arXiv subjects

Ethan Cotterill

Publications and source records attributed to Ethan Cotterill.

At least 19 recordsLinked to original sources

arXiv2026

The relation type of point configurations in the projective plane

We study the {\it relation type} of ideals of finite reduced sets of points in the projective plane; for a given ideal, this is the maximal $T$-degree of a minimal generator of the defining ideal of the Rees algebra. Our main focus is on point configurations whose defining ideals are not necessarily linearly presented, with an emphasis on almost collinear configurations. We prove that $\rt(X)\in\{1,3\}$ whenever $X\subseteq \PP^2_k$ is a finite set of at most ten points; and we characterize the configurations of relation type $3$ in this range. We then show that a configuration of eleven points in generic position has relation type $5$, thereby yielding the first occurrence of relation type larger than $3$. Finally, we exhibit a configuration of $17$ points with relation type $4$ and we formulate some questions regarding the spectrum of admissible relation types of point configurations.

math.AC↗

arXiv2025

On gap sets in arbitrary Kummer extensions of $K(x)$

Let $K$ be an algebraically closed field, and let $F/K(x)$ be a Kummer extension of function fields of genus $g$. We provide a compact and explicit description of the gap set $G(Q)$ at any totally ramified place $Q$ of the extension $F/K(x)$. As a consequence, we deduce structural properties of the Weierstrass semigroup $H(Q)$; in particular, we determine a generating set for $H(Q)$, and we characterize its symmetry in certain cases. We also generalize a formula due to Towse that describes the asymptotic behavior of the sum of the Weierstrass weights at all totally ramified places of the extension $F/K(x)$ relative to $g^3-g$.

math.AG↗

arXiv2024

Arithmetic inflection of superelliptic curves

In this paper, we explore the inflectionary behavior of linear series on superelliptic curves $X$ over fields of arbitrary characteristic. Here we give a precise description of the inflection of linear series over the ramification locus of the superelliptic projection; and we initiate a study of those inflectionary varieties that parameterize the inflection points of linear series on $X$ supported away from the superelliptic ramification locus that is predicated on the behavior of their Newton polytopes.

math.AG↗

arXiv2024

Matroids and semirings attached to toric singularity arrangements

Curve singularities are classical objects of study in algebraic geometry. The key player in their combinatorial structure is the {\it value semigroup}, or its compactification, the {\it value semiring}. One natural problem is to explicitly determine the value semirings of distinguished infinite classes of singularities, with a view to understanding their asymptotic properties. In this paper, we establish a matroidal framework for resolving this problem for singularities determined by arrangements of toric branches; and we obtain precise quantitative results in the case of line arrangements. Our results have implications for the topology of Severi varieties of unisingular rational curves in projective space.

math.AG↗

arXiv2023

Certified Severi dimensions for hyperelliptic and supersymmetric cusps

In a previous paper, the first three authors formulated a precise conjecture about the dimension of the {\it generalized Severi variety} $M^n_{d,g; {\rm S}, {\bf k}}$ of degree-$d$ holomorphic maps $\mathbb{P}^1 \rightarrow \mathbb{P}^n$ whose images' singularities are singleton cusps with value semigroups ${\rm S}$ and ramification profiles ${\bf k}$. In this paper, we prove that an adjusted form of the conjecture holds for generic profiles ${\bf k}$ associated with two distinguished (infinite) classes of semigroups ${\rm S}$.

math.AG↗

arXiv2023

Cusps in $\mathbb{C}^3$ with prescribed ramification

We study value semigroups associated to germs of maps $\mathbb{C} \rightarrow \mathbb{C}^3$ with fixed ramification profiles in a distinguished point. We then apply our analysis to deduce that Severi varieties of unicuspidal rational fixed-degree curves with value semigroup ${\rm S}$ in $\mathbb{P}^3$ are often reducible when ${\rm S}$ is either 1) the semigroup of a generic cusp whose ramification profile is a supersymmetric triple; or 2) a supersymmetric semigroup with ramification profile given by a supersymmetric triple. In doing so, we uncover new connections with additive combinatorics and number theory.

