the end of this section.

We will work with the following description of the toric orbits of in terms of faces of the orthant : the standard torus acts via

Given a face , define the reduction of an -tuple , of e.g. numbers, along as

where denote the canonical basis vectors. Then, one has a bijection

Note, that in the notation of [12], is the -orbit through the distinguished point corresponding to the dual face .

Definition 2.1.

Let be an ideal. A face of the positive orthant is an -face if .

If is the zero set of the ideal , then the -faces correspond exactly to the -orbits intersecting nontrivially. The computation of -faces will be discussed in section 3.

We are ready to introduce GIT-chambers and the GIT-fan. Assume that the defining ideal of is monomial-free and homogeneous with respect to a -grading

Then the corresponding action of the torus on leaves the zero set invariant. Let be the matrix with columns . We assume that the cone is of dimension .

A projected -face is a cone with an -face. In [6] these are called orbit cones. Write for the set of all projected -faces.

Definition 2.2.

The GIT-chamber of a vector is the convex, polyhedral cone

The GIT-fan of the -action on is the set of all GIT-chambers.

As the name suggests, is indeed a fan in with as its support, see [2, Thm. III.1.2.8]. Note, however, that the cones of the GIT fan need not be pointed in general. The set of -dimensional cones of will be denoted by .

We turn to the computation of GIT-chambers. Let be the set of projected faces of and let be the subset of -dimensional cones. Similarly, is the subset of -dimensional projected -faces. We have

where the first containment is due to the fact that faces of projected -faces are again projected -faces, see [6, Cor. 2.4]. Given a vector in the relative interior , set for the collection of all that contain . The next algorithm determines the associated GIT-chamber .

Remark 2.3.
  1. The set is computed directly by taking cones over suitable subsets of .

  2. The computation of can be sped up via point location [18], i.e. we only consider cones with at least one generator lying on the same side as of a random hyperplane subdividing .

  3. For an efficient computation of , one reduces the amount of -face tests as follows. Check for any if some with is an -face. As soon as such a face has been found, all other faces projecting to may be ignored in subsequent tests.

Algorithm 2.4 (GIT-chamber).

Let be given. Assume that and are known.

  1. for each

  2. if and all facets of are in

  3. return

Lemma 2.5.

Let be a pure -dimensional fan with convex support and let be such that . Then is the intersection over all satisfying .

Lemma 2.6.

Let and . If then .

Proof.

Suppose . Choose and . Then lies on some facet . By construction, . Since holds, is not a GIT-chamber; a contradiction. ∎

Proof of Algorithm 2.4.

The algorithm terminates with a cone containing the given and our task is to show that holds. For this we establish

The first equality is due to the algorithm. The third one follows from Lemma 2.5. Moreover, in the middle one, the inclusion “” follows from . Thus we are left with verifying “” of the middle equality.

First suppose that is of full dimension. Then, for any with , we obtain , because holds. Lemma 2.6 shows . Thus, we obtain . The case of then follows from the observation that is the intersection over all fulldimensional chambers with , see Lemma 2.5. ∎

Working with -dimensional projected -faces in Algorithm 2.4 simplifies the necessary -face tests compared to the following naive variant of the algorithm using -dimensional ones.

Algorithm 2.7 (GIT-chamber, v2).

Let be given and assume that is known.

  1. for each

  2. if and there is an -face with

  3. return

The naive variant 2.7, in contrast, involves fewer convex geometric operations as 2.4 and thus can be more efficient if the latter ones are limiting the computation. See Remark 2.11 for a more concrete comparison of complexity aspects.

We turn to the GIT-fan. Given a full-dimensional cone , we denote by the set of all facets of that intersect the relative interior . Moreover, for two sets , we shortly write for . The following algorithm computes the set of maximal cones of the GIT-fan .

Algorithm 2.8 (GIT-fan).
  1. with a random full-dimensional GIT-chamber

  2. while there is

  3. Compute the full-dimensional GIT-chamber with

  4. return

Remark 2.9.

