Complete intersections in toric ideals

Cattani, Eduardo; Curran, Raymond; Dickenstein, Alicia

doi:10.1090/s0002-9939-06-08513-3

Cited by 10 publications

(9 citation statements)

References 23 publications

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“…Remark 3.22. We know from [19] that the intersection of an affine toric subvariety of A N (k) with the ambient torus is a complete intersection and from [12] that its ideal in k[U1, . .…”

Section: Given a Regular Conementioning

confidence: 99%

Overweight deformations of affine toric varieties and local uniformization

Teissier

2014

EMS Series of Congress Reports

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Given an equicharacteristic complete noetherian local domain R with algebraically closed residue field k, we first present a combinatorial proof of embedded local uniformization for zero-dimensional valuations of R whose associated graded ring gr ν R with respect to the filtration defined by the valuation is a finitely generated k-algebra. The main idea here is that some of the birational toric maps which provide embedded pseudo-resolutions for the affine toric variety corresponding to gr ν R also provide local uniformizations for ν on R. These valuations are necessarily Abhyankar (for zero-dimensional valuations this means that the value group is Z r with r = dimR). In a second part we show that conversely, given an excellent noetherian equicharacteristic local domain R with algebraically closed residue field, if the zero-dimensional valuation ν of R is Abhyankar, there are local domains R ′ which are essentially of finite type over R and dominated by the valuation ring Rν (ν-modifications of R) such that the semigroup of values of ν on R ′ is finitely generated, and therefore so is the k-algebra gr ν R ′ . Combining the two results and using the fact that Abhyankar valuations behave well under completion gives a proof of local uniformization for rational Abhyankar valuations and, by a specialization argument, for all Abhyankar valuations.As a by-product we obtain a description of the valuation ring of a rational Abhyankar valuation as an inductive limit indexed by N of birational toric maps of regular local rings. One of our main tools, the valuative Cohen theorem, is then used to study the extensions of rational monomial Abhyankar valuations of the ring k[[x 1 , . . . , xr]] to monogenous integral extensions and the nature of their key polynomials. In the conclusion we place the results in the perspective of local embedded resolution of singularities by a single toric modification after an appropriate re-embedding. 1

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“…Remark 3.22. We know from [19] that the intersection of an affine toric subvariety of A N (k) with the ambient torus is a complete intersection and from [12] that its ideal in k[U1, . .…”

Section: Given a Regular Conementioning

confidence: 99%

Overweight deformations of affine toric varieties and local uniformization

Teissier

2014

EMS Series of Congress Reports

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“…where u, v ∈ N m . In particular, beginning with the work of Herzog [16] and Delorme [6] the question of classifying complete intersection toric ideals (and the corresponding semigroup algebras) has been extensively studied by many authors [1,4,11,12,13,26]. A key step in many of these works is the study of the ideal generated by binomials x ui − x vi associated with a Z-basis of the kernel of A.…”

Section: Applicationsmentioning

confidence: 99%

Counting solutions to binomial complete intersections

Cattani

Dickenstein

2007

Journal of Complexity

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We study the problem of counting the total number of affine solutions of a system of n binomials in n variables over an algebraically closed field of characteristic zero. We show that we may decide in polynomial time if that number is finite. We give a combinatorial formula for computing the total number of affine solutions (with or without multiplicity) from which we deduce that this counting problem is #P -complete. We discuss special cases in which this formula may be computed in polynomial time; in particular, this is true for generic exponent vectors.

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“…We know from the work by Cattani, Curran and Dickenstein [8] that the defining ideal of every projective monomial curve in P 3 contains a complete intersection ideal of height 2. Our Proposition 3 makes this result more precise: it establishes that this ideal can be chosen in such a way that it has the same radical as the determinantal ideal generated by the 2-minors of a 2 × 3 simple matrix with monomial entries.…”

Section: Remarkmentioning

confidence: 99%

“…Our Proposition 3 makes this result more precise: it establishes that this ideal can be chosen in such a way that it has the same radical as the determinantal ideal generated by the 2-minors of a 2 × 3 simple matrix with monomial entries. In [8] the authors consider the curve known as the twisted cubic…”

Section: Remarkmentioning

confidence: 99%