math.AG↗

arXiv2023

Weierstrass semigroups from cyclic covers of hyperelliptic curves

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 $π: (C,p) \rightarrow (B,q)$ 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.

math.AG↗

arXiv2023

Towards Brill--Noether theory for cuspidal curves

Understanding when an abstract complex curve of given genus comes equipped with a map of fixed degree to a projective space of fixed dimension is a foundational question; and Brill--Noether theory addresses this question via linear series, which algebraically codify maps to projective targets. Classical Brill--Noether theory, which focuses on smooth curves, has been intensively explored; but much less is known for singular curves, particularly for those with non-nodal singularities. In a one-parameter family of smooth curves specializing to a singular curve $C_0$, one expects certain aspects of the global geometry of the smooth fibers to ``specialize" to the local geometry of the singularities of $C_0$. Making this expectation quantitatively precise involves analyzing the arithmetic and combinatorics of semigroups ${\rm S}$ attached to discrete valuations defined on (the local rings of) these singularities. In this largely-expository note we focus primarily on Brill--Noether-type results for curves with {\it cusps}, i.e., unibranch singularities; in this setting, the associated semigroups are {\it numerical} semigroups with finite complement in $\mathbb{N}$.

math.AG↗

arXiv2022

Arithmetic inflection formulae for linear series on hyperelliptic curves

Over the complex numbers, Plücker's formula computes the number of inflection points of a linear series of projective dimension $r$ and degree $d$ on a curve of genus $g$. Here we explore the geometric meaning of a natural analogue of Plücker's formula in $\mathbb{A}^1$-homotopy theory for certain linear series on hyperelliptic curves defined over an arbitrary field.

math.AG↗

arXiv2021

Severi dimensions for unicuspidal curves

We study parameter spaces of linear series on projective curves in the presence of unibranch singularities, i.e. {\it cusps}; and to do so, we stratify cusps according to value semigroup. We show that {\it generalized Severi varieties} of maps $\mathbb{P}^1 \rightarrow \mathbb{P}^n$ with images of fixed degree and arithmetic genus are often {\it reducible} whenever $n \geq 3$. We also prove that the Severi variety of degree-$d$ maps with a hyperelliptic cusp of delta-invariant $g \ll d$ is of codimension at least $(n-1)g$ inside the space of degree-$d$ holomorphic maps $\mathbb{P}^1 \rightarrow \mathbb{P}^n$; and that for small $g$, the bound is exact, and the corresponding space of maps is the disjoint union of unirational strata. Finally, we conjecture a generalization for unicuspidal rational curves associated to an {\it arbitrary} value semigroup.

math.AG↗

arXiv2021

Exploring tropical differential equations

The purpose of this paper is fourfold. The first is to develop the theory of tropical differential algebraic geometry from scratch; the second is to present the tropical fundamental theorem for differential algebraic geometry, and show how it may be used to extract combinatorial information about the set of power series solutions to a given system of differential equations, both in the archimedean (complex analytic) and in the non-archimedean (e.g., $p$-adic) settings. A third and subsidiary aim is to show how tropical differential algebraic geometry is a natural application of semiring theory, and in so doing, contribute to the valuative study of differential algebraic geometry. We use this formalism to extend the fundamental theorem of partial differential algebraic geometry to the differential fraction field of the ring of formal power series in arbitrarily (finitely) many variables; in doing so we produce new examples of non-Krull valuations that merit further study in their own right.

math.AG↗

arXiv2020

The Strong Maximal Rank Conjecture and higher rank Brill--Noether theory

In this paper, we compute the cohomology class of certain "special maximal-rank loci" originally defined by Aprodu and Farkas. By showing that such classes are nonzero, we are able to verify the non-emptiness portion of the Strong Maximal Rank Conjecture in a wide range of cases. As an application, we obtain new evidence for the existence portion of a well-known conjecture due to Bertram, Feinberg and independently Mukai in higher-rank Brill--Noether theory.

