Computing GIT-fans with symmetry and the Mori chamber decomposition of \overline{𝑀}_{0,6}

Abstract

We propose an algorithm to compute the GIT-fan for torus actions on affine varieties with symmetries. The algorithm combines computational techniques from commutative algebra, convex geometry and group theory. We have implemented our algorithm in the Singular library gitfan.lib. Using our implementation, we compute the Mori chamber decomposition of Mov(M 0,6). IntroductionDolgachev/Hu [10] and Thaddeus [18] assigned to an algebraic variety with the action of an algebraic group the GIT-fan, a polyhedral fan enumerating the GIT-quotients in the sense of Mumford [16]. The case of the action of an algebraic torus H on an affine variety X has been treated by Berchtold/Hausen [3]. Based on their construction, an algorithm to compute the GIT-fan in this setting has been proposed in [15]. Note that this setting is essential for many applications, since the torus case can be used to investigate the GIT-variation of the action of a connected reductive group G, see [2].In many important examples, X is symmetric under the action of a finite group which either is known directly from its geometry or can be computed, e.g., using [13]. A prominent instance is the Deligne-Mumford compactification M 0,6 of the moduli space of 6-pointed stable curves of genus zero, which has a natural action of the symmetric group S 6 . In this paper, we address two main problems:• to develop an efficient algorithm computing GIT-fans, which makes use of symmetries, and • to determine the Mori chamber decomposition of the cone of movable divisor classes of M 0,6 .We first describe an algorithm that determines the GIT-fan by computing exactly one representative in each orbit of maximal cones. Each cone is represented by a single integer. The algorithm relies on Gröbner basis techniques, convex geometry and actions of finite symmetry groups. It demonstrates the strength of cross-boarder methods in computer algebra, and the efficiency of the algorithms implemented in all involved areas. The algorithm is also suitable for parallel computations. We provide an implementation in the library gitfan.lib [6] for the computer algebra system Singular 1 [9]. The implementation is an interesting use case for the current efforts to connect different Open Source computer algebra systems, see [5, Sec. 2.4].We then turn to M 0,6 , which is known to be a Mori dream space, that is, its Cox ring Cox(M 0,6 ) is finitely generated, see [14]. Castravet [8] has determined generators for Cox(M 0,6 ) and Bernal Guillén [4] the relations as well as an explicit description of the symmetry group action. An interesting open problem is the computation of the Mori chamber decomposition of the cone of movable divisor classes Mov(M 0,6 ) ⊆ Eff(M 0,6 ), see [12] for a description of these cones in terms of generators. This fan is the decomposition of Mov(M 0,6 ) into chambers of the GIT-fan of the action of the characteristic torus on its total coordinate space; it characterizes the birational geometry of M 0,6 . In Section 6, we solve the mentioned problem and obtain the following r...

Massively Parallel Computations in Algebraic Geometry

“…It is important to remark that, in many practical examples, it is crucial to do the traversal modulo the action of a finite symmetry group. A detailed description of the approach can be found in [10]. The randomized nature of the execution of the Petri net results in a very well-behaved queue, which in many examples allows one to predict the total computation time and the size of the final output early in the computation, see [9].…”

“…this variation of GIT-quotients. In [12] a parallel algorithm for computing GIT-fans, which is also designed to make use of symmetries of the setup, has been described. The algorithm is based on symbolic methods from commutative algebra (Gröbner bases), convex geometry (double description) and group theory (orbit decomposition according to the finite symmetry group).…”

A software package to compute automorphisms of graded algebras

“…Then Aut H ( X) is obtained from G by choosing only those elements (A B , B, J B ) of the list stabExported where B ∈ Aut(Ω S ) fixes λ(w). In our library, you can compute it with (making use of gitfan.lib [5]) > intvec w = 1,9,16,0; // drawn in blue > setring R; // from before > def RR = autXhat(I, w, TOR); > setring RR;…”

“…Then Aut H ( X) is obtained from G by choosing only those elements (A B , B, J B ) of the list stabExported where B ∈ Aut(Ω S ) fixes λ(w). In our library, you can compute it with (making use of gitfan.lib [5])…”

“…We are interested in the automorphism group Aut K (R): it consists of all pairs (ϕ, ψ) such that ϕ : R → R is an automorphism of C-algebras, ψ : K → K is an automorphism of groups and ϕ(R w ) = R ψ(w) holds for all w ∈ K. Note that Aut K (R) not only is an important invariant of the algebra R, the methods to compute it can by applied to compute symmetries of homogenous ideals I. Once given explicitely, the knowledge of the latter largely accelerates further computations involving I, see [11,5,14] for examples. This note presents an implementation autgradalg.lib of the algorithms from [10] to compute Aut K (R).…”