Mustafin varieties

Cartwright, Dustin; Häbich, Mathias; Sturmfels, Bernd; Werner, Annette

doi:10.1007/s00029-011-0075-x

Cited by 23 publications

(37 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…In this section, we recall some of the relations between Bruhat-Tits buildings and tropical convexity. For a short summary of Bruhat-Tits buildings, we refer to Section 2 of [5], for more details on the relation between buildings and tropical convexity see e.g. [27].…”

Section: Bruhat-tits Buildings and Tropical Convexitymentioning

confidence: 99%

“…The following Lemma essentially goes back to [16], is a special case of Lemma 21 in [13] and can be found in this version as Lemma 4.1 in [5].…”

Section: Bruhat-tits Buildings and Tropical Convexitymentioning

confidence: 99%

“…In his original work, Mustafin studied the case of so-called convex subsets of B d as defined in Definition 2.12. An approach to study arbitrary subsets of B d was developed in [5], where the total space of this type of degenerations was named Mustafin variety for the first time. There it was proved that if the lattices in the subset of B d have diagonal form with respect to a common basis (i.e.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Mustafin varieties, moduli spaces and tropical geometry

Hahn

2020

manuscripta math.

View full text Add to dashboard Cite

Mustafin varieties are flat degenerations of projective spaces, induced by a set of lattices in a vector space over a non-archimedean field. They were introduced by Mustafin (Math USSR-Sbornik 34(2):187, 1978) in the 70s in order to generalise Mumford's groundbreaking work on the unformisation of curves to higher dimension. These varieties have a rich combinatorial structure as can be seen in pioneering work of Cartwright et al. (Selecta Math 17(4):757-793, 2011). In this paper, we introduce a new approach to Mustafin varieties in terms of images of rational maps, which were studied in Li (IMRN, 2017). Applying tropical intersection theory and tropical convex hull computations, we use this method to give a new combinatorial description of the irreducible components of the special fibers of Mustafin varieties. Finally, we outline a first application of our results in limit linear series theory.

show abstract

Section: Bruhat-tits Buildings and Tropical Convexitymentioning

confidence: 99%

“…The following Lemma essentially goes back to [16], is a special case of Lemma 21 in [13] and can be found in this version as Lemma 4.1 in [5].…”

Section: Bruhat-tits Buildings and Tropical Convexitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Mustafin varieties, moduli spaces and tropical geometry

Hahn

2020

manuscripta math.

View full text Add to dashboard Cite

show abstract

“…Let I be the ideal generated by polynomials {−3x 1 x 4 + 6x 3 x 4 + 3x 1 x 5 + 92x 2 x 5 + 2x 3 x 5 −23x 2 x 6 −2x 3 x 6 , x 1 x 8 + 7x 2 x 8 −4x 3 x 8 −6x 1 x 9 −3x 2 x 9 , x 4 x 8 + 3x 5 x 8 − 3x 6 x 8 − 24x 5 x 9 − 3x 6 x 9 , −x 2 x 4 − 4x 3 x 4 + x 2 x 5 + 4x 3 x 5 + 23x 2 x 6 + 2x 3 x 6 , −13x 1 x 7 − 4x 3 x 7 + 7x 2 x 8 + 28x 3 x 8 − 65x 1 x 9 − 3x 2 x 9 − 32x 3 x 9 , x 4 x 7 + 27x 5 x 7 − 9x 6 x 8 + 5x 4 x 9 + 135x 5 x 9 − 9x 6 x 9 , −4x 2 x 5 − 16x 3 x 5 + 3x 1 x 6 + x 2 x 6 − 2x 3 x 6 , 13x 2 x 7 − 8x 3 x 7 + x 2 x 8 + 4x 3 x 8 + 59x 2 x 9 − 64x 3 x 9 , 8x 5 x 7 + x 6 x 7 − 3x 6 x 8 + 40x 5 x 9 + 5x 6 x 9 , 4x 2 x 5 x 8 + 16x 3 x 5 x 8 + 20x 2 x 6 x 8 −10x 3 x 6 x 8 −24x 2 x 5 x 9 −96x 3 x 5 x 9 −3x 2 x 6 x 9 −12x 3 x 6 x 9 }. This is the general fiber of a Mustafin variety in the sense of [CHSW11]. Its special fiber is the initial ideal with respect to w = 0.…”

Section: Implementation Issuesmentioning

confidence: 99%

Gröbner bases over fields with valuations

Chan

Maclagan

2018

Math. Comp.

