Multiplier ideals and modules on toric varieties

A formula computing the multiplier ideal of a monomial ideal on an arbitrary affine toric variety is given. Variants for the multiplier module and test ideals are also treated.Mathematics Subject Classification (2000): 14J17, 13A35The author is grateful to Nobuo Hara for interesting discussions and thanks the referee for a careful reading and thoughtful comments.in final form: 02 November 2003     

Degenerations of toric ideals and toric varieties

Toric degenerations of toric varieties and toric ideals are important both in theory and in applications. In this paper, we study the correspondence between degenerations of toric variety and of toric ideal when the weight admits a regular subdivision.     

On the ideals and singularities of secant varieties of Segre varieties

We find minimal generators for the ideals of secant varietiesof Segre varieties in the cases of _k(¹ x ⁿ x ^m) for all k, n,m, ₂(ⁿ x ^m x ^p x ^r) for all n, m, p, r (GSS conjecture for fourfactors), and ₃(ⁿ x ^m x ^p) for all n, m, p and prove they arenormal with rational singularities in the first case and arithmeticallyCohen–Macaulay in the second two cases.     

Approximate varieties, approximate ideals and dimension reduction

We consider a method to determine, for a given finite set of points, an algebraic variety of small degree which contains these points. In contrast to most other algorithms in Computer Algebra, this one is adapted to numerical, inexact computations. Dedicated to Walter Gautschi, an admirable and outstanding scientist and gentleman.     

On generators of ideals defining projective toric varieties

 For any ample line bundle L on a projective toric variety of dimension n, it is proved that the line bundle L ^⊗i is normally generated if i is greater than or equal to n−1, and examples showing that this estimate is best possible are given. Moreover we prove an estimate for the degree of the generators of the ideals defining projective toric varieties. In particular, when L is normally generated, the defining ideal of the variety embedded by the global sections of L has generators of degree at most n+1. When the variety is embedded by the global sections of L ^⊗(n−1) , then the defining ideal has generators of degree at most three. Received: 11 July 2001 / Revised version: 17 December 2001     

Toric varieties as spectra of homogeneous prime ideals

We describe the construction of a class of toric varieties as spectra of homogeneous prime ideals.      

The correspondence between multivariate spline ideals and piecewise algebraic varieties

As a piecewise polynomial with a certain smoothness, the spline plays an important role in computational geometry. The algebraic variety is the most important subject in classical algebraic geometry. As the zero set of multivariate splines, the piecewise algebraic variety is a generalization of the algebraic variety. In this paper, the correspondence between piecewise algebraic varieties and spline ideals is discussed. Furthermore, Hilbert’s Nullstellensatz for the piecewise algebraic variety is also studied.     

The correspondence between multivariate spline ideals and piecewise algebraic varieties

As a piecewise polynomial with a certain smoothness, the spline plays an important role in computational geometry. The algebraic variety is the most important subject in classical algebraic geometry. As the zero set of multivariate splines, the piecewise algebraic variety is a generalization of the algebraic variety. In this paper, the correspondence between piecewise algebraic varieties and spline ideals is discussed. Furthermore, Hilbert’s Nullstellensatz for the piecewise algebraic variety is also studied.     

On subtractive varieties IV: Definability of principal ideals

This paper deals with notions of (equational) definability of principal ideals in subtractive varieties. These notions are first characterized in several different ways. The strongest notion (EDPI) is then further investigated. We introduce the variety of MINI algebras (a generalization of Hilbert algebras) and we show that they are a paradigm for subtractive EDPI varieties. Finally we deal with principal ideal operations, and in particular with the cases of meet and join of principal ideals being equationally definable. Received November 7, 1996; accepted in final form December 17, 1997.     

On subtractive varieties III: From ideals to congruences

As a sequel to [2] and [15] we investigate ideal properties focusing on subtractive varieties. Here we probe the relations between congruences and ideals in subtractive varieties, in order to give some means to recover the congruence structure from the ideal structure. To do so we consider mainly two operators from the ideal lattice to the congruence lattice of a given algebra and we classify subtractive varieties according to various properties of these operators. In the last section several examples are discussed in details. Received May 23, 1996; accepted in final form November 25, 1996.     

Initial ideals of tangent cones to Schubert varieties in orthogonal Grassmannians

We compute the initial ideals, with respect to certain conveniently chosen term orders, of ideals of tangent cones at torus fixed points to Schubert varieties in orthogonal Grassmannians. The initial ideals turn out to be square-free monomial ideals and therefore define Stanley-Reisner face rings of simplicial complexes. We describe these complexes. The maximal faces of these complexes encode certain sets of non-intersecting lattice paths.     

Principal Borel ideals and Gotzmann ideals

In this paper we characterize all principal Borel ideals with Borel generator up to degree 4 which are Gotzmann. We also classify principal Borel ideals with a Borel generator of degree d which are lexsegment and we describe the shadows of principal Borel ideals. Finally, we discuss the corresponding results for squarefree monomial ideals.Received: 10 May 2002     

Monomial ideals with tiny squares and Freiman ideals

Czechoslovak Mathematical Journal - We provide a construction of monomial ideals in R = K[x, y] such that μ(I2) < μ(I), where μ denotes the least number of generators. This...     

Decomposition of ideals into pseudo-irreducible ideals

A proper ideal of a commutative ring is called pseudo-irreducible if it cannot be written as a product of two comaximal proper ideals. In this paper, we give a necessary and su?cient condition for every proper ideal of a commutative ring to be a product of pairwise comaximal pseudo-irreducible ideals. Examples of such rings include Laskerian rings, or more generally J-Noetherian rings and ZD-rings. We study when certain classes of rings satisfy this condition.