The gonality of curves on a Hirzebruch surface

Martens, Gerriet

doi:10.1007/bf01197600

Cited by 34 publications

(29 citation statements)

References 3 publications

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“…Additionally, we recover the description of the minimal pencils in these cases, which has already been known before (for plane curve, see, for example [ACGH], for curves on a Hirzebruch surface we refer to [Ma1]). …”

Section: Introduction Main Resultsmentioning

confidence: 99%

“…The obvious inequality H k,m .X ≥ 2g + 1, together with the existence of a g 1 k given by the ruling, show that the gonality conjecture is verified for X (the fact that X is k-gonal was previously proved in [Ma1], by using completely different methods): Theorem 6.3. Let e ≥ 0, k ≥ 2, and m ≥ max{ke, k+e} be three integers, and X ≡ kC 0 + mf be a curve on the Hirzebruch surface Σ e .…”

Section: Syzygies Of Curves On a Hirzebruch Surfacementioning

confidence: 99%

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On the vanishing of higher syzygies of curves

Aprodu¹

2002

Math. Z.

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The present paper is related to a conjecture made by Green and Lazarsfeld concerning 1-linear syzygies of curves embedded by complete linear systems of sufficiently large degrees. Given a smooth, irreducible, complex, projective curve $X$, we prove that the least integer $q$ for which the property $(M_q)$ fails for a line bundle $L$ on $X$ does not depend on $L$ as soon as its degree becomes sufficiently large. Consequently, this number is an invariant of the curve, and the statement of Green-Lazarsfeld's conjecture is equivalent to saying that this invariant equals the gonality of the curve. We verify the conjecture for plane curves, curves lying on Hirzebruch surfaces, and for generic curves having the genus sufficiently large compared to the gonality. We conclude the paper by proving that Green's canonical conjecture holds for curves lying on Hirzebruch surfaces.Comment: to appear in Math.

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Section: Introduction Main Resultsmentioning

confidence: 99%

Section: Syzygies Of Curves On a Hirzebruch Surfacementioning

confidence: 99%

On the vanishing of higher syzygies of curves

Aprodu¹

2002

Math. Z.

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“…In our opinion in characteristic zero the best results are still obtained using [7] or the case e = 0 of [10] and [6], Remark 2 on page 351. To get Corollary 1 and related results we need first to work over an algebraically closed field K and study low degree linear series on smooth models of singular curves on a smooth quadric surface Q (see section 2).…”

Section: We Have Y = 2a If and Only If Y (Kmentioning

confidence: 89%

Gonality of curves with a singular model on an elliptic quadric surface

Ballico¹

2014

Turk J Math

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“…It suffices to check that D has no g 1 6 (since if D has a g 1 3 it has a g 1 6 ). But it follows from a theorem of Martens [18] that if D has a g 1 k then k ≥ 2d.…”

Section: An Irrational Examplementioning

confidence: 99%