Related papers: A Macaulay2 package for computations with rational…

Numerical Algebraic Geometry uses numerical data to describe algebraic varieties. It is based on the methods of numerical polynomial homotopy continuation, an alternative to the classical symbolic approaches of computational algebraic…

We introduce the Macaulay2 package HomologicalShiftIdeals. It allows to compute the homological shift ideals of a monomial ideal, and to check the homological shift properties, including having linear resolution, having linear quotients, or…

We introduce the AdditiveToricVarieties package for Macaulay2, a software system for algebraic geometry and commutative algebra, with methods for working with additive group actions on complete toric varieties. More precisely, we implement…

Algebraic methods have a long history in statistics. The most prominent manifestation of modern algebra in statistics can be seen in the field of algebraic statistics, which brings tools from commutative algebra and algebraic geometry to…

We present simple examples of rational maps of the complex projective plane with equal first and second dynamical degrees and no invariant foliation.

We develop a link between degree estimates for rational sphere maps and compressed sensing. We provide several new ideas and many examples, both old and new, that amplify connections with linear programming. We close with a list of ten open…

One develops {\em ab initio} the theory of rational/birational maps over reduced, but not necessarily irreducible, projective varieties in arbitrary characteristic. A numerical invariant of a rational map is introduced, called the Jacobian…

We give an algorithm for computing Segre classes of subschemes of arbitrary projective varieties by computing degrees of a sequence of linear projections. Based on the fact that Segre classes of projective varieties commute with…

Symbolic powers are a classical commutative algebra topic that relates to primary decomposition, consisting, in some circumstances, of the functions that vanish up to a certain order on a given variety. However, these are notoriously…

Are Fourier-Mukai equivalent cubic fourfolds birationally equivalent? We obtain an affirmative answer to this question for very general cubic fourfolds of discriminant 20, where we produce birational maps via the Cremona transformation…

In this work, we study a family of Cremona transformations of weighted projective planes which generalize the standard Cremona transformation of the projective plane. Starting from special plane projective curves we construct families of…

In this paper we present algorithms for computing the topology of planar and space rational curves defined by a parametrization. The algorithms given here work directly with the parametrization of the curve, and do not require to compute or…

We investigate the birational geometry (in the sense of Mori's program) of the moduli space of rank 2 semistable parabolic vector bundles on a rational curve. We compute the effective cone of the moduli space and show that all birational…

We present an approach to a large class of enumerative problems concerning rational curves in projective spaces. This approach uses analysis to obtain topological information about moduli spaces of stable maps. We demonstrate it by…

We give a description of a new Macaulay2 package called SimplicialPosets. This package provides functions for working with simplicial posets and calculating their generalized Stanley-Reisner ideals. For practical purposes, we also introduce…

We describe a Macaulay2 package for computing Schur complexes. This package expands on the ChainComplexOperations package by David Eisenbud.

In this article we show how to compute a matrix representation and the implicit equation by means of the method developed in [Botbol: arXiv:1007.3437], using the computer algebra system Macaulay2 \cite{M2}. As it is probably the most…

We give a complete characterization of degree two rational maps with potential good reduction over local fields. We show this happens exactly when the map corresponds to an integral point in the moduli space. We detail an algorithm by which…

We consider the dynamics of rational semigroups (semigroups of rational maps) on the Riemann sphere. We provide proof that a random backward iteration algorithm to draw the pictures of the Julia sets, previously proven to work in the…

We present the Macaulay2 package TropicalToric.m2 for toric intersection theory computations using tropical geometry.