On the homology of two-dimensional elimination

Hong, Jooyoun; Simis, Aron; Vasconcelos, Wolmer V.

doi:10.1016/j.jsc.2007.10.010

Cited by 42 publications

(58 citation statements)

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“…2 Two comments are in order here. First, the hypothesis V (U 0 , U 1 , U 2 ) = ∅, implicitly assumed in [HSV08,Proposition 4.9], is not superfluous since otherwise there exist some counterexamples. Also, we mention that this latter condition corresponds to the geometric property that there is no singular point on the curve C of multiplicity d − 2, the maximum possible value for a singular point on C in this case by [SCG07,Theorem 3].…”

Section: The Case μ =mentioning

confidence: 99%

On the equations of the moving curve ideal of a rational algebraic plane curve

Busé

2009

Journal of Algebra

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International audienceGiven a parametrization of a rational plane algebraic curve C, some explicit adjoint pencils on C are described in terms of determinants. Moreover, some generators of the Rees algebra associated to this parametrization are presented. The main ingredient developed in this paper is a detailed study of the elimination ideal of two homogeneous polynomials in two homogeneous variables that form a regular sequence

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Section: The Case μ =mentioning

confidence: 99%

On the equations of the moving curve ideal of a rational algebraic plane curve

Busé

2009

Journal of Algebra

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“…In Section 1 we recall the implicitization problem and the Rees algebra. In Section 2 we provide an algorithm of finding the minimal generators of K , the kernel of the map h, and prove a conjecture of Hong, Simis and Vasconcelos [12]. In Section 3 we relate our results to the theory of adjoint curves and prove a special case of a conjecture of Cox [6].…”

Section: Introductionmentioning

confidence: 99%

Syzygies and the Rees algebra

Cox

Hoffman

Wang

2008

Journal of Pure and Applied Algebra

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Let a, b, c be linearly independent homogeneous polynomials in the standard Z-graded ring R := k[s, t] with the same degree d and no common divisors. This defines a morphism P 1 → P 2 . The Rees algebra Rees(I ) = R⊕I ⊕I 2 ⊕· · · of the ideal I = a, b, c is the graded R-algebra which can be described as the image of an R-algebra homomorphism h: R[x, y, z] → Rees(I ). This paper discusses one result concerning the structure of the kernel of the map h and its relation to the problem of finding the implicit equation of the image of the map given by a, b, c. In particular, we prove a conjecture of Hong, Simis and Vasconcelos. We also relate our results to the theory of adjoint curves and prove a special case of a conjecture of Cox. Published by Elsevier B.V.

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“…Also, we return to this question in Section 6. This question is of interest because there is much recent work concerning the equations that define the Rees algebra of ideals which are primary to the maximal ideal; see, for example, [14,10,6,11]. The driving force behind this work is the desire to understand the singularities of parameterized curves or surfaces; see [19,9,7,2,8] and especially [11].…”

Section: 4mentioning

confidence: 99%

The explicit minimal resolution constructed from a Macaulay inverse system

Khoury

Kustin

2015

Journal of Algebra

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ABSTRACT. Let A be a standard-graded Artinian Gorenstein algebra of embedding codimension three over a field k k k. In the generic case, the minimal homogeneous resolution, G, of A, by free Sym k k k• (A 1 ) modules, is Gorenstein-linear. Fix a basis x, y, z for the k k k-vector space A 1 . If G is Gorenstein linear, then the socle degree of A is necessarily even, and, if n is the least index with dim k k k A n less than dim k k k Sym k k k n (A 1 ), then the socle degree of A is 2n − 2. Let • (U * ) be the graded S-module of graded k k k-linear homomorphisms from S to k k k. In his 1916 paper [16], Macaulay proved that each element Φ of D determines (in our language) an Artinian Gorenstein ring A Φ = S/ ann(Φ); furthermore, each Artinian Gorenstein quotient of S is obtained in this manner. Of course, Φ determines everything about the quotient A Φ ; so in particular, when Φ is a homogeneous element of D, then Φ determines a minimal resolution of A Φ by free S-modules. The standard way to find this minimal resolution is to first solve some equations in order to determine a minimal generating set for ann(Φ) and then to use Gröbner basis techniques in order to find a minimal resolution of A Φ by free S-modules. We are interested in by-passing all of the intermediate steps. We aim to describe a minimal resolution of A Φ directly (and in a polynomial manner) in terms of the coefficients of Φ, at least in the generic case. In [12], we proved that if Φ is homogeneous of even degree 2n − 2 and the pairing, is perfect, then a minimal resolution for A Φ may be read directly, and in a polynomial manner, from the coefficients of Φ. Furthermore, there is one such resolution for The paper [12] proves the existence of a unique generic Gorenstein-linear resolution for each pair (d, n); but exhibits this resolution only for the pair (d, n) = (3, 2). In the present paper, we exhibit this resolution when d = 3 and n ≥ 2 is arbitrary. Indeed, once n ≥ 2 is fixed, we exhibit an explicit complex (B, b) (see Definition 2.7 or Observation 4.4 or Proposition 5.5, depending upon your tolerance for, and/or need to see, explicitness). If U is a vector space over k k k of dimensionWe preview the complex B. (Complete details are given in Section 2.) This complex is built over Z. Let U be a free Z-module of rank 3 and R be the ring Sym(No harm is done by choosing bases x, y, z for U and {t m | m is a monomial of degree 2n − 2 in x, y, z} for Sym Z 2n−2 U and viewing R as the polynomial ring Z[x, y, z, {t m }]; although we will not officially choose these bases until Section 5. Until Section 5, we will keep the calculation as coordinate-free as possible.) In the complex B, one of the basis elements of U (we call this element x) is given a distinguished role. The complex B is symmetric in the complementary basis elements (we call these elements y and z) of U . Let U 0 be the free Z-summand Zy ⊕ Zz of U = Zx ⊕ Zy ⊕ Zz. The complex B is, where the Z-module homomorphismsall are defined in terms of the classical adjoint of the mapwhich is induced...

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On the homology of two-dimensional elimination

Cited by 42 publications

References 20 publications

On the equations of the moving curve ideal of a rational algebraic plane curve

On the equations of the moving curve ideal of a rational algebraic plane curve

Syzygies and the Rees algebra

The explicit minimal resolution constructed from a Macaulay inverse system

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