• Corpus ID: 119140473

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

@article{Boehm2016ComputingGW,
  title={Computing GIT-fans with symmetry and the Mori chamber decomposition of \$\bar\{M\}\_\{0,6\}\$},
  author={Janko Boehm and Simon Keicher and Yue Ren},
  journal={arXiv: Algebraic Geometry},
  year={2016}
}
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 the cone of movable divisors of $\bar{M}_{0,6}$. 
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References

SHOWING 1-10 OF 24 REFERENCES
Computing automorphisms of Mori dream spaces
TLDR
An algorithm to compute the automorphism group of a Mori dream space by studying graded automorphisms of affine algebras graded by a finitely generated abelian groups and applying the results to the Cox ring is presented.
Geometric invariant theory via Cox rings
Abstract We consider actions of reductive groups on a variety with finitely generated Cox ring, e.g., the classical case of a diagonal action on a product of projective spaces. Given such an action,…
Mori dream spaces and GIT.
The main goal of this paper is to study varieties with the best possible Mori theoretic properties (measured by the existence of a certain decomposition of the cone of effective divisors). We call…
The Cox ring of \overline{}_{0,6}
We prove that the Cox ring of the moduli space V 0,6 , of stable rational curves with 6 marked points, is finitely generated by sections corresponding to the boundary divisors and divisors which are…
Lower and upper bounds for nef cones
The nef cone of a projective variety Y is an important and often elusive invariant. In this paper we construct two polyhedral lower bounds and one polyhedral upper bound for the nef cone of Y using…
GIT-Equivalence beyond the ample cone
Given an algebraic torus action on a normal projective variety with finitely generated total coordinate ring, we study the GIT-equivalence for not necessarily ample linearized divisors, and we…
THE COX RING OF M0,6
We prove that the Cox ring of the moduli space M0,6, of stable rational curves with 6 marked points, is finitely generated by sections corresponding to the boundary divisors and divisors which are…
Geometric invariant theory and flips
We study the dependence of geometric invariant theory quotients on the choice of a linearization. We show that, in good cases, two such quotients are related by a flip in the sense of Mori, and…
Gröbner bases and convex polytopes
Grobner basics The state polytope Variation of term orders Toric ideals Enumeration, sampling and integer programming Primitive partition identities Universal Grobner bases Regular triangulations The…
Computing the GIT-Fan
  • S. Keicher
  • Mathematics, Computer Science
    Int. J. Algebra Comput.
  • 2012
TLDR
An algorithm to compute the GIT-fan of algebraic torus actions on affine varieties using LaSalle's inequality to solve the inequality of the following type: For α ≥ 1, α ≥ 2, β ≥ 3, etc.
...
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2
3
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