## LIMIT CANONICAL SYSTEMS ON CURVES WITH TWO COMPONENTS (2000)

Citations: | 9 - 2 self |

### BibTeX

@MISC{Esteves00limitcanonical,

author = {Eduardo Esteves and Nivaldo Medeiros},

title = {LIMIT CANONICAL SYSTEMS ON CURVES WITH TWO COMPONENTS},

year = {2000}

}

### OpenURL

### Abstract

Abstract. In the 80’s D. Eisenbud and J. Harris considered the following problem: “What are the limits of Weierstrass points in families of curves degenerating to stable curves? ” But for the case of stable curves of compact type, treated by them, this problem remained wide open since then. In the present article, we propose a concrete approach to this problem, and give a quite explicit solution for stable curves with just two irreducible components meeting at points in general position. Contents

### Citations

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48 |
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(Show Context)
Citation Context ...Some history. Limits of ramification points and linear systems were studied by Eisenbud and Harris in the 80’s, when they developed the theory of limit linear series for curves C of compact type; see =-=[EH1]-=-. (As observed in [HM, p. 256], the theory works more generally when C is treelike, that is, when the normalization of C at all its irreducible nodes is of compact type.) Many important applications o... |

18 |
Compactifying the relative Jacobian over families of reduced curves
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(Show Context)
Citation Context ...nding smoothing of C. (Recall that a sheaf L on C is simple if End(L) = k.) So we need only consider sheaves that are simple on C to obtain a projective variety V. This is the same situation found in =-=[E3]-=-, where a compactification of the (relative) Jacobian of a family of curves was constructed using only simple sheaves. Theorem 10.1 asserts the importance of the data αµ, βµ, Iµ, Jµ: they determine Vµ... |

10 |
Existence, decomposition and limits of certain Weierstrass points
- Eisenbud, Harris
(Show Context)
Citation Context ...7. Regeneration, I 20 8. Regeneration, II 24 9. Boundary of tori orbits on Grassmannians 29 10. The variety of limit canonical systems 35 11. Miscellany 46 References 52 1. Introduction 1.1. Goal. In =-=[EH2]-=- D. Eisenbud and J. Harris asked: “What are the limits of Weierstrass points in families of curves degenerating to stable curves not of compact type?” As announced in [EM], in the present article we a... |

10 | Linear systems and ramification points on reducible nodal curves, math.AG/9808069, Algebra Meeting (Rio de Janeiro
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- 1996
(Show Context)
Citation Context ...linear series [R], but the crucial relationship with degenerations of ramification points was yet to be established. It was only recently that this relationship was established by the first author in =-=[E2]-=-, building up on the work of Ran (or rather rediscovering it) and using the substitutes for the sheaves of (relative) principal parts discovered in [E1]. It is apparently possible to generalize the th... |

9 | Progress in the theory of complex algebraic curves - Eisenbud, Harris - 1989 |

6 |
Wronski algebra systems on families of singular curves
- Esteves
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(Show Context)
Citation Context ...tionship was established by the first author in [E2], building up on the work of Ran (or rather rediscovering it) and using the substitutes for the sheaves of (relative) principal parts discovered in =-=[E1]-=-. It is apparently possible to generalize the theory of limit linear series of [EH1] starting from results in [E2]. Though a more general theory of limit linear series is yet to be developed, it is al... |

4 |
Degenerations of linear series and binary curves
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(Show Context)
Citation Context ... R. Pandharipande [P] who takes a Geometric Invariant Theory approach to proving the existence of what he calls generalized linear series for certain degenerations. Second, there is A. Bruno’s thesis =-=[B1]-=-, where he studies degenerations of linear systems to a stable curve with two components. It’s particularly interesting the conditions he gives for such a curve to be a limit of smooth plane quintics.... |

4 | Pluricanonical-Gorenstein-curves, in “Enumerative geometry and classical algebraic geometry - Catanese - 1981 |

4 |
Moduli space of enriched stable curves
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(Show Context)
Citation Context ... and obtain partial results. In contrast with the methods used in the present article, they use admissible covