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We analyze a family of graphs known as banana graphs, with two marked vertices, through the lens of Hurwitz-Brill-Noether theory. As an application, we construct explicit new examples of finite graphs which are Brill-Noether general. These are the first such examples since the analysis of chains of loops by Cools, Draisma, Payne and Robeva. The graphs constructed are chains of loops and "theta graphs," which are banana graphs of genus 2. We also demonstrate that almost all banana graphs of genus at least 3 cannot be used for this purpose, due either to failure of a submodularity condition or to the presence of far too many inversions in certain permutations associated to divisors called transmission permutations., Comment: 34 pages. Preliminary version, comments welcome

Coxeter groups possess an associative operation, called variously the Demazure, greedy, or $0$-Hecke product. For symmetric groups, this product has an amusing formulation as matrix multiplication in the min-plus (tropical) semiring of two matrices associated to the permutations. We prove that this min-plus formulation extends to furnish a Demazure product on a much larger group of integer permutations, consisting of all permutations that change the sign of finite many integers. We prove several alternative descriptions of this product and some useful properties of it. These results were developed in service of future applications to Brill-Noether theory of algebraic and tropical curves; the connection is surveyed in an appendix., Comment: 26 pages. Preliminary version; comments welcome

This paper gives a novel and compact proof that a metric graph consisting of a chain of loops of torsion order $0$ is Brill-Noether general (a theorem of Cools-Draisma-Payne-Robeva), and a finite or metric graph consisting of a chain of loops of torsion order $k$ is Hurwitz-Brill-Noether general in the sense of splitting loci (a theorem of Cook-Powell-Jensen). In fact, we prove a generalization to (metric) graphs with two marked points, that behaves well under vertex gluing. The key construction is a way to associate permutations to divisors on twice-marked graphs, simultaneously encoding the ranks of every twist of the divisor by the marked points. Vertex gluing corresponds to the Demazure product, which can be formulated via tropical matrix multiplication., Comment: 14 pages. Comments welcome. v2: Added examples and exposition; minor corrections

The moduli space $\mathcal{G}^r_{g,d} \to \mathcal{M}_g$ parameterizing algebraic curves with a linear series of degree $d$ and rank $r$ has expected relative dimension $\rho = g - (r+1)(g-d+r)$. Classical Brill-Noether theory concerns the case $\rho \geq 0$; we consider the non-surjective case $\rho < 0$. We prove the existence of components of this moduli space with the expected relative dimension when $0 > \rho \geq -g+3$, or $0 > \rho \geq -C_r g + \mathcal{O}(g^{5/6})$, where $C_r$ is a constant depending on the rank of the linear series such that $C_r \to 3$ as $r \to \infty$. These results are proved via a two-marked-point generalization suitable for inductive arguments, and the regeneration theorem for limit linear series., Comment: 27 pages. v2: added Appendix A, on connections to Hurwitz-Brill-Noether theory, and made minor corrections

The {\it 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 ${\rm S} \subset \mathbb{N}$ with finite complement are {\it realizable} as Weierstrass semigroups ${\rm S}= {\rm S}(C,p)$. In this note, we establish realizability results for cyclic covers $\pi: (C,p) \rightarrow (B,q)$ 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., Comment: 30 pages. Final version, to appear in Bulletin of the Brazilian Mathematical Society

A Brill-Noether degeneracy locus is closure in $\Pic^d(C)$ of the locus of line bundles with a specified rank function $r(a,b) = h^0(C,L(-ap-bq))$. These loci generalize the classical Brill-Noether loci $W^r_d(C)$ as well as Brill-Noether loci with imposed ramification. For general $(C,p,q)$ we determine the dimension, singular locus, and intersection class of Brill-Noether degeneracy loci, generalizing classical results about $W^r_d(C)$. The intersection class has a combinatorial interpretation in terms of the number of reduced words for a permutation associated to the rank function, or alternatively the number of saturated chains in the Bruhat order. The essential tool is a versality theorem for a certain pair of flags on $\Pic^d(C)$, conjectured by Melody Chan and the author., Comment: 20 pages. v3: proofs and exposition condensed, minor corrections. To appear in Algebraic Geometry

