A study of singularities on rational curves via syzygies

Cox, David A.; Kustin, Andrew R.; Polini, Claudia; Ulrich, Bernd

doi:10.1090/s0065-9266-2012-00674-5

Cited by 28 publications

(64 citation statements)

References 31 publications

(52 reference statements)

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“…In principle, one could now actually calculate the implicit equation, but the matrix M ν is easier to get and well suited for numerical methods [ACGVS07]. As we remarked in the surface case, testing whether a point p lies on the curve only requires computing the rank of M ν evaluated in p. Also, the singularities of C can be read off from M ν [JG09,CKPU13,BD12].…”

Section: The Main Algorithm Based On Linear Syzygiesmentioning

confidence: 99%

Implicitization of rational hypersurfaces via linear syzygies: A practical overview

Botbol

Dickenstein

2016

Journal of Symbolic Computation

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Abstract. We unveil in concrete terms the general machinery of the syzygybased algorithms for the implicitization of rational surfaces in terms of the monomials in the polynomials defining the parametrization, following and expanding our joint article with M. Dohm. These algebraic techniques, based on the theory of approximation complexes due to J. Herzog, A, Simis and W. Vasconcelos, were introduced for the implicitization problem by J.-P. Jouanolou, L. Busé, and M. Chardin. Their work was inspired by the practical method of moving curves, proposed by T. Sederberg and F. Chen, translated into the language of syzygies by D. Cox. Our aim is to express the theoretical results and resulting algorithms into very concrete terms, avoiding the use of the advanced homological commutative algebra tools which are needed for their proofs.

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Section: The Main Algorithm Based On Linear Syzygiesmentioning

confidence: 99%

Implicitization of rational hypersurfaces via linear syzygies: A practical overview

Botbol

Dickenstein

2016

Journal of Symbolic Computation

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“…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]. One of the key steps in [11] is the decomposition of the space of 3 × 2 matrices with homogeneous entries from k k k [x, y] of a fixed degree into disjoint orbits under the action of GL 3 k k k × GL 2 k k k. A successful answer to question (0.4) would have an immediate interpretation in terms of the defining equations of Rees algebras. Eventually, the Rees algebra result would have an interpretation in terms of singularities on parameterized surfaces.…”

Section: 4mentioning

confidence: 99%

“…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].…”

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confidence: 99%

“…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]. One of the key steps in [11] is the decomposition of the space of 3 × 2 matrices with homogeneous entries from k k k [x, y] of a fixed degree into disjoint orbits under the action of GL 3 k k k × GL 2 k k k. A successful answer to question (0.4) would have an immediate interpretation in terms of the defining equations of Rees algebras.…”

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confidence: 99%

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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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“…A homogeneous basis of S is usually called a μ-basis of I and there has been recent work relating properties of such a μ-basis to properties of the curve C, see [2]. In the sequel, once the parameterization of C has been fixed, we will say syzygies and μ-basis of C for syzygies and μ-basis of I.…”

Section: Introductionmentioning

confidence: 99%