arXiv subjects

Jessica Sidman

Publications and source records attributed to Jessica Sidman.

At least 19 recordsLinked to original sources

arXiv2025

A geometric view of formation control with application to directed sensing

We propose a geometric approach to distance-based formation control modeled on a minimum-norm lifting of Riemannian gradient descent in edge-space to node-space. This yields a unified family of controllers, including the classical gradient controller and its directed variant. For the directed case, we give a simple numerical test for local convergence that applies to any directed graph and target. We show that persistence is neither necessary nor sufficient for local convergence of our directed controller and propose an alternative that is necessary and more easily checked.

math.OC↗

arXiv2025

Terracini matroids: algebraic matroids of secants and embedded joins

Applications of algebraic geometry have sparked much recent work on algebraic matroids. An algebraic matroid encodes algebraic dependencies among coordinate functions on a variety. We study the behavior of algebraic matroids under joins and secants of varieties. Motivated by Terracini's lemma, we introduce the notion of a Terracini union of matroids, which captures when the algebraic matroid of a join coincides with the matroid union of the algebraic matroids of its summands. We illustrate applications of our results with a discussion of the implications for toric surfaces and threefolds.

math.CO↗

arXiv2025

Linearizing Algebraic Matroids

Although algebraic matroids were discovered in the 1930s, interest in them was largely dormant until their recent use in applications of algebraic geometry. Because nonlinear algebra is computationally challenging, it is easier to work with an isomorphic linear matroid if one exists. We describe an explicit construction that produces a linear representation over an algebraically closed field of characteristic zero starting with the data used in applications. We will also discuss classical examples of algebraic matroids in the modern language of polynomial ideals, illustrating how the existence of an isomorphic linear matroid depends on properties of the field.

math.AC↗

arXiv2023

Geometric aspects of the Jacobian of a hyperplane arrangement

An embedding of the complete bipartite graph $K_{3,3}$ in $\mathbb{P}^2$ gives rise to both a line arrangement and a bar-and-joint framework. For a generic placement of the six vertices, the graded Betti numbers of the logarithmic module of derivations of the line arrangement are constant, but an example due to Ziegler shows that the graded Betti numbers are different when the points lie on a conic. Similarly, in rigidity theory a generic embedding of $K_{3,3}$ in the plane is an infinitesimally rigid bar-and-joint framework, but the framework is infinitesimally flexible when the points lie on a conic. In this paper we develop the theory of weak perspective representations of hyperplane arrangements to formalize and generalize the striking connection between hyperplane arrangements and rigidity theory that this example suggests. In particular, we seek to understand how the interplay of combinatorics and geometry influence algebraic structures associated to an arrangement, such as the saturation of the Jacobian ideal of the arrangement. We make connections between examples and constructions from rigidity theory and interesting phenomena in the study of hyperplane arrangements.

math.AC↗

arXiv2020

Geometric Equations for Matroid Varieties

Each point $x$ in Gr$(r,n)$ corresponds to an $r \times n$ matrix $A_x$ which gives rise to a matroid $M_x$ on its columns. Gel'fand, Goresky, MacPherson, and Serganova showed that the sets $\{y \in \mathrm{Gr}(r,n) | M_y = M_x\}$ form a stratification of Gr$(r,n)$ with many beautiful properties. However, results of Mnëv and Sturmfels show that these strata can be quite complicated, and in particular may have arbitrary singularities. We study the ideals $I_x$ of matroid varieties, the Zariski closures of these strata. We construct several classes of examples based on theorems from projective geometry and describe how the Grassmann-Cayley algebra may be used to derive non-trivial elements of $I_x$ geometrically when the combinatorics of the matroid is sufficiently rich.

math.AG↗

arXiv2019

Algebraic matroids in action

In recent years, various notions of algebraic independence have emerged as a central and unifying theme in a number of areas of applied mathematics, including algebraic statistics and the rigidity theory of bar-and-joint frameworks. In each of these settings the fundamental problem is to determine the extent to which certain unknowns depend algebraically on given data. This has, in turn, led to a resurgence of interest in algebraic matroids, which are the combinatorial formalism for algebraic (in)dependence. We give a self-contained introduction to algebraic matroids together with examples highlighting their potential application.

math.CO↗

arXiv2015

Algorithms for detecting dependencies and rigid subsystems for CAD

Geometric constraint systems underly popular Computer Aided Design soft- ware. Automated approaches for detecting dependencies in a design are critical for developing robust solvers and providing informative user feedback, and we provide algorithms for two types of dependencies. First, we give a pebble game algorithm for detecting generic dependencies. Then, we focus on identifying the "special positions" of a design in which generically independent constraints become dependent. We present combinatorial algorithms for identifying subgraphs associated to factors of a particular polynomial, whose vanishing indicates a special position and resulting dependency. Further factoring in the Grassmann- Cayley algebra may allow a geometric interpretation giving conditions (e.g., "these two lines being parallel cause a dependency") determining the special position.

cs.CG↗

arXiv2015

Line arrangements modeling curves of high degree: equations, syzygies and secants

We study curves consisting of unions of projective lines whose intersections are given by graphs. Under suitable hypotheses on the graph, these so-called \emph{graph curves} can be embedded in projective space as line arrangements. We discuss property $N_p$ for these embeddings and are able to produce products of linear forms that generate the ideal in certain cases. We also briefly discuss questions regarding the higher-dimensional subspace arrangements obtained by taking the secant varieties of graph curves.