In the fourth line of the algorithm, let be the already found GIT-chamber with facet . Then can be calculated with Algorithm 2.4, where for some and with a suitably small . One possibly must reduce until .

Proof of Algorithm 2.8.

Write for the union over all and for the union over all . Then, in each passage of the loop, a full-dimensional chamber of is added to and, after adapting, is the boundary of with respect to . The set is empty if and only if equals . This shows that the algorithm terminates with the collection of maximal cones of as output. ∎

Note that Algorithm 2.8 traverses a spanning tree of the (implicitly known) dual graph of which has the maximal cones as its vertices and any two are connected by an edge if they share a common facet. Another traversal method for implicitly known graphs is reverse search by Avis and Fukuda [4], which also might be applied to our problem by the following observation.

Proposition 2.10.

The GIT-fan is the normal fan of a polyhedron. If , then can be enumerated using reverse search.

Proof.

The first statement is [1, Cor. 10.4]. The second claim follows from the first one and [11, Sec. 3]. ∎

Remark 2.11.

We compare the usage of Algorithm 2.4 (in 2.8) to that of 2.7. As a test, we compute the GIT-fans of the maximal torus action on the (affine cones over the) Grassmannians and , using a Maple/convex implementation. The following table lists the total number of -face tests and the total number of cones entering the fourth line of Algorithms 2.4 and 2.7

 

Algorithm 2.8 with 2.4 Algorithm  2.8 with 2.7
-face-tests cones -face-tests cones

 

Note that in Algorithm 2.4, the -face tests concern faces of lower dimension than in Algorithm 2.7 and thus are even faster.

Remark 2.12.
  1. Intermediate storage of occurring cones and their intersections in Algorithms 2.4 and 2.7 saves time.

  2. The traversal of the GIT-fan can take advantage of symmetries as explained in [15, Ch. 3.1].

3. Computing -faces

Let be the zero set of an ideal . Here we compute torus orbits of intersecting . In the notation of section 2, this means to determine the -faces of the orthant .

Given a face and a polynomial , we write where , i.e. we replace each with zero if . Let . A direct -face test is the following, based on a radical membership problem.

Remark 3.1.

A face is an -face if and only if .

This leads to a Gröbner based way to decide whether a given is an -face. The main aim of this section is speed up this direct approach by dividing out all possible torus symmetry. This is done in A lgorithm 3.5. Further possible improvements are discussed at the end of the section.

First consider any torus and an ideal . Let be the maximal subgroup leaving invariant and denote by the quotient map. To describe explicitly, we use the correspondence between integral matrices and homomorphisms of algebraic tori: every matrix defines a homomorphism by sending to where the are the rows of .

Remark 3.2.

The map is given by any matrix of full rank satisfying

where to we assign the matrix with rows .

Remark 3.3.

Fix a generating set of . Let be the stack matrix, i.e. the vertical concatenation, of . Compute the Hermite normal form with a unimodular matrix . Choose as the matrix consisting of the upper non-zero rows of . Then describes .

Proof.

Clearly, is of full rank. Since the exponent vectors of each are linear combinations of the exponent vectors of , we have

A push forward of under is a satisfying for some monomial ; we simply write for any such and

Remark 3.4.

Let be as in 3.2 and let . Compute a Smith normal form with unimodular matrices . Define where is a rational section for and is as in 3.2. Then there is such that

Proof.

Let and be the maps of tori defined by the matrices and . Clearly, for and some . Each is -homogeneous. This implies , so there is a unique integral matrix such that . In particular, is integral. Therefore, . ∎

We now specialize to the case of -face-verification. Given , let be the maximal subgroup leaving invariant. Our approach reduces the dimension of the problem by using

Algorithm 3.5 (-face verification).

Let be an ideal and let . Set and .

  1. Use 3.3 to compute a matrix representing

  2. Apply 3.4 to to obtain

  3. if

  4. return false

  5. return true

Proof.

The map is a good quotient for the -action on . Consequently, we have

by standard properties of good quotients [17, p. 96]. This shows that if and only if . ∎

Remark 3.6.

