Moduli of Weighted Hyperplane Arrangements

Alexeev, Valery

doi:10.1007/978-3-0348-0915-3

Cited by 26 publications

(35 citation statements)

References 35 publications

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“…His main example (P d ) n / / Ch SL d+1 provides a finer compactification than Mumford's of point configurations in P d , and in the case d = 1 this quotient is isomorphic to the ubiquitous Grothendieck-Knudsen compactification M 0,n [16,Theorem 4.1.8] (see also [11]). Thus, Kapranov's spaces (P d ) n / / Ch SL d+1 are seen as higher-dimensional generalizations of M 0,n ; they appear in a variety of settings (e.g., [1,13,20,25]).…”

Section: Background and Motivationmentioning

confidence: 99%

Modular Interpretation of a Non-Reductive Chow Quotient

Gallardo¹,

Giansiracusa²

2018

Proceedings of the Edinburgh Mathematical Society

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The space of n distinct points and a disjoint parameterized hyperplane in projective d-space up to projectivity-equivalently, configurations of n distinct points in affine d-space up to translation and homothety-has a beautiful compactification introduced by Chen-Gibney-Krashen. This variety, constructed inductively using the apparatus of Fulton-MacPherson configuration spaces, is a parameter space of certain pointed rational varieties whose dual intersection complex is a rooted tree. This generalizes M 0,n and shares many properties with it. In this paper, we prove that the normalization of the Chow quotient of (P d ) n by the diagonal action of the subgroup of projectivities fixing a hyperplane, pointwise, is isomorphic to this Chen-Gibney-Krashen space T d,n . This is a non-reductive analogue of Kapranov's famous quotient construction of M 0,n , and indeed as a special case we show that M 0,n is the Chow quotient of (P 1 ) n−1 by an action of G m G a . arXiv:1509.03608v2 [math.AG]

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Section: Background and Motivationmentioning

confidence: 99%

Modular Interpretation of a Non-Reductive Chow Quotient

Gallardo¹,

Giansiracusa²

2018

Proceedings of the Edinburgh Mathematical Society

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“…In what follows, we discuss the stable replacement of shas with multiple points which will be relevant for us (see [Ale13, Chapter 5] for a complete discussion).…”

Section: Definition Of R W (Q)mentioning

confidence: 99%

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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“…, 1) was constructed by Kapranov [Kap93] and Hacking-Keel-Tevelev [HKT06]. A compact moduli space M w (P d , n) that contains the moduli space of n hyperplanes with weights w was constructed by Alexeev [Ale13]. The space M w (P d , n) can be arbitrarily singular, and can contain many irreducible components when d ≥ 2.…”

Section: Comparing Pmentioning

confidence: 99%

“…The two shas must be parametrized by different components of B I because degenerations of the pair (X, D) correspond to pairs (X , D ) where X is a further degeneration of X. Then, the strict transform of H I cannot be in the closure of the loci parametrizing (X, D) (see [Ale13,Sec 4]).…”

Section: [I]mentioning

confidence: 99%

Wonderful compactifications of the moduli space of points in affine and projective space

Gallardo¹,

Routis

2017

European Journal of Mathematics

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ABSTRACT. We introduce and study smooth compactifications of the moduli space of n labeled points with weights in projective space, which have normal crossings boundary and are defined as GIT quotients of the weighted Fulton MacPherson compactification. We show that the GIT quotient of a wonderful compactification is also a wonderful compactification under certain hypotheses. We also study a weighted version of the configuration spaces parametrizing n points in affine space up to translation and homothety. In dimension one, the above compactifications are isomorphic to Hassett's moduli space of rational weighted stable curves.

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Moduli of Weighted Hyperplane Arrangements

Cited by 26 publications

References 35 publications

Modular Interpretation of a Non-Reductive Chow Quotient

Modular Interpretation of a Non-Reductive Chow Quotient

A generic slice of the moduli space of line arrangements

Wonderful compactifications of the moduli space of points in affine and projective space

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