Abel maps and limit linear series

Esteves, Eduardo; Osserman, Brian

doi:10.1007/s12215-013-0111-0

Cited by 17 publications

(50 citation statements)

References 7 publications

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“…1.4, p. 4034, at least if X is general. Furthermore, limit linear series that are actual limits of linear series along families of smooth curves degenerating to X are exact, as long as the total space of the family is regular; see , Section 5, p. 90. The space

G_{d}^{r, Oss} false(X false)

has a defect similar to that of

G_{d}^{r} false(X false)

though: There are non‐exact limit linear series that are limits of linear series along (nonregular) smoothings of X .…”

Section: Introductionmentioning

confidence: 99%

“…Given a family of linear series

frakturg

of degree d on smooth curves degenerating to a singular curve, we may ask: What is the limit of the

P (g)

on the d ‐th symmetric product of the singular curve? In Osserman and the first author considered the corresponding subscheme

P (g)

X^{false(d false)}

associated to an Osserman limit linear series

frakturg

on X . It is defined as for smooth curves but, as some sections of the linear series of

frakturg

may vanish on a whole component of X , the subscheme

P (g)

is actually the closure of the locus of divisors of zeros of all the other sections.…”

Section: Introductionmentioning

confidence: 99%

“…Finally, following , we associate with each level‐δ limit linear series

frakturg

a subscheme

P (g)

X^{false(d false)}

. The association is natural through the several levels: If

g^{'}

is a level‐

δ^{'}

limit linear series such that

ρ_{δ^{'}, δ} false(g^{'} false) = g

then

P false(frakturg false) \subseteq P false(g^{'} false)

; see Theorem .…”

Section: Introductionmentioning

confidence: 99%

“…In particular, we construct a projective moduli space

G_{d, δ}^{r} false(X false)

parameterizing level‐δ limit linear series of rank r and degree d on X , and show that it is a new compactification, for each δ, of the moduli space of Osserman exact limit linear series. Finally, we generalize by associating with each exact level‐δ limit linear series

frakturg

on X a closed subscheme

P false(frakturg false) \subseteq X^{(d)}

of the d th symmetric product of X , and showing that, if

frakturg

is a limit of linear series on the smooth curves degenerating to X , then

P (g)

is the limit of the corresponding spaces of divisors. In short, we describe completely limits of divisors along degenerations to such a curve X .…”

mentioning

confidence: 99%

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Level‐ limit linear series

Esteves

Nigro

Rizzo

2017

Mathematische Nachrichten

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We consider all one‐parameter families of smooth curves degenerating to a singular curve X and describe limits of linear series along such families. We treat here only the simplest case where X is the union of two smooth components meeting transversely at a point P. We introduce the notion of level‐δ limit linear series on X to describe these limits, where δ is the singularity degree of the total space of the degeneration at P. If the total space is regular, that is, δ=1, we recover the limit linear series introduced by Osserman in . So we extend his treatment to a more general setup. In particular, we construct a projective moduli space Gd,δrfalse(Xfalse) parameterizing level‐δ limit linear series of rank r and degree d on X, and show that it is a new compactification, for each δ, of the moduli space of Osserman exact limit linear series. Finally, we generalize by associating with each exact level‐δ limit linear series frakturg on X a closed subscheme Pfalse(frakturgfalse)⊆X(d) of the dth symmetric product of X, and showing that, if frakturg is a limit of linear series on the smooth curves degenerating to X, then P(g) is the limit of the corresponding spaces of divisors. In short, we describe completely limits of divisors along degenerations to such a curve X.

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G_{d}^{r, Oss} false(X false)

has a defect similar to that of

G_{d}^{r} false(X false)

though: There are non‐exact limit linear series that are limits of linear series along (nonregular) smoothings of X .…”

Section: Introductionmentioning

confidence: 99%

“…Given a family of linear series

frakturg

of degree d on smooth curves degenerating to a singular curve, we may ask: What is the limit of the

P (g)

on the d ‐th symmetric product of the singular curve? In Osserman and the first author considered the corresponding subscheme

P (g)

X^{false(d false)}

associated to an Osserman limit linear series

frakturg

on X . It is defined as for smooth curves but, as some sections of the linear series of

frakturg

may vanish on a whole component of X , the subscheme

P (g)

is actually the closure of the locus of divisors of zeros of all the other sections.…”

Section: Introductionmentioning

confidence: 99%

“…Finally, following , we associate with each level‐δ limit linear series

frakturg

a subscheme

P (g)

X^{false(d false)}

. The association is natural through the several levels: If

g^{'}

is a level‐

δ^{'}

limit linear series such that

ρ_{δ^{'}, δ} false(g^{'} false) = g

then

P false(frakturg false) \subseteq P false(g^{'} false)

; see Theorem .…”

Section: Introductionmentioning

confidence: 99%

“…In particular, we construct a projective moduli space

G_{d, δ}^{r} false(X false)

frakturg

on X a closed subscheme

P false(frakturg false) \subseteq X^{(d)}

of the d th symmetric product of X , and showing that, if

frakturg

is a limit of linear series on the smooth curves degenerating to X , then

P (g)

is the limit of the corresponding spaces of divisors. In short, we describe completely limits of divisors along degenerations to such a curve X .…”

mentioning

confidence: 99%

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Level‐ limit linear series

Esteves

Nigro

Rizzo

2017

Mathematische Nachrichten

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“…Also, Abel maps of any degree were constructed for stable curves of compact type in [10], and for nodal curves with two components in [1]. It is worth mentioning as well the relation established in [13] between limit linear series and fibers of Abel maps for two-component curves of compact type.…”

Section: Introductionmentioning

confidence: 99%

Brill–Noether Locus of Rank 1 and Degreeg− 1 on a Nodal Curve

Coelho

Esteves

2015

Communications in Algebra

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In this paper we consider the Brill-Noether locus W d C of line bundles of multidegree d of total degree g − 1 having a nonzero section over a connected nodal curve C of genus g. We give an explicit description of the irreducible components of W d C for a semistable multidegree d. As a consequence we show that, if two semistable multidegrees of total degree g − 1 over a curve with no rational components differ by a twister, then the corresponding Brill-Noether loci are isomorphic.

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The resolution of the degree‐2 Abel‐Jacobi map for nodal curves‐I

Pacini

2014

Mathematische Nachrichten

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Let f:C→B be a regular local smoothing of a nodal curve. In this paper, we find a modular description of the Néron maps associated to Abel‐Jacobi maps with values in Esteves's fine compactified Jacobian and with source the B‐smooth locus of either the double product of scriptC over B or the degree‐2 Hilbert scheme of the family f. This is the first of a series of two papers dedicated to the construction of a resolution of the degree‐2 Abel‐Jacobi map for a regular smoothing of a nodal curve.

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Abel maps and limit linear series

Cited by 17 publications

References 7 publications

Level‐ limit linear series

Level‐ limit linear series

Brill–Noether Locus of Rank 1 and Degreeg− 1 on a Nodal Curve

The resolution of the degree‐2 Abel‐Jacobi map for nodal curves‐I

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