math.AG↗

arXiv2020

Inflection divisors of linear series on an elliptic curve

In this largely-expository note, we describe a class of divisors on elliptic curves that index the inflection points of linear series arising (as subspaces of holomorphic sections) from line bundles on $\mathbb{P}^1$ via pullback along the canonical 2-to-1 projection. Associated to each inflection divisor on an elliptic curve $E_λ: y^2= x(x-1)(x-λ)$, there is an associated {\it inflectionary curve} in (the projective compactification of) the affine plane in coordinates $x$ and $λ$. These inflectionary curves have remarkable features; among other things, they lead directly to an explicit conjecture for the number of {\it real} inflection points of linear series on $E_λ$ whenever the Legendre parameter $λ$ is real.

math.AG↗

arXiv2020

Secant planes of a general curve via degenerations

We study linear series on a general curve of genus g, whose images are exceptional with respect to their secant planes. Each such exceptional secant plane is algebraically encoded by an included linear series, whose number of base points computes the incidence degree of the corresponding secant plane. With enumerative applications in mind, we construct a moduli scheme of inclusions of limit linear series with base points over families of nodal curves of compact type, which we then use to compute combinatorial formulas for the number of secant-exceptional linear series when the spaces of linear series and of inclusions are finite.

math.AG↗

arXiv2020

Weight bounds for $(3,γ)$-hyperelliptic curves

{\it $(N,γ)$-hyperelliptic} semigroups were introduced by Fernando Torres to encapsulate the most salient properties of Weierstrass semigroups associated to totally-ramified points of $N$-fold covers of curves of genus $γ$. Torres characterized $(2,γ)$-hyperelliptic semigroups of maximal weight whenever their genus is large relative to $γ$. Here we do the same for $(3,γ)$-hyperelliptic semigroups, and we formulate a conjecture about the general case whenever $N \geq 3$ is prime.

math.CO↗

arXiv2019

Dimension counts for singular rational curves via semigroups

We study singular rational curves in projective space, deducing conditions on their parametrizations from the value semigroups $\sss$ of their singularities. In particular, we prove that a natural heuristic for the codimension of the space of nondegenerate rational curves of arithmetic genus $g>0$ and degree $d$ in $\mb{P}^n$, viewed as a subspace of all degree-$d$ rational curves in $\mb{P}^n$, holds whenever $g$ is small. On the other hand, we show that this heuristic fails in general, by exhibiting an infinite family of examples of Severi-type varieties of rational curves containing "excess" components of dimension strictly larger than the space of $g$-nodal rational curves.

math.AG↗

arXiv2019

$K$-weight bounds for $γ$-hyperelliptic semigroups

In this note, we show that {\it $γ$-hyperelliptic} numerical semigroups of genus $g \gg γ$ satisfy a refinement of a well-known characteristic weight inequality due to Torres. The refinement arises from substituting the usual notion of weight by an alternative version, the $K$-weight, which we previously introduced in the course of our study of unibranch curve singularities.

math.CO↗

arXiv2018

Real inflection points of real hyperelliptic curves

Given a real hyperelliptic algebraic curve $X$ with non-empty real part and a real effective divisor $\mc{D}$ arising via pullback from $\mathbb{P}^1$ under the hyperelliptic structure map, we study the real inflection points of the associated complete real linear series $|\mc{D}|$ on $X$. To do so we use Viro's patchworking of real plane curves, recast in the context of some Berkovich spaces studied by M. Jonsson. Our method gives a simpler and more explicit alternative to limit linear series on metrized complexes of curves, as developed by O. Amini and M. Baker, for curves embedded in toric surfaces.

math.AG↗