If the total number of terms occurring among the generators is low compared to the number of variables in the sense that in the first line of Algorithm 3.5, then we might speed up the algorithm using linear algebra as follows. Each term is linear by construction. Solve the linear system of equations . Then is an -face if and only if there is a solution in .

Let us briefly recall the connection to tropical geometry, compare e.g. [7]. Given a monomial-free ideal , its tropical variety is

where is the support of the codimension one skeleton of the normal fan of the Newton polytope of . By [21],

(1)

Fixing a fan structure on , this can be turned into a computable criterion. Note however that usually carries more information than needed to determine the -faces and is in general harder to compute (see [7] for an algorithm).

Remark 3.7.

To compute all -faces, the number of calls to Algorithm 3.5 can be reduced by any of the following ideas.

  1. The tropical prevariety of a generating set of is the coarsest common refinement where is the one-codimensional skeleton of the normal fan of the Newton polytope of . Then each face whose dual face does not satisfy equation (1) w.r.t is not an -face.

  2. A face is not an -face if and only if there is such that exactly one vertex of the newton polytope of lies in ; also compare [5, Prop. 9.3]. Choosing any subset of , we may identify some faces that are no -faces.

  3. Veronese embedding: Let be such that there are (classically) homogeneous generators of of degree . The images of the under

    give a linear system of equations with coefficient matrix . If a Gauss-Jordan normal form of contains a row with exactly one non-zero entry, is no -face. Adding redundant generators to refines this procedure.

  4. Let be a permutation of (the indices of) the variables that keeps the set of generators of invariant. Then

    Some of those permutations can be computed by assigning a both edge- and vertex-colored graph to the generators of and computing its automorphism group, e.g. using [19].

Remark 3.8.

The efficiency of Algorithm 3.5 depends on the algorithms used for both Gröbner bases and Smith normal forms. An implementation using the respective built in functions of Maple gave the following timings.

 

remark 3.1 algorithm 3.5 with 3.7(2)
-faces of  s  s
-faces of  min  s
-faces of days  h
-faces of  h  min

 

There, stands for the respective Plücker ideal and denotes the defining ideal of the Cox ring of the space of complete rank two collineations [13, Thm. 1].

4. Examples

We consider torus actions on the affine cone over the Grassmannian induced by a diagonal action on the Plücker coordinate space , where . Such actions will be encoded by assigning the variable the -th column of a matrix . Moreover, we write for the Plücker ideal.

We compute both, the GIT-fan of the torus action on as well as the GIT fan of the ambient space . The latter coincides with the so-called Gelfand Kapranov Zelevinsky decomposition , i.e. the coarsest common refinement of all normal fans having their rays among the cones over the columns of . In general, the Gelfand Kapranov Zelevinsky decomposition is a refinement of the GIT-fan. See [8] for a toric background.

Below, the drawings show (projections of) the intersections of the respective fans with the standard simplex.

Example 4.1.

For , the ideal is homogeneous with respect to

Using Algorithm 2.8, we obtain the four maximal GIT-chambers of . The finer fan has twelve maximal cones.

Example 4.2.

For , the ideal is homogeneous with respect to

By Algorithm 2.8, there are twelve four-dimensional cones in whereas contains such cones.

Example 4.3.

For , the ideal is homogeneous with respect to

Using Algorithm 2.8, we obtain the five-dimensional cones of . The fan has such cones.

References

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  • [2] I. Arzhantsev, U. Derenthal, J. Hausen, A. Laface: Cox rings. Preprint, arXiv:1003.4229; see also the authors’ webpages.
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  • [7] T. Bogart, A. Jensen, D. Speyer, B. Sturmfels, R. Thomas: Computing tropical varieties. Journal of Symbolic Computation 42(1-2) (2007), 54–73.
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  • [10] M. Franz: Convex – a Maple package for convex geometry. (Available at http://www.math.uwo.ca/ mfranz/convex/)
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  • [22] M. Thaddeus: Geometric Invariant Theory and Flips. J. Amer. Math. Soc. 9(3) (1996), 691–723.
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