Related papers: Equations defining secant varieties: geometry and …
We consider the genus-one curves which arise in the cuts of the sunrise and in the elliptic double-box Feynman integrals. We compute and compare invariants of these curves in a number of ways, including Feynman parametrization, lightcone…
Graph drawing research traditionally focuses on producing geometric embeddings of graphs satisfying various aesthetic constraints. After the geometric embedding is specified, there is an additional step that is often overlooked or ignored:…
Interpreting tangency as a limit of two transverse intersections, we obtain a concrete formula to enumerate smooth degree $d$ plane curves tangent to a given line at multiple points with arbitrary order of tangency. Extending that idea, we…
The Separation of Variables theory for the Hamilton-Jacobi equation is 'by definition' related to the use of special kinds of coordinates, for example Jacobi coordinates on the ellipsoid or St\"ackel systems in the Euclidean space. However,…
We discuss recent progress on syzygies of curves, including proofs of Green's and Gonality Conjectures as well as applications of Koszul cycles to the study of the birational geometry of various moduli spaces of curves. We prove a number of…
In projective algebraic geometry, there are classical and fundamental results that describe the structure of geometry and syzygies, and many of them characterize varieties of minimal degree and del Pezzo varieties. In this paper, we…
We survey both old and new developments in the theory of algorithms in real algebraic geometry -- starting from effective quantifier elimination in the first order theory of reals due to Tarski and Seidenberg, to more recent algorithms for…
Phylogenetic algebraic geometry is concerned with certain complex projective algebraic varieties derived from finite trees. Real positive points on these varieties represent probabilistic models of evolution. For small trees, we recover…
Inside the symmetric product of a very general curve, we consider the codimension-one subvarieties of symmetric tuples of points imposing exceptional secant conditions on linear series on the curve of fixed degree and dimension. We compute…
We define and study graphs associated to hexagon decompositions of surfaces by curves and arcs. One of the variants is shown to be quasi-isometric to the pants graph, whereas the other variant is quasi-isometric to (a Cayley graph of) the…
One considers geometry with the intransitive equaivalence relation. Such a geometry is a physical geometry, i.e. it is described completely by the world function, which is a half of the squared distance function. The physical geometry…
The formal group law of an elliptic curve has seen recent applications to computational algebraic geometry in the work of Couveignes to compute the order of an elliptic curve over finite fields of small characteristic. The purpose of this…
I survey methods from differential geometry, algebraic geometry and representation theory relevant for the permanent v. determinant problem from computer science, an algebraic analog of the P v. NP problem.
We study evolutes and involutes of space curves. Although much of the material presented is not new and can be found in classic treatises, we believe that a modern and unified treatment, complemented with several novel observations, may be…
In this paper we present a way of computing the degree of the secant (resp., tangent) variety of a smooth projective surface, under the assumption that the divisor giving the embedding in the projective space is $3$-very ample. This method…
We study the computational complexity of sequences of projective varieties. We define analogues of the complexity classes P and NP for these and prove the NP-completeness of a sequence called the universal circuit resultant. This is the…
The aim of these notes is to present an accessible overview of some topics in classical algebraic geometry which have applications to aspects of discrete integrable systems. Precisely, we focus on surface theory on the algebraic geometry…
We determine set theoretic defining equations for the third secant variety of the Segre product of $n$ projective spaces, and from the proof of the main statement we derive an upper bound for the degrees of these equations.
In this paper we study a construction of algebraic curves from combinatorial data. In the study of algebraic curves through degeneration, graphs usually appear as the dual intersection graph of the central fiber. Properties of such graphs…
This friendly introduction to tropical geometry is meant to be accessible to first year students in mathematics. The topics discussed here are basic tropical algebra, tropical plane curves, some tropical intersections, and Viro's…