We give a combinatorial expansion of the stable Grothendieck polynomials of skew Young diagrams in terms of skew Schur functions, using a new row insertion algorithm for set-valued semistandard tableaux of skew shape. This expansion unifies some previous results: it generalizes a combinatorial formula obtained in earlier joint work with L\'opez Mart\'in and Teixidor i Bigas concerning Brill-Noether curves, and it generalizes a 2000 formula of Lenart and a recent result of Reiner-Tenner-Yong to skew shapes. We also give an expansion in the other direction: expressing skew Schur functions in terms of skew Grothendieck polynomials., Comment: 13 pages, to appear in Algebraic Combinatorics. arXiv admin note: substantial text overlap with arXiv:1708.09378

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 generalize 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 generalize 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 generalization, in the flag variety case, of a theorem of Knutson-Woo-Yong; we also generalize this result to intersections of arbitrarily many relative Schubert varieties. We give an application to Brill-Noether varieties on elliptic curves, and a conjectural generalization to higher genus curves., Comment: 21 pages. v3: minor improvements and corrections

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., Comment: 17 pages, revised version published in Bull. London Math. Soc

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., Comment: 18 pages, 2 figures. v3: minor corrections and updated references. Final version published in Transactions of the AMS

We define a class of numerical semigroups S, which we call Castelnuovo semigroups, and study the subvariety $M^S_{g,1}$ 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 $M^S_{g,1}$ is reducible and non-equidimensional., Comment: 18 pages. v3: minor typographical corrections

We give an upper bound on the codimension in $M_{g,1}$ of the variety $M^S_{g,1}$ of marked curves $(C,p)$ with a given Weierstrass semigroup. The bound is a combinatorial quantity which we call the effective weight of the semigroup; it is a refinement of the weight of the semigroup, and differs from it precisely when the semigroup is not primitive. We prove that whenever the effective weight is less than g, the variety $M^S_{g,1}$ is nonempty and has a component of the predicted codimension. These results extend previous results of Eisenbud, Harris, and Komeda to the case of non-primitive semigroups. We also survey other cases where the codimension of $M^S_{g,1}$ is known, as evidence that the effective weight estimate is correct in much wider circumstances., Comment: 28 pages. v2: minor typographical corrections and expository improvements

We consider a general curve of fixed gonality k and genus g. We propose an estimate for the dimension of the variety $W^r_d(C)$ 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 is at least g/5 + 2, and is an upper bound in all other cases. We also completely characterize the cases in which $W^r_d(C)$ has the same dimension as for a general curve of genus g., Comment: 17 pages. v2: statement of Theorem 1.2 improved; various typographical corrections

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., Comment: 25 pages. v2: various corrections and expository improvements

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 $\alpha$ and $\beta$ 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., Comment: 36 pages. v2: Re-organized and improved exposition. To appear in Transactions of the AMS

We study smooth tropical plane quartic curves and show that they satisfy certain properties analogous to (but also different from) smooth plane quartics in algebraic geometry. For example, we show that every such curve admits either infinitely many or exactly 7 bitangent lines. We also prove that a smooth tropical plane quartic curve cannot be hyperelliptic., Comment: 13 pages, 9 figures. Minor revisions; accepted for publication in Mathematische Zeitschrift

Brill-Noether theory studies the existence and deformations of curves in projective spaces; its basic object of study is $\mathcal{W}^r_{d,g}$, the moduli space of smooth genus $g$ curves with a choice of degree $d$ line bundle having at least $(r+1)$ independent global sections. The Brill-Noether theorem asserts that the map $\mathcal{W}^r_{d,g} \rightarrow \mathcal{M}_g$ is surjective with general fiber dimension given by the number $\rho = g - (r+1)(g-d+r)$, under the hypothesis that $0 \leq \rho \leq g$. One may naturally conjecture that for $\rho < 0$, this map is generically finite onto a subvariety of codimension $-\rho$ in $\mathcal{M}_g$. This conjecture fails in general, but seemingly only when $-\rho$ is large compared to $g$. This paper proves that this conjecture does hold for at least one irreducible component of $\mathcal{W}^r_{d,g}$, under the hypothesis that $0 < -\rho \leq \frac{r}{r+2} g - 3r+3$. We conjecture that this result should hold for all $0 < -\rho \leq g + C$ for some constant $C$, and we give a purely combinatorial conjecture that would imply this stronger result., Comment: 16 pages

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 with two elements, 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 the finite field with two elements. 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 relation 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.