On bundles of rank 3 computing Clifford indices

Lange, Herbert; Newstead, P. E.

doi:10.1215/21562261-1966062

Cited by 7 publications

(17 citation statements)

References 18 publications

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“…When r = 3 and g = 9, the fact that a general curve has Cliff 3 (C) = 10 3 was known according to the results in [16]. When r = 3 and g = 11, our result implies that a general curve has Cliff 3 (C) = 14 3 , which improves the result 11 3 ≤ Cliff 3 (C) ≤ 14 3 in [14,Theorem 3.6].…”

Section: Introductionsupporting

confidence: 57%

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Higher rank Clifford indices of curves on a K3 surface

Feyzbakhsh

2021

Sel. Math. New Ser.

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Let (X, H) be a polarized K3 surface with $$\mathrm {Pic}(X) = \mathbb {Z}H$$ Pic ( X ) = Z H , and let $$C\in |H|$$ C ∈ | H | be a smooth curve of genus g. We give an upper bound on the dimension of global sections of a semistable vector bundle on C. This allows us to compute the higher rank Clifford indices of C with high genus. In particular, when $$g\ge r^2\ge 4$$ g ≥ r 2 ≥ 4 , the rank r Clifford index of C can be computed by the restriction of Lazarsfeld–Mukai bundles on X corresponding to line bundles on the curve C. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher rank vector bundles. We also apply the same method to the projective plane and show that the rank r Clifford index of a degree $$d(\ge 5)$$ d ( ≥ 5 ) smooth plane curve is $$d-4$$ d - 4 , which is the same as the Clifford index of the curve.

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Section: Introductionsupporting

confidence: 57%

“…In particular, we show that the first part of the Mercat's conjecture [24] holds for smooth plane curves: The result Cliff 2 (C) = l − 4 for plane curves first appeared in [15,Proposition 8.1]. Further discussions for the rank 3 case appeared in [16]. In particular, the result Cliff 3 (C) = l − 4 was known for l ≤ 6.…”

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confidence: 76%

Higher rank Clifford indices of curves on a K3 surface

Feyzbakhsh

2021

Sel. Math. New Ser.

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“…It follows that h 0 (E H ⊗ E H ) ≤ 9 (in fact h 0 (E H ⊗ E H ) = 9). This gives a negative answer to [13,Question 8.4] in this case. Comment 9.2.…”

Section: Bundles Of Degree 12 With 5 Sectionsmentioning

confidence: 95%

“…It can be proved from Proposition 6.1 that, if π : C → Γ is the normalization of an irreducible plane curve of degree 7 such that C has genus 13 or 14, then Cliff(C) = 3. In fact, this can be extended to the case when C has genus ≥ 9, provided the only singularities of Γ are ordinary nodes or cusps (see [ (see [13,Remark 5.10]). Remark 6.6.…”

Section: Genus 13 and 14mentioning

confidence: 99%

Bundles of rank 3 on curves of Clifford index 3

Lange

Newstead

2013

Journal of Symbolic Computation

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Two definitions of the Clifford index for vector bundles on a smooth projective curve $C$ of genus $g\ge4$ were introduced in a previous paper by the authors. In another paper the authors obtained results on one of these indices for bundles of rank 3. In this paper we extend these results in the case where $C$ has classical Clifford index 3. In particular we prove Mercat's conjecture for bundles of rank 3 for $g \leq 8$ and $g \geq 13$ when $C$ has classical Clifford index 3. We obtain complete results in the case of genus 7.Comment: Final version. Typos corrected and minor style change

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“…Many properties of these indices have been obtained in [5] and subsequent papers. It is an interesting question to determine the bundles which compute the Clifford indices; for results in ranks 2 and 3, see [7,8].…”

Section: Introductionmentioning

confidence: 99%