A Riemann–Kempf singularity theorem for higher rank Brill–Noether loci

AbstractLet (X, H) be a polarized K3 surface with $$\mathrm {Pic}(X) = \mathbb {Z}H$$ Pic ( X ) = Z H , and let $$C\in |H|$$ C ∈ | H | be a smooth curve of genus g. We give an upper bound on the dimension of global sections of a semistable vector bundle on C. This allows us to compute the higher rank Clifford indices of C with high genus. In particular, when $$g\ge r^2\ge 4$$ g ≥ r 2 ≥ 4 , the rank r Clifford index of C can be computed by the restriction of Lazarsfeld–Mukai bundles on X corresponding to line bundles on the curve C. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher rank vector bundles. We also apply the same method to the projective plane and show that the rank r Clifford index of a degree $$d(\ge 5)$$ d ( ≥ 5 ) smooth plane curve is $$d-4$$ d - 4 , which is the same as the Clifford index of the curve.

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Processing of novel food reveals payoff and rank-biased social learning in a wild primate

Scientific Reports ◽

10.1038/s41598-021-88857-6 ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Charlotte Canteloup ◽

Mabia B. Cera ◽

Brendan J. Barrett ◽

Erica van de Waal

Keyword(s):

Social Learning ◽

Learning Strategies ◽

Individual Learning ◽

Diffusion Experiment ◽

Vervet Monkeys ◽

Dynamic Learning ◽

Novel Food ◽

Bias Attention ◽

Higher Rank ◽

Learning From Others

AbstractSocial learning—learning from others—is the basis for behavioural traditions. Different social learning strategies (SLS), where individuals biasedly learn behaviours based on their content or who demonstrates them, may increase an individual’s fitness and generate behavioural traditions. While SLS have been mostly studied in isolation, their interaction and the interplay between individual and social learning is less understood. We performed a field-based open diffusion experiment in a wild primate. We provided two groups of vervet monkeys with a novel food, unshelled peanuts, and documented how three different peanut opening techniques spread within the groups. We analysed data using hierarchical Bayesian dynamic learning models that explore the integration of multiple SLS with individual learning. We (1) report evidence of social learning compared to strictly individual learning, (2) show that vervets preferentially socially learn the technique that yields the highest observed payoff and (3) also bias attention toward individuals of higher rank. This shows that behavioural preferences can arise when individuals integrate social information about the efficiency of a behaviour alongside cues related to the rank of a demonstrator. When these preferences converge to the same behaviour in a group, they may result in stable behavioural traditions.

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Higher rank FZZ-dualities

Journal of High Energy Physics ◽

10.1007/jhep02(2021)140 ◽

2021 ◽

Vol 2021 (2) ◽

Author(s):

Thomas Creutzig ◽

Yasuaki Hikida

Keyword(s):

Field Theory ◽

Reduction Method ◽

Operator Algebras ◽

The Self ◽

Liouville Theory ◽

Conformal Field ◽

Self Duality ◽

Higher Rank ◽

Corner Vertex ◽

Liouville Field Theory

Abstract We examine strong/weak dualities in two dimensional conformal field theories by generalizing the Fateev-Zamolodchikov-Zamolodchikov (FZZ-)duality between Witten’s cigar model described by the $$ \mathfrak{sl}(2)/\mathfrak{u}(1) $$ sl 2 / u 1 coset and sine-Liouville theory. In a previous work, a proof of the FZZ-duality was provided by applying the reduction method from $$ \mathfrak{sl}(2) $$ sl 2 Wess-Zumino-Novikov-Witten model to Liouville field theory and the self-duality of Liouville field theory. In this paper, we work with the coset model of the type $$ \mathfrak{sl}\left(N+1\right)/\left(\mathfrak{sl}(N)\times \mathfrak{u}(1)\right) $$ sl N + 1 / sl N × u 1 and investigate the equivalence to a theory with an $$ \mathfrak{sl}\left(N+\left.1\right|N\right) $$ sl N + 1 N structure. We derive the duality explicitly for N = 2, 3 by applying recent works on the reduction method extended for $$ \mathfrak{sl}(N) $$ sl N and the self-duality of Toda field theory. Our results can be regarded as a conformal field theoretic derivation of the duality of the Gaiotto-Rapčák corner vertex operator algebras Y0,N,N+1[ψ] and YN,0,N+1[ψ−1].

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