View full text Add to dashboard Cite

Let K be a field with a valuation and let S be the polynomial ring S := K[x 1 , . . . , x n ]. We discuss the extension of Gröbner theory to ideals in S, taking the valuations of coefficients into account, and describe the Buchberger algorithm in this context. In addition we discuss some implementation and complexity issues. The main motivation comes from tropical geometry, as tropical varieties can be defined using these Gröbner bases, but we also give examples showing that the resulting Gröbner bases can be substantially smaller than traditional Gröbner bases. In the case K = Q with the p-adic valuation the algorithms have been implemented in a Macaulay 2 package.

show abstract

“…The existence of such an isomorphism is well-known (see [Has03], [Li09], and [CHSW10]), and the claim about pull-backs of basis elements could be proven by a modular interpretation of the Kapranov basis, but a direct proof seems preferable.…”

Section: Background and Notationmentioning

confidence: 99%

Permutohedral spaces and the Cox ring of the moduli space of stable pointed rational curves

Larsen

2012

Geom Dedicata

View full text Add to dashboard Cite

We study the Cox ring of the moduli space of stable pointed rational curves, M 0,n , via the closely related permutohedral (or Losev-Manin) spaces L n−2 . Our main result establishes n 2 polynomial subrings of Cox(M 0,n ), thus giving collections of boundary variables that intersect the ideal of relations of Cox(M 0,n ) trivially. As applications, we give a combinatorial way to partially solve the Riemann-Roch problem for M 0,n , and we show that all relations in degrees of Cox(M 0,6 ) arising from certain pull-backs from projective spaces are generated by the Plücker relations.Date: October 29, 2018.Since for each n there correspond n 2 different permutohedral spaces according to a choice of poles (see Section 3), we obtain n 2 polynomial subrings of Cox(M 0,n ). An immediate corollary is that the corresponding subrings of the ring of generators meet the ideal of relations trivially. In addition, Theorem 1.1 provides a partial solution to the Riemann-Roch problem for M 0,n : for any divisor D with a nontrivial section in the image of f * above, the dimension of global sections h 0 (M 0,n , D) equals that of its preimage in L n−2 , which can be calculated by counting lattice points in a corresponding polytope.Theorem 1.1 follows from a detailed study of Kapranov's blow-up construction of M 0,n . The key technical result states that the pull-back and proper-transforms of the centers being blown up coincide, even for n ≥ 6 when the blow-up centers are not all disjoint (see Proposition 3.5).As an application, in Section 4 we exhibit degrees of Cox(M 0,n ) that always contain relations and show that, for n = 6, these relations are generated by Plücker relations. The proof follows similar lines to those used in [LV09] to prove the Batyrev-Popov conjecture that the Cox rings of del Pezzo surfaces (including M 0,5 ) are quadratic algebras. Theorem 1.1 provides the bridge between del Pezzo surfaces and M 0,n . We hope that this analogy can be extended to find a presentation of the Cox ring of M 0,6 .The remainder of the paper is organized as follows. In Section 2, we introduce the spaces M 0,n and L n−2 , present basic facts and notation from toric geometry, and then define the Cox ring of a projective variety. Section 3 contains the proof of Theorem 1.1, obtained by studying the blow-up constructions of L n−2 and M 0,n , with attention focused on how divisor classes in L n−2 pull back to M 0,n . Of particular use is the language of 'clean intersections.' Originally defined by Bott in [Bot56] in the context of differential geometry, we use the more algebraic formulation of [Li09]. Section 4 then contains the application to relations in Cox(M 0,6 ). Definition 3.12. For J ⊆ {1, . . . , n − 2}, 1 ≤ |J| ≤ n − 3, let ∆ ′ J∪{n} be the torus-invariant divisor V ( ρ J ≥0 ).If 1 ≤ |J| ≤ n − 4, then ∆ ′ J∪{n} = E ′ J , while for |J| = n − 3, ∆ ′ J∪{n} is the proper transform of the line containing all l J ′ ⊆ P n−3 , with J ′ J. As with boundary divisors in M 0,n , we will identify ∆ ′ J and ∆ ′ J c . Corollary 3.13. For every bo...

show abstract

Mustafin varieties

Cited by 23 publications

References 22 publications

Mustafin varieties, moduli spaces and tropical geometry

Mustafin varieties, moduli spaces and tropical geometry

Gröbner bases over fields with valuations

Permutohedral spaces and the Cox ring of the moduli space of stable pointed rational curves

Contact Info

Product

Resources

About