math.AG↗

arXiv2013

Hyperbanana Graphs

A bar-and-joint framework is a finite set of points together with specified distances between selected pairs. In rigidity theory we seek to understand when the remaining pairwise distances are also fixed. If there exists a pair of points which move relative to one another while maintaining the given distance constraints, the framework is flexible; otherwise, it is rigid. Counting conditions due to Maxwell give a necessary combinatorial criterion for generic minimal bar-and-joint rigidity in all dimensions. Laman showed that these conditions are also sufficient for frameworks in R^2. However, the flexible "double banana" shows that Maxwell's conditions are not sufficient to guarantee rigidity in R^3. We present a generalization of the double banana to a family of hyperbananas. In dimensions 3 and higher, these are (infinitesimally) flexible, providing counterexamples to the natural generalization of Laman's theorem.

math.CO↗

arXiv2012

Combinatorics and the Rigidity of CAD Systems

We study the rigidity of body-and-cad frameworks which capture the majority of the geometric constraints used in 3D mechanical engineering CAD software. We present a combinatorial characterization of the generic minimal rigidity of a subset of body-and-cad frameworks in which we treat 20 of the 21 body-and-cad constraints, omitting only point-point coincidences. While the handful of classical combinatorial characterizations of rigidity focus on distance constraints between points, this is the first result simultaneously addressing coincidence, angular, and distance constraints. Our result is stated in terms of the partitioning of a graph into edge-disjoint spanning trees. This combinatorial approach provides the theoretical basis for the development of deterministic algorithms (that will not depend on numerical methods) for analyzing the rigidity of body-and-cad frameworks.

cs.DM↗

arXiv2011

Linear determinantal equations for all projective schemes

We prove that every projective embedding of a connected scheme determined by the complete linear series of a sufficiently ample line bundle is defined by the 2-minors of a 1-generic matrix of linear forms. Extending the work of Eisenbud-Koh-Stillman for integral curves, we also provide effective descriptions for such determinantally presented ample line bundles on products of projective spaces, Gorenstein toric varieties, and smooth n-folds.

math.AG↗

arXiv2011

Equations defining secant varieties: geometry and computation

In the 1980's, work of Green and Lazarsfeld helped to uncover the beautiful interplay between the geometry of the embedding of a curve and the syzygies of its defining equations. Similar results hold for the first secant variety of a curve, and there is a natural conjectural picture extending to higher secant varieties as well. We present an introduction to the algebra and geometry used in previous work of the authors to study syzygies of secant varieties of curves with an emphasis on examples of explicit computations and elementary cases that illustrate the geometric principles at work.

math.AG↗

arXiv2009

Syzygies of the secant variety of a curve

We show that the secant variety of a linearly normal smooth curve of degree at least 2g+3 is arithmetically Cohen-Macaulay, and we use this information to study the graded Betti numbers of the secant variety.

math.AG↗

arXiv2008

Prolongations and computational algebra

We explore the geometric notion of prolongations in the setting of computational algebra, extending results of Landsberg and Manivel which relate prolongations to equations for secant varieties. We also develop methods for computing prolongations which are combinatorial in nature. As an application, we use prolongations to derive a new family of secant equations for the binary symmetric model in phylogenetics.

math.AC↗

arXiv2006

Secant varieties of toric varieties

Let $X_P$ be a smooth projective toric variety of dimension $n$ embedded in $\PP^r$ using all of the lattice points of the polytope $P$. We compute the dimension and degree of the secant variety $\Sec X_P$. We also give explicit formulas in dimensions 2 and 3 and obtain partial results for the projective varieties $X_A$ embedded using a set of lattice points $A \subset P\cap\ZZ^n$ containing the vertices of $P$ and their nearest neighbors.

math.AG↗

arXiv2005

Multigraded regularity: coarsenings and resolutions

Let S = k[x_1,...,x_n] be a Z^r-graded ring with deg (x_i) = a_i \in Z^r for each i and suppose that M is a finitely generated Z^r-graded S-module. In this paper we describe how to find finite subsets of Z^r containing the multidegrees of the minimal multigraded syzygies of M. To find such a set, we first coarsen the grading of M so that we can view M as a Z-graded S-module. We use a generalized notion of Castelnuovo-Mumford regularity, which was introduced by D. Maclagan and G. Smith, to associate to M a number which we call the regularity number of M. The minimal degrees of the multigraded minimal syzygies are bounded in terms of this invariant.

math.AC↗

arXiv2005

Multigraded regularity: syzygies and fat points

The Castelnuovo-Mumford regularity of a graded ring is an important invariant in computational commutative algebra, and there is increasing interest in multigraded generalizations. We study connections between two recent definitions of multigraded regularity with a view towards a better understanding of the multigraded Hilbert function of fat point schemes in P^{n_1} x ... x P^{n_k}.

math.AC↗

arXiv2005

Generic initial ideals of points and curves

Let I be the defining ideal of a smooth irreducible complete intersection space curve C with defining equations of degrees a and b. We use the partial elimination ideals introduced by Mark Green to show that the lexicographic generic initial ideal of I has Castelnuovo-Mumford regularity 1+ab(a-1)(b-1)/2 with the exception of the case a=b=2, where the regularity is 4. Note that ab(a-1)(b-1)/2 is exactly the number of singular points of a general projection of C to the plane. Additionally, we show that for any term ordering tau, the generic initial ideal of a generic set of points in P^r is a tau-segment ideal.

math.AC↗