Genera of Brill-Noether curves and staircase paths in Young tableaux

Chan, Melody; Martín, Alberto López; Pflueger, Nathan; Bigas, Montserrat Teixidor i

doi:10.1090/tran/7044

Cited by 31 publications

(66 citation statements)

References 29 publications

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“…It was first discovered in the context of the algebraic geometry of curves: establishing that the distributive lattice of order ideals of the rectangle is CDE was the key combinatorial result that Chan, López Martín, Pflueger, and Teixidor i Bigas [13] needed to reprove a product formula for the genus of one-dimensional Brill-Noether loci. Later, Chan, Haddadan, Hopkins, and Moci [12] generalized the work of [13] by introducing the "toggle perspective." The toggle perspective, based on the notion of "toggling" elements into and out of order ideals, is the principal tool we have for establishing that posets are CDE.…”

Section: Introductionmentioning

confidence: 99%

Minuscule Doppelgängers, The Coincidental down-Degree Expectations Property, and Rowmotion

Hopkins

2020

Experimental Mathematics

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We relate Reiner, Tenner, and Yong's coincidental down-degree expectations (CDE) property of posets to the minuscule doppelgänger pairs studied by Hamaker, Patrias, Pechenik, and Williams. Via this relation, we put forward a series of conjectures which suggest that the minuscule doppelgänger pairs behave "as if" they had isomorphic comparability graphs, even though they do not. We further explore the idea of minuscule doppelgänger pairs pretending to have isomorphic comparability graphs by considering the rowmotion operator on order ideals. We conjecture that the members of a minuscule doppelgänger pair behave the same way under rowmotion, as they would if they had isomorphic comparability graphs. Moreover, we conjecture that these pairs continue to behave the same way under the piecewise-linear and birational liftings of rowmotion introduced by Einstein and Propp. This conjecture motivates us to study the homomesies (in the sense of Propp and Roby) exhibited by birational rowmotion. We establish the birational analog of the antichain cardinality homomesy for the major examples of posets known or conjectured to have finite birational rowmotion order (namely: minuscule posets and root posets of coincidental type).

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Section: Introductionmentioning

confidence: 99%

Minuscule Doppelgängers, The Coincidental down-Degree Expectations Property, and Rowmotion

Hopkins

2020

Experimental Mathematics

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“…Thus, we are in particular giving a new proof of the Gieseker-Petri theorem (in characteristic 0). As an immediate consequence of Theorem 1.4, the twice-pointed Brill-Noether curves studied in [4] as well as twice-pointed Brill-Noether surfaces [1,6] are smooth. The singular locus of G r d (X, (P, a • ), (Q, b • )) can also be described using hook removals in Young diagrams; see Remark 3.4.…”

Section: Introductionmentioning

confidence: 84%

The Gieseker–Petri theorem and imposed ramification

Chan

Osserman

Pflueger

2019

Bull. London Math. Soc.

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We prove a smoothness result for spaces of linear series with prescribed ramification on twice‐marked elliptic curves. In characteristic 0, we then apply the Eisenbud–Harris theory of limit linear series to deduce a new proof of the Gieseker–Petri theorem, along with a generalization to spaces of linear series with prescribed ramification at up to two points. Our main calculation involves the intersection of two Schubert cycles in a Grassmannian associated to almost‐transverse flags.

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“…We assume that the square just to the right of the first row and the square just below the first column are also outside corners. It should be noticed that the outside corners of µ are also called outer boxes of µ, see Stanley The notion of left turns and right turns of a lattice path in λ was introduced by Chan, López Martín, Pflueger and Teixidor i Bigas [2] for the computation of the genera of the Brill-Noether curves. To be more specific, a left turn of a lattice path in λ is an east step immediately followed by a north step, and a right turn is a north step immediately followed by an east step with the additional requirement that these two consecutive steps are borders of a square of λ.…”

Section: The Formula Of Chan-haddadan-hopkins-mocimentioning

confidence: 99%

Proof of a Conjecture of Reiner--Tenner--Yong on Barely Set-Valued Tableaux

Fan¹,

Guo²,

Sun³

2019

SIAM J. Discrete Math.

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The notion of a barely set-valued semistandard Young tableau was introduced by Reiner, Tenner and Yong in their study of the probability distribution of edges in the Young lattice of partitions. Given a partition λ and a positive integer k, let BSSYT(λ, k) (respectively, SYT(λ, k)) denote the set of barely setvalued semistandard Young tableaux (respectively, ordinary semistandard Young tableaux) of shape λ with entries in row i not exceeding k + i. In the case when λ is a rectangular staircase partition δ d (b a ), Reiner, Tenner and Yong conjectured that |BSSYT(λ, k)| = kab(d−1) (a+b) |SYT(λ, k)|. In this paper, we establish a connection between barely set-valued tableaux and reverse plane partitions with designated corners. We show that for any shape λ, the expected jaggedness of a subshape of λ under the weak probability distribution can be expressed as 2|BSSYT(λ,k)| k|SYT(λ,k)| . On the other hand, when λ is a balanced shape with r rows and c columns, Chan, Haddadan, Hopkins and Moci proved that the expected jaggedness of a subshape in λ under the weak distribution equals 2rc/(r + c). Hence, for a balanced shape λ with r rows and c columns, we establish the relation that |BSSYT(λ, k)| = krc (r+c) |SYT(λ, k)|. Since a rectangular staircase shape δ d (b a ) is a balanced shape, we confirm the conjecture of Reiner, Tenner and Yong.

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Genera of Brill-Noether curves and staircase paths in Young tableaux

Cited by 31 publications

References 29 publications

Minuscule Doppelgängers, The Coincidental down-Degree Expectations Property, and Rowmotion

Minuscule Doppelgängers, The Coincidental down-Degree Expectations Property, and Rowmotion

The Gieseker–Petri theorem and imposed ramification

Proof of a Conjecture of Reiner--Tenner--Yong on Barely Set-Valued Tableaux

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