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Maps of toric varieties in Cox coordinates

Gavin Brown, Jarosław Buczyński (2013)

Fundamenta Mathematicae

The Cox ring provides a coordinate system on a toric variety analogous to the homogeneous coordinate ring of projective space. Rational maps between projective spaces are described using polynomials in the coordinate ring, and we generalise this to toric varieties, providing a unified description of arbitrary rational maps between toric varieties in terms of their Cox coordinates. Introducing formal roots of polynomials is necessary even in the simplest examples.

Minimal resolutions of lattice ideals and integer linear programming.

Emilio Briales-Morales, Antonio Campillo-López, Pilar Pisón-Casares, Alberto Vigneron-Tenorio (2003)

Revista Matemática Iberoamericana

A combinatorial description of the minimal free resolution of a lattice ideal allows us to the connection of Integer Linear Programming and Al1gebra. The non null reduced homology spaces of some simplicial complexes are the key. The extremal rays of the associated cone reduce the number of variables.

Modified Nash triviality of a family of zero-sets of real polynomial mappings

Toshizumi Fukui, Satoshi Koike, Masahiro Shiota (1998)

Annales de l'institut Fourier

In this paper we introduce the notion of modified Nash triviality for a family of zero sets of real polynomial map-germs as a desirable one. We first give a Nash isotopy lemma which is a useful tool to show triviality.Then, using it, we prove two types of modified Nash triviality theorem and a finite classification theorem for this triviality. These theorems strengthen similar topological results.

Moment-angle complexes from simplicial posets

Zhi Lü, Taras Panov (2011)

Open Mathematics

We extend the construction of moment-angle complexes to simplicial posets by associating a certain T m-space Z S to an arbitrary simplicial poset S on m vertices. Face rings ℤ[S] of simplicial posets generalise those of simplicial complexes, and give rise to new classes of Gorenstein and Cohen-Macaulay rings. Our primary motivation is to study the face rings ℤ[S] by topological methods. The space Z S has many important topological properties of the original moment-angle complex Z K associated to...

Multi-Harnack smoothings of real plane branches

Pedro Daniel González Pérez, Jean-Jacques Risler (2010)

Annales scientifiques de l'École Normale Supérieure

Let Δ ⊂ 𝐑 2 be an integral convex polygon. G. Mikhalkin introduced the notion ofHarnack curves, a class of real algebraic curves, defined by polynomials supported on Δ and contained in the corresponding toric surface. He proved their existence, viaViro’s patchworkingmethod, and that the topological type of their real parts is unique (and determined by Δ ). This paper is concerned with the description of the analogous statement in the case of a smoothing of a real plane branch ( C , 0 ) . We introduce the class...

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