Some results on fat points whose support is a
      complete intersection minus a point

Ciliberto, Ciro; Geramita, Antony V.; Harbourne, Brian; Miró‐Roig, Rosa M.; Ranestad, Kristian

doi:10.1515/9783110199703.257

Cited by 7 publications

(6 citation statements)

References 6 publications

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“…This result can be viewed as a Cayley-Bacharach type of result since a set of reduced points has the Cayley-Bacharach property if and only if the degree of every point in X is the same. The results of this section extend our understanding of fat points in special position (see, for example, [13,14] and references there within).…”

Section: Introductionsupporting

confidence: 66%

“…Thus, for any P ∈ Supp(Z), Theorems 5.4 and 6.4 give deg Z (P ) = (13 − 3, 14 − 3, 15 − 3, 15 − 3, 16 − 3, 17 − 3) = (10,11,12,12,13,14).…”

Section: Application: a Cayley-bacharach Type Of Resultsmentioning

confidence: 97%

“…We use Theorem 5.4 to produce a Cayley-Bacharach (to be defined below) type of result for homogeneous sets of fat points in P n whose support is a complete intersection (see [13,14] and references there within, for more on these special configurations). In particular, we show that if Z is a homogeneous fat point scheme whose support is a complete intersection, then deg Z (P ) is the same for every point P ∈ Supp(Z).…”

Section: Application: a Cayley-bacharach Type Of Resultsmentioning

confidence: 99%

“…By results found in [13,14], the minimal graded free resolution of Thus, the minimal separating set of the fat point 2P 36 is the set DEG Z (2P 36 ) = {( 12), (10, 13)}.…”

Section: The Degree Of a Separator And The Minimal Resolutionmentioning

confidence: 91%

See 3 more Smart Citations

Separators of fat points in P^n

Guardo,

Marino,

Van Tuyl

2009

Preprint

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show abstract

Section: Introductionsupporting

confidence: 66%

“…Thus, for any P ∈ Supp(Z), Theorems 5.4 and 6.4 give deg Z (P ) = (13 − 3, 14 − 3, 15 − 3, 15 − 3, 16 − 3, 17 − 3) = (10,11,12,12,13,14).…”

Section: Application: a Cayley-bacharach Type Of Resultsmentioning

confidence: 97%

Section: Application: a Cayley-bacharach Type Of Resultsmentioning

confidence: 99%

“…By results found in [13,14], the minimal graded free resolution of Thus, the minimal separating set of the fat point 2P 36 is the set DEG Z (2P 36 ) = {( 12), (10, 13)}.…”

Section: The Degree Of a Separator And The Minimal Resolutionmentioning

confidence: 91%

See 2 more Smart Citations

Separators of fat points in P^n

Guardo,

Marino,

Van Tuyl

2009

Preprint

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“…But it is not, for example, known which functions arise as Hilbert functions of points taken with higher multiplicity, even if the induced reduced subscheme consists of generic points; see, for example, [Ci] and its bibliography, [Gi], [H5], [Hi] and [Ro]. The paper [GMS] asks what can be said about Hilbert functions and graded Betti numbers of symbolic powers I (m) of the ideal I of a finite set of reduced points in P 2 , particularly when m = 2, while the papers [GuV1] and [GuV2] study I (m) for larger m, but only when I defines a reduced subscheme of projective space whose support is close to a complete intersection. Other work has focused on what one knows as a consequence of knowing the Hilbert function; e.g., [BGM] shows how the growth of the Hilbert function of a set of points influences its geometry, [Cam] studies how the Hilbert function constrains the graded Betti numbers in case I has height 2, and [ER] studies graded Betti numbers but more generally for graded modules M over R.…”

Section: Introductionmentioning

confidence: 99%

Erratum to: Classifying Hilbert functions of fat point subschemes in $${\mathbb{P}^{2}}$$

Geramita

Harbourne²,

Migliore

2011

Collect. Math.

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The paper [GMS] raised the question of what the possible Hilbert functions are for fat point subschemes of the form 2p1 + • • • + 2pr, for all possible choices of r distinct points in P 2 . We study this problem for r points in P 2 over an algebraically closed field k of arbitrary characteristic in case either r ≤ 8 or the points lie on a (possibly reducible) conic. In either case, it follows from [H2] and [H3] that there are only finitely many configuration types of points, where our notion of configuration type is a generalization of the notion of a representable combinatorial geometry, also known as a representable simple matroid. (We say p1, . . . , pr and p ′ 1 , . . . , p ′ r have the same configuration type if for all choices of nonnegative integers mi, Z = m1p1r have the same Hilbert function.) Assuming either that 7 ≤ r ≤ 8 (see [GuH] for the cases r ≤ 6) or that the points pi lie on a conic, we explicitly determine all the configuration types, and show how the configuration type and the coefficients mi determine (in an explicitly computable way) the Hilbert function (and sometimes the graded Betti numbers) of Z = m1p1 + • • • + mrpr. We demonstrate our results by explicitly listing all Hilbert functions for schemes of r ≤ 8 double points, and for each Hilbert function we state precisely how the points must be arranged (in terms of the configuration type) to obtain that Hilbert function.

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Classifying Hilbert functions of fat point subschemes in ℙ2

2009

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The paper [10] raised the question of what the possible Hilbert functions are for fat point subschemes of the form 2p 1 +· · · + 2p r , for all possible choices of r distinct points in P 2 . We study this problem for r points in P 2 over an algebraically closed field k of arbitrary characteristic in case either r ≤ 8 or the points lie on a (possibly reducible) conic. In either case, it follows from [17,18] that there are only finitely many configuration types of points, where our notion of configuration type is a generalization of the notion of a representable combinatorial geometry, also known as a representable simple matroid. (We say p 1 , . . . , p r and p 1 , . . . , p r have the same configuration type if for all choices of nonnegative integers m i , Z = m 1 p 1 + · · · + m r p r and Z = m 1 p 1 + · · · + m r p r have 159 Collectanea Mathematica (electronic version): http://www.collectanea.ub.edu Geramita, Harbourne and Migliore the same Hilbert function.) Assuming either that 7 ≤ r ≤ 8 (see [12] for the cases r ≤ 6) or that the points p i lie on a conic, we explicitly determine all the configuration types, and show how the configuration type and the coefficients m i determine (in an explicitly computable way) the Hilbert function (and sometimes the graded Betti numbers) of Z = m 1 p 1 + · · · + m r p r . We demonstrate our results by explicitly listing all Hilbert functions for schemes of r ≤ 8 double points, and for each Hilbert function we state precisely how the points must be arranged (in terms of the configuration type) to obtain that Hilbert function.

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Some results on fat points whose support is a complete intersection minus a point

Cited by 7 publications

References 6 publications

Separators of fat points in P^n

Separators of fat points in P^n

Erratum to: Classifying Hilbert functions of fat point subschemes in $${\mathbb{P}^{2}}$$

Classifying Hilbert functions of fat point subschemes in ℙ2

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