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Erschienen in:
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2013 | OriginalPaper | Buchkapitel

Lazarsfeld–Mukai Bundles and Applications

verfasst von : Marian Aprodu

Erschienen in: Commutative Algebra

Verlag: Springer New York

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Abstract

We survey the development of the notion of Lazarsfeld-Mukai bundles together with various applications, from the classification of Mukai manifolds to Brill-Noether theory and syzygies of K3 sections. To see these techniques at work, we present a short proof of a result of M. Reid on the existence of elliptic pencils.

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Nächstes Kapitel Some Applications of Commutative Algebra to String Theory
Fußnoten
1
In fact, we do have a Harder–Narasimhan filtration, but we cannot control all the factors.
 
2
This ingenious procedure is an efficient replacement of the base-point-free pencil trick; “it has killed the base-point-free pencil trick,” to quote Enrico Arbarello.
 
3
Some authors consider that Mukai manifolds have dimension four or more.
 
4
In genus 11, it is actually birational [25].
 
5
The gonality gon(C) of a curve C is the minimal degree of a morphism from C to the projective line.
 
6
It is conjectured that the only other examples should be some half-canonical curves of even genus and maximal gonality [8]; however, this conjecture seems to be very difficult.
 
7
The indices p and q are usually forgotten when defining Koszul cohomology.
 
8
The dimension of K 1, q indicates the number of generators of degree (q + 1) in the homogeneous ideal.
 
9
Duality for Koszul cohomology of curves follows from Serre’s duality. For higher-dimensional manifolds, some supplementary vanishing conditions are required [11, 13].
 
10
Voisin’s and Teixidor’s cases complete each other quite remarkably.
 
11
A curvilinear subscheme is defined locally, in the classical topology, by \(x_{1} = \cdots = x_{s-1} = x_{s}^{k} = 0\); equivalently, it is locally embedded in a smooth curve.
 
12
The connectedness of X c [n] follows from the observation that a curvilinear subscheme is a deformation of a reduced subscheme.
 
13
We see one advantage of working on X c [n]: subtraction makes sense only for curvilinear subschemes.
 
14
The gonality for a singular stable curve is defined in terms of admissible covers [14].
 
15
Higher-rank Brill–Noether theory is a major, rapidly growing research field, and it deserves a separate dedicated survey.
 
16
For any line bundle A, we have γ(A  ⊕ n ) = Cliff(A).
 
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Metadaten
Titel
Lazarsfeld–Mukai Bundles and Applications
verfasst von
Marian Aprodu
Copyright-Jahr
2013
Verlag
Springer New York
Buch
Commutative Algebra

Print ISBN: 978-1-4614-5291-1

Electronic ISBN: 978-1-4614-5292-8

Copyright-Jahr: 2013

DOI
https://doi.org/10.1007/978-1-4614-5292-8_1