Modular Interpretation of a Non-Reductive Chow Quotient

Gallardo, Patricio; Giansiracusa, Noah

doi:10.1017/s0013091517000293

Cited by 3 publications

(3 citation statements)

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“…Since this H-action extends to the standard GL d -action, we can apply Question 6.2 and ask whether this non-reductive Chow quotient is isomorphic to the reductive Chow quotient GL d /H × (P d−1 ) n−d+1 // GL d . The following lemma describes GL d /H and the induced group actions and implies that this reductive Chow quotient is precisely the one appearing in our generalized Gelfand-MacPherson correspondence, the right side of Equation (4), and hence as claimed that the Krashen question is a specific instance of Question 6.2: The left vertical equality (up to normalization) here is [GG18], the right vertical equality is the following lemma together with the Gelfand-MacPherson isomorphism, the top horizontal equality is the Krashen question, and the bottom horizontal equality is a special instance of Question 6.2. Certainly the most natural projective completion to take for the space of full rank matrices is its Zariski closure in the space of all matrices, hence GL d /H = P Hom(k d−1 , k d ).…”

Section: The Borel Transfer Principle and The Chen-gibney-krashen Mod...mentioning

confidence: 92%

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Chow quotients of Grassmannians by diagonal subtori

Giansiracusa¹,

Wang²

2019

Preprint

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The literature on maximal torus orbits in the Grassmannian is vast; in this paper we initiate a program to extend this to diagonal subtori. Our main focus is generalizing portions of Kapranov's seminal work on Chow quotient compactifications of these orbit spaces. This leads naturally to discrete polymatroids, generalizing the matroidal framework underlying Kapranov's results. By generalizing the Gelfand-MacPherson isomorphism, these Chow quotients are seen to compactify spaces of arrangements of parameterized linear subspaces, and a generalized Gale duality holds here. A special case is birational to the Chen-Gibney-Krashen moduli space of pointed trees of projective spaces, and we show that the question of whether this birational map is an isomorphism is a specific instance of a much more general question that hasn't previously appeared in the literature, namely, whether the geometric Borel transfer principle in non-reductive GIT extends to an isomorphism of Chow quotients.

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Section: The Borel Transfer Principle and The Chen-gibney-krashen Mod...mentioning

confidence: 92%

“…On the other hand, the Chen-Gibney-Krashen moduli space T d−1,n−d+1 is a compactification of the same configuration space[CGK09], and Krashen's question is whether these are isomorphic. In[GG18] it is shown that T d−1,n−d+1 is isomorphic to the normalization of the Chow quotient (P d−1 ) n−d+1 // Ch H, whereH ∼ = G 2 m ⋊ G d−1 a…”

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confidence: 99%

Chow quotients of Grassmannians by diagonal subtori

Giansiracusa¹,

Wang²

2019

Preprint

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“…. , p n ) ∈ (P d ) n is automorphism-free (meaning that there is no non-trivial automorphism of P d fixing all the p i ), then p is strongly non degenerate (see also [GG18,Proposition 3.2]). Notice that the converse of this statement is false.…”

Section: The Veronese Compactification In Higher Dimensionsmentioning

confidence: 99%

Equations for point configurations to lie on a rational normal curve

Caminata

Giansiracusa

Moon

et al. 2018

Advances in Mathematics

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The parameter space of n ordered points in projective d-space that lie on a rational normal curve admits a natural compactification by taking the Zariski closure in (P d ) n . The resulting variety was used to study the birational geometry of the moduli space M 0,n of n-tuples of points in P 1 . In this paper we turn to a more classical question, first asked independently by both Speyer and Sturmfels: what are the defining equations? For conics, namely d = 2, we find scheme-theoretic equations revealing a determinantal structure and use this to prove some geometric properties; moreover, determining which subsets of these equations suffice set-theoretically is equivalent to a well-studied combinatorial problem. For twisted cubics, d = 3, we use the Gale transform to produce equations defining the union of two irreducible components, the compactified configuration space we want and the locus of degenerate point configurations, and we explain the challenges involved in eliminating this extra component. For d ≥ 4 we conjecture a similar situation and prove partial results in this direction.

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A generic slice of the moduli space of line arrangements

Ascher

Gallardo

2018

Alg. Number Th.

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We study the compactification of the locus parametrizing lines having a fixed intersection with a given line, inside the moduli space of line arrangements in the projective plane constructed for weight one by Hacking-Keel-Tevelev and Alexeev for general weights. We show that this space is smooth, with normal crossing boundary, and that it has a morphism to the moduli space of marked rational curves which can be understood as a natural continuation of the blow up construction of Kapranov. In addition, we prove that our space is isomorphic to a closed subvariety inside a non-reductive Chow quotient.arXiv:1602.08958v3 [math.AG] 19 Aug 2017 Definition and basic propertiesWe work only over C for convenience. We begin with the necessary background on the moduli space M w (P 2 , n + 1), see [HKT06] and [Ale13] for a full exposition.Configurations of (n + 1) labeled lines (l 1 , ..., l n+1 ) in P 2 up to projective equivalence are parametrized by the open moduli space M (P 2 , n + 1), which has a family of geometric compactifications M β (P 2 , n + 1) depending on a weight vector β := (β 1 , . . . , β n+1 ) (see [Ale13, Theorem 5.4.2]).The weight domain of possible weights β is

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Modular Interpretation of a Non-Reductive Chow Quotient

Cited by 3 publications

References 30 publications

Chow quotients of Grassmannians by diagonal subtori

Chow quotients of Grassmannians by diagonal subtori

Equations for point configurations to lie on a rational normal curve

A generic slice of the moduli space of line arrangements

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