Linear Series with an N-Fold Point on a General Curve

Abstract This paper aims at settling the issue of the validity of the de Jonquières formulas. Consider the space of divisors with prescribed multiplicity, or de Jonquières divisors, contained in a linear series on a smooth projective curve. Under the assumption that this space is zero dimensional, the de Jonquières formulas compute the expected number of de Jonquières divisors. Using degenerations to nodal curves we show that, for a general curve equipped with a complete linear series, the space is of expected dimension, which shows that the counts are in fact true. This implies that in the case of negative expected dimension a general linear series on a general curve does not admit de Jonquières divisors of the expected type.

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Geometry of curves with exceptional secant planes: linear series along the general curve

Mathematische Zeitschrift ◽

10.1007/s00209-009-0635-3 ◽

2009 ◽

Vol 267 (3-4) ◽

pp. 549-582 ◽

Cited By ~ 5

Author(s):

Ethan Cotterill

Keyword(s):

Linear Series ◽

General Curve

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Divisor Varieties of Symmetric Products

International Mathematics Research Notices ◽

10.1093/imrn/rnaa313 ◽

2021 ◽

Author(s):

John Sheridan

Keyword(s):

Algebraic Curves ◽

Linear Series ◽

Symmetric Product ◽

Fixed Degree ◽

Symmetric Products ◽

Projective Varieties ◽

General Curve ◽

Higher Dimensional ◽

Detailed Picture

Abstract The geometry of divisors on algebraic curves has been studied extensively over the years. The foundational results of this Brill-Noether theory imply that on a general curve, the spaces parametrizing linear series (of fixed degree and dimension) are smooth, irreducible projective varieties of known dimension. For higher dimensional varieties, the story is less well understood. Our purpose in this paper is to study in detail one class of higher dimensional examples where one can hope for a quite detailed picture, namely (the spaces parametrizing) divisors on the symmetric product of a curve.

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Linear stability and stability of syzygy bundles

International Journal of Mathematics ◽

10.1142/s0129167x18500805 ◽

2018 ◽

Vol 29 (11) ◽

pp. 1850080 ◽

Cited By ~ 1

Author(s):

Abel Castorena ◽

H. Torres-López

Keyword(s):

Exact Sequence ◽

Linear Stability ◽

Commutative Diagram ◽

Linear Series ◽

Stability Conditions ◽

Stable Bundle ◽

Projective Curve ◽

General Curve ◽

The Stability ◽

Natural Way

Let [Formula: see text] be a smooth irreducible projective curve and let [Formula: see text] be a complete and generated linear series on [Formula: see text]. Denote by [Formula: see text] the kernel of the evaluation map [Formula: see text]. The exact sequence [Formula: see text] fits into a commutative diagram that we call the Butler’s diagram. This diagram induces in a natural way a multiplication map on global sections [Formula: see text], where [Formula: see text] is a subspace and [Formula: see text] is the dual of a subbundle [Formula: see text]. When the subbundle [Formula: see text] is a stable bundle, we show that the map [Formula: see text] is surjective. When [Formula: see text] is a Brill–Noether general curve, we use the surjectivity of [Formula: see text] to give another proof of the semistability of [Formula: see text], moreover, we fill up a gap in some incomplete argument by Butler: With the surjectivity of [Formula: see text] we give conditions to determine the stability of [Formula: see text], and such conditions imply the well-known stability conditions for [Formula: see text] stated precisely by Butler. Finally we obtain the equivalence between the (semi)stability of [Formula: see text] and the linear (semi)stability of [Formula: see text] on [Formula: see text]-gonal curves.

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