Related papers: Graphs and obstruction theory for algebraic curves

We consider Abel maps for regular smoothing of nodal curves with values in the Esteves compactified Jacobian. In general, these maps are just rational, and an interesting question is to find an explicit resolution. We translate this problem…

Metric graphs are important models for capturing the structure of complex data across various domains. While much effort has been devoted to extracting geometric and topological features from graph data, computational aspects of metric…

In the last few years there has been a growing interest towards methods for statistical inference and learning based on computational geometry and, notably, tropical geometry, that is, the study of algebraic varieties over the min-plus…

We introduce the notion of resultant of two planar curves in the tropical geometry framework. We prove that the tropicalization of the algebraic resultant can be used to compute the stable intersection of two tropical plane curves. It is…

We develop a homotopical variant of the classic notion of an algebraic theory as a tool for producing deformations of homotopy theories. From this, we extract a framework for constructing and reasoning with obstruction theories and spectral…

To a compact tropical variety of arbitrary dimension, we associate a collection of intermediate Jacobians defined in terms of tropical homology and tropical monodromy. We then develop an Abel-Jacobi theory in the tropical setting by…

Tropical geometry is sensitive to embeddings of algebraic varieties inside toric varieties. The purpose of this paper is to advertise tropical modifications as a tool to locally repair bad embeddings of plane curves, allowing the…

In the previous works, the rational function semifields of abstract tropical curves were characterized. In this paper, we give a contravariant categorical equivalence between the category of abstract tropical curves with morphisms and the…

The traditional study of plane and space algebraic curves by looking at their tangent vectors, curvatures and torsions provides geometric, but unfortunately not sufficient information about individual curves in order to be able to…

In the branch of mathematics known as graph theory, graphs are considered as a set of points, called vertices, with connections between these points, called edges. The purpose of this paper is to study mappings between two graphs that have…

Complex algebraic varieties become easy piecewise-linear objects after passing to the so-called tropical limit. Geometry of these limiting objects is known as tropical geometry. In this short survey we take a look at motivation and…

In tropical geometry, given a curve in a toric variety, one defines a corresponding graph embedded in Euclidean space. We study the problem of reversing this process for curves of genus zero and one. Our methods focus on describing curves…

We construct a tangent-obstruction theory for Azumaya algebras equipped with a quadratic pair. Under the assumption that either 2 is a global unit or the algebra is of degree 2, we show how the deformation theory of these objects reduces to…

We study a class of complex polynomial equations on a finite graph with a view to understanding how holistic phenomena emerge from combinatorial structure. Particular solutions arise from orthogonal projections of regular polytopes,…

Tropical geometry is a degeneration of classical geometry which loose the property of unique factorization for polynomials. In this paper we explore a structure that is known to be a semi-degeneration between the classical algebra and the…

The divisor theory for graphs is compared to the theory of linear series on curves through the correspondence associating a curve to its dual graph. An algebro-geometric interpretation of the combinatorial rank is proposed, and proved in…

In this paper we prove that a graph is a string graph (the intersection graph of curves in the plane) if and only if it admits a drawing in the plane with certain properties. This also allows us to define an algebraic obstruction, similar…

We introduce a new class of matroids, called graph curve matroids. A graph curve matroid is associated to a graph and defined on the vertices of the graph as a ground set. We prove that these matroids provide a combinatorial description of…

In this paper, we introduce tiled graphs as models of learning and maturing processes. We show how tiled graphs can combine graphs of learning spaces or antimatroids (partial hypercubes) and maturity models (total orders) to yield models of…

We propose a generalization of tropical curves by dropping the rationality and integrality requirements while preserving the balancing condition. An interpretation of such curves as critical points of a certain quadratic functional allows…