On varieties of almost minimal degree in small codimension

Brodmann, Markus; Schenzel, Peter

doi:10.1016/j.jalgebra.2006.03.027

Cited by 14 publications

(20 citation statements)

References 9 publications

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“…It is guessed by Brodmann and Schenzel that the type of the rational normal scroll that contains X determines the Betti diagram "near the beginning of the resolution". See Example 4.8 in [BS1] and Example 9.1 in [BS2]. And our result gives an affirmative answer for their expectation.…”

Section: Introductionsupporting

confidence: 61%

Smooth varieties of almost minimal degree

Park

2007

Journal of Algebra

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In this article we study non-linearly normal smooth projective varieties X ⊂ P r of deg(X) = codim(X, P r ) + 2. We first give geometric characterizations for X (Theorem 1.1). Indeed X is the image of an isomorphic projection of smooth varieties X ⊂ P r+1 of minimal degree. Also if X is not the Veronese surface, then there exists a smooth rational normal scroll Y ⊂ P r which contains X as a divisor linearly equivalent to H + 2F where H is the hyperplane section of Y and F is a fiber of the projection morphism π : Y → P 1 . By using these characterizations, (1) we determine all the possible types of Y from the type of X (Theorem 1.2), and (2) we investigate the relation between the Betti diagram of X and the type of Y (Theorem 1.3). In particular, we clarify the relation between the number of generators of the homogeneous ideal of X and the type of Y . As an application, we construct non-linearly normal examples where the converse to Theorem 1.1 in [D. Eisenbud, M. Green, K. Hulek, S. Popescu, Restriction linear syzygies: Algebra and geometry, Compos. Math. 141 (2005) 1460-1478] fails to hold (Remark 2).

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Section: Introductionsupporting

confidence: 61%

Smooth varieties of almost minimal degree

Park

2007

Journal of Algebra

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“…Also when X is nonarithmetically Cohen-Macaulay, there have been several partial results concerned with the minimal free resolution of X (e.g. [1], [2], [7], [10], [11], [12]). In particular, M. Brodmann and P. Schenzel [1] gave a complete list of all occurring Betti diagrams in the case where codim(X, P r K ) ≤ 4.…”

Section: Introductionmentioning

confidence: 99%

“…[1], [2], [7], [10], [11], [12]). In particular, M. Brodmann and P. Schenzel [1] gave a complete list of all occurring Betti diagrams in the case where codim(X, P r K ) ≤ 4. However, in general, it remains open to determine the graded Betti numbers of X.…”

Section: Introductionmentioning

confidence: 99%

Classification of Betti Diagrams of Varieties of Almost Minimal Degree

Lee¹,

Park²

2011

Journal of the Korean Mathematical Society

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Abstract. In this article we study the problem to determine all occurring Betti diagrams of varieties X ⊂ P r of almost minimal degree, i.e., deg(X) = codim(X, P r ) + 2. We describe a realistic picture of how many different kind of Betti diagrams exist at all (Theorem 3.1). By means of the computer algebra system "SINGULAR", we obtain a complete list of all occurring Betti diagrams in the cases where codim(X, P r ) ≤ 8.

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“…X = π P ( X) where X ⊂ P d is a rational normal curve and P ∈ X 2 \ X. 3. X = π P ( X) where X ⊂ P d is a rational normal curve and P ∈ P d \ X 2 .…”

Section: Introductionmentioning

confidence: 99%

“…Also they prove that in the second case, if the variety of minimal degree is a rational normal scroll, then X lies on a rational normal scroll as a divisor. In [3], they guess that if X is a proper projection of a rational normal scroll, then the type of the rational normal scroll that contains X determines the Betti diagram "near the beginning of the resolution". For n = 1, Theorem 1.1 provides an affirmative answer to their expectation.…”

Section: Introductionmentioning

confidence: 99%

Projective curves of degree=codimension+2

Park

2007

Math. Z.

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In this article we study nondegenerate projective curves X ⊂ P d−1 of degree d which are not arithmetically Cohen-Macaulay. Note that X = π P ( X) for a rational normal curve X ⊂ P d and a point P ∈ P d \ X 2 . Our main result is about the relation between the geometric properties of X and the position of P with respect to X. We show that the graded Betti numbers of X are uniquely determined by the rank rk X P of P with respect to X. In particular, X satisfies property N 2,p if and only if p ≤ rk X P − 3. Therefore property N 2,p of X is controlled by rk X P and conversely rk X P can be read off from the minimal free resolution of X. This result provides a non-linearly normal example for which the converse to Theorem 1.1 in (Eisenbud et al., Compositio Math 141:1460-1478, 2005 holds. Also our result implies that for nondegenerate projective curves X ⊂ P d−1 of degree d which are not arithmetically Cohen-Macaulay, there are exactly d−2 2 distinct Betti tables.

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On varieties of almost minimal degree in small codimension

Cited by 14 publications

References 9 publications

Smooth varieties of almost minimal degree

Smooth varieties of almost minimal degree

Classification of Betti Diagrams of Varieties of Almost Minimal Degree

Projective curves of degree=codimension+2

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