Gonality of non-Gorenstein curves of genus five

Feital, Lia; Martins, Renato Vidal

doi:10.1007/s00574-014-0067-5

Cited by 6 publications

(6 citation statements)

References 14 publications

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“…Conversely, assume gon(C) = 4 and let us prove that C ′ lies on a threefold scroll but not on a surface scroll. By [FM,p. 10], the gonality of C is computed by a sheaf F := O 1, t n for some nonzero n ∈ Z.…”

Section: Rational Curves On Threefold Scrollsmentioning

confidence: 98%

“…Conversely, assume gon(C) = 4 and let us prove that C ′ lies on a threefold scroll but not on a surface scroll. By [FM,p. 10] For the remainder we will analyze the possibilities for S according to each n. For this, we will split the semigroup into blocks of consecutive integers, that is, write (d) the elements of S are black, the ones of K \ S are double-circled, and the remaining numbers are white.…”

Section: Canonical Models On Threefold Scrollsmentioning

confidence: 98%

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Curves with canonical models on scrolls

Lara¹,

Marchesi

Martins

2016

Int. J. Math.

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Abstract. Let C be an integral and projective curve whose canonical model C ′ lies on a rational normal scroll S of dimension n. We mainly study some properties on C, such as gonality and the kind of singularities, in the case where n = 2 and C is non-Gorenstein, and in the case where n = 3, the scroll S is smooth, and C ′ is a local complete intersection inside S. We also prove that a rational monomial curve with just one singular point lies on a surface scroll if and only if its gonality is at most 3, and that it lies on a threefold scroll if and only if its gonality is at most 4.

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Section: Rational Curves On Threefold Scrollsmentioning

confidence: 98%

Section: Canonical Models On Threefold Scrollsmentioning

confidence: 98%

Curves with canonical models on scrolls

Lara¹,

Marchesi

Martins

2016

Int. J. Math.

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“…Indeed, that deg(F) = 4 follows from (3.3) and the fact that (((S ∪ (S + 2)) \ S) = {2, 9}. On the other hand, the gonality of a unicuspidal rational monomial curve is always realized by a sheaf of the form O⟨1, t µ ⟩ [9]. Applying (3.3) with µ = 1, 3 and 4 yields degrees 5, 5 and 7 respectively; meanwhile, for every µ ≥ 5, it follows immediately from the right-hand side of…”

Section: Gonalities From Canonical Modelsmentioning

confidence: 99%

Towards Brill–Noether theory for cuspidal curves

Cotterill,

Martins

2024

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Understanding when an abstract complex curve of given genus comes equipped with a map of fixed degree to a projective space of fixed dimension is a foundational question; and Brill-Noether theory addresses this question via linear series, which algebraically codify maps to projective targets. Classical Brill-Noether theory, which focuses on smooth curves, has been intensively explored; but much less is known for singular curves, particularly for those with non-nodal singularities. In a one-parameter family of smooth curves specializing to a singular curve C0, one expects certain aspects of the global geometry of the smooth fibers to "specialize" to the local geometry of the singularities of C0. Making this expectation quantitatively precise involves analyzing the arithmetic and combinatorics of semigroups S attached to discrete valuations defined on (the local rings of) these singularities. In this largelyexpository note we focus primarily on Brill-Noether-type results for curves with cusps, i.e., unibranch singularities; in this setting, the associated semigroups are numerical semigroups with finite complement in N.

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“…The description of more complex non-Gorenstein curves via gonality is a subject still little explored in the literature. In [9], a study of curves of genus at most 5 is carried out based on a systematic analysis of all possible semigroups for such singularities. This is the point of departure of the present work.…”

Section: Introductionmentioning

confidence: 99%

On Gonality and Canonical Models of Unicuspidal Rational Curves

Galdino¹,

Nicolau²,

Martins³

2022

Preprint

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The goal of this work is the study of the gonality of a rational unicuspidal curve C. We are mainly interested in the case where C is non-Gorenstein. We classify such curves via different notions of gonality, and by its canonical model C , up to genus 6. We do it by means of more general families of curves of arbitrary genus. Afterwards, we get a general formula for the dimension of the space of hypersurfaces of a fixed degree containing C , which we apply to some particular cases.

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Gonality of non-Gorenstein curves of genus five

Cited by 6 publications

References 14 publications

Curves with canonical models on scrolls

Curves with canonical models on scrolls

Towards Brill–Noether theory for cuspidal curves

On Gonality and Canonical Models of Unicuspidal Rational Curves

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