Cross-ratio degrees and perfect matchings

Silversmith, Rob

doi:10.1090/proc/16016

Cited by 2 publications

(4 citation statements)

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“…The basic connection to trees via the boundary stratification (see Section 2) is long established. Enumerative questions have been of particular interest recently, including examining (as in this paper) many intersection products and structure constants on M g,n [CL21,Sil22], tautological relations [CJ18, PP21,Pix13], and Schubert calculus involving limit linear series [CP21,EH86]. Other topics of interest include the S n action on H * (M 0,n ) over C [BM13, Get95, RS22] and R [Rai06], Chern classes of vector bundles on M 0,n associated to sl r [DGT22,GKM02], explicit projective equations for M 0,n [MR17], and similar questions pertaining to a number of closely-related moduli spaces [CDH + 22, CLQ22, Fry19, LLV20, Sha19].…”

Section: Introductionmentioning

confidence: 99%

Lazy tournaments and multidegrees of a projective embedding of \(\overline{M}_{0,n}\)

Gillespie¹,

T.²,

Levinson³

2023

Combinatorial Theory

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We consider the (iterated) Kapranov embedding Ω n : M 0,n+3 → P 1 × • • • × P n , where M 0,n+3 is the moduli space of stable genus 0 curves with n + 3 marked points. In 2020, Gillespie, Cavalieri, and Monin gave a recursion satisfied by the multidegrees of Ω n and showed, using two combinatorial insertion algorithms on certain parking functions, that the total degree of Ω n is (2n − 1)!! = (2n − 1)In this paper, we give a new proof of this fact by enumerating each multidegree by a set of boundary points of M 0,n+3 , via an algorithm on trivalent trees that we call a lazy tournament. The advantages of this new interpretation are twofold: first, these sets project to one another under the forgetting maps used to derive the multidegree recursion. Second, these sets naturally partition the complete set of boundary points on M 0,n+2 , of which there are (2n − 1)!!, giving an immediate proof of the total degree formula.

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Section: Introductionmentioning

confidence: 99%

Lazy tournaments and multidegrees of a projective embedding of \(\overline{M}_{0,n}\)

Gillespie¹,

T.²,

Levinson³

2023

Combinatorial Theory

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“…The empty product is set to be 1. Then, we have that In the case of cross-ratio degrees, the upper bound in Theorem A(a) was proven in [Sil22] using Gromov-Witten theory. We prove Theorem A(a) in Section 3 by comparing the Kapranov degree with the intersection number of the pushforwards of the X S,i 's under the map f : M 0,n → (P 1 ) n−3 given by taking the cross-ratio of the points marked by {j, p, q, r} for j ∈ [n] \ {p, q, r}.…”

Section: Introductionmentioning

confidence: 93%

“…• When |S j | = 4 for all j, Kapranov degrees in this case were studied by Silversmith [Sil22] under the name of cross-ratio degrees, because the divisor class X S,i when |S| = 4 is the pullback of the hyperplane class under the map M 0,n → P 1 given by taking the cross-ratio of the four points marked by S. That is, the Kapranov degree in this case can be interpreted as the number of ways to choose n marked points on P 1 with n − 3 prescribed cross-ratios.…”

Section: Introductionmentioning

confidence: 99%

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Rhetorikikonographie

Brakensiek¹

Historisches Wörterbuch Der Rhetorik Online

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The moduli space of stable rational curves with marked points has two distinguished families of maps: the forgetful maps, given by forgetting some of the markings, and the Kapranov maps, given by complete linear series of ψ-classes. The collection of all these maps embeds the moduli space into a product of projective spaces. We call the multidegrees of this embedding "Kapranov degrees," which include as special cases the work of Witten, Silversmith, Castravet-Tevelev, Postnikov, Cavalieri-Gillespie-Monin, and Gillespie-Griffins-Levinson. We establish, in terms of a combinatorial matching condition, upper bounds for Kapranov degrees and a characterization of their positivity. The positivity characterization answers a question of Silversmith. We achieve this by proving a recursive formula for Kapranov degrees and by using tools from the theory of error correcting codes.

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Cross-ratio degrees and perfect matchings

Cited by 2 publications

References 12 publications

Lazy tournaments and multidegrees of a projective embedding of \(\overline{M}_{0,n}\)

Lazy tournaments and multidegrees of a projective embedding of \(\overline{M}_{0,n}\)

Rhetorikikonographie

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