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The Weierstrass semigroup of pole orders of meromorphic functions in a point p of a smooth algebraic curve C is a classical object of study; a celebrated problem of Hurwitz is to characterize which semigroups S ⊂ N with finite complement are realizable as Weierstrass semigroups S = S (C , p) . In this note, we establish realizability results for cyclic covers π : (C , p) → (B , q) of hyperelliptic targets B marked in hyperelliptic Weierstrass points; and we show that realizability is dictated by the behavior under j-fold multiplication of certain divisor classes in hyperelliptic Jacobians naturally associated to our cyclic covers, as j ranges over all natural numbers. [ABSTRACT FROM AUTHOR]

A Richardson variety in a flag variety is an intersection of two Schubert varieties defined by transverse flags. We define and study relative Richardson varieties , which are defined over a base scheme with a vector bundle and two flags. To do so, we generalise transversality of flags to a relative notion, versality, that allows the flags to be non-transverse over some fibers. Relative Richardson varieties share many of the geometric properties of Richardson varieties. We generalise several geometric and cohomological facts about Richardson varieties to relative Richardson varieties. We also prove that the local geometry of a relative Richardson variety is governed, in a precise sense, by the two intersecting Schubert varieties, giving a generalisation, in the flag variety case, of a theorem of Knutson–Woo–Yong; we also generalise this result to intersections of arbitrarily many relative Schubert varieties. We give an application to Brill–Noether varieties on elliptic curves, and a conjectural generalisation to higher genus curves. [ABSTRACT FROM AUTHOR]

The moduli space G g , d r → M g parameterizing algebraic curves with a linear series of degree 푑 and rank 푟 has expected relative dimension ρ = g − (r + 1) ⁢ (g − d + r) . Classical Brill–Noether theory concerns the case ρ ≥ 0 ; we consider the non-surjective case ρ < 0 . We prove the existence of components of this moduli space with the expected relative dimension when 0 > ρ ≥ − g + 3 , or 0 > ρ ≥ − C r ⁢ g + O ⁢ (g 5 / 6) , where C r is a constant depending on the rank of the linear series such that C r → 3 as r → ∞ . These results are proved via a two-marked-point generalization suitable for inductive arguments, and the regeneration theorem for limit linear series. [ABSTRACT FROM AUTHOR]

We define a class of numerical semigroups S , which we call Castelnuovo semigroups, and study the subvariety M g , 1 S of M g , 1 consisting of marked smooth curves with Weierstrass semigroup S. We determine the number of irreducible components of these loci and determine their dimensions. Curves with these Weierstrass semigroups are always Castelnuovo curves, which provides the basic tool for our argument. This analysis provides examples of numerical semigroups for which M g , 1 S is reducible and non-equidimensional. [ABSTRACT FROM AUTHOR]

We prove an enumerative formula for the algebraic Euler characteristic of Brill-Noether varieties, parametrizing degree d and rank r linear series on a general genus g curve, with ramification profiles specified at up to two general points. Up to sign, this Euler characteristic is the number of standard set-valued tableaux of a certain skew shape with g labels. We use a flat degeneration via the Eisenbud-Harris theory of limit linear series, relying on moduli-theoretic advances of Osserman and Murray-Osserman; the count of set-valued tableaux is an explicit enumeration of strata of this degeneration. [ABSTRACT FROM AUTHOR]

We completely describe all Brill–Noether loci on metric graphs consisting of a chain of g cycles with arbitrary edge lengths, generalizing work of Cools, Draisma, Payne, and Robeva. The structure of these loci is determined by displacement tableaux on rectangular partitions, which we define. More generally, we fix a marked point on the rightmost cycle, and completely analyze the loci of divisor classes with specified ramification at the marked point, classifying them using displacement tableaux. Our results give a tropical proof of the generalized Brill–Noether theorem for general marked curves, and serve as a foundation for the analysis of general algebraic curves of fixed gonality. [ABSTRACT FROM AUTHOR]

We consider a general curve of fixed gonality k and genus g . We propose an estimate ρ ‾ g , k ( d , r ) for the dimension of the variety W d r ( C ) of special linear series on C , by solving an analogous problem in tropical geometry. Using work of Coppens and Martens, we prove that this estimate is exactly correct if k ≥ 1 5 g + 2 , and is an upper bound in all other cases. We also completely characterize the cases in which W d r ( C ) has the same dimension as for a general curve of genus g . [ABSTRACT FROM AUTHOR]

Abstract: We describe a graph reduction operation, generalizing three graph reduction operations related to gene assembly in ciliates. The graph formalization of gene assembly considers three reduction rules, called the positive rule, double rule, and negative rule, each of which removes one or two vertices from a graph. The graph reductions we define consist precisely of all compositions of these rules. We study graph reductions in terms of the adjacency matrix of a graph over the finite field , and show that they are path invariant, in the sense that the result of a sequence of graph reductions depends only on the vertices removed. The binary rank of a graph is the rank of its adjacency matrix over . We show that the binary rank of a graph determines how many times the negative rule is applied in any sequence of positive, double, and negative rules reducing the graph to the empty graph, resolving two open problems posed by Harju, Li, and Petre. We also demonstrate the close relationship between graph reductions and the matrix pivot operation, both of which can be studied in terms of the poset of subsets of vertices of a graph that can be removed by a graph reduction. [Copyright &y& Elsevier]

In this paper, we compute the genus of the variety of linear series of rank r and degree d on a general curve of genus g, with ramification at least α and β at two given points, when that variety is 1-dimensional. Our proof uses degenerations and limit linear series along with an analysis of random staircase paths in Young tableaux, and produces an explicit scheme-theoretic description of the limit linear series of fixed rank and degree on a generic chain of elliptic curves when that scheme is itself a curve. [ABSTRACT FROM AUTHOR]