Computing automorphisms of Mori dream spaces

Hausen, Juergen; Keicher, Simon; Wolf, Rüdiger

doi:10.48550/arxiv.1511.05059

Cited by 1 publication

(3 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 4.2. Symmetries of a homogeneous ideal as in Definition 4.1 can be computed with the methods of [13].…”

Section: Computing Git-fans With Symmetrymentioning

confidence: 99%

“…and the i-th column of the matrix Q is the degree deg(T i ) ∈ Z 5 ; this determines the Z 5grading of R. Using, e.g., [13,Example 5.5], we observe that there is an S 5 -symmetry for the H ∼ = (K * ) 5 -action on V (a) where the symmetry group S 5 ∼ = G ⊆ S 10 is generated by (2, 3)(5, 6)(9, 10), (1, 5, 9, 10, 3)(2, 7, 8, 4, 6) ∈ S 10 .…”

Section: Examplesmentioning

confidence: 99%

“…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 .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

2020

Self Cite

View full text Add to dashboard Cite

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...

show abstract

“…Remark 4.2. Symmetries of a homogeneous ideal as in Definition 4.1 can be computed with the methods of [13].…”

Section: Computing Git-fans With Symmetrymentioning

confidence: 99%

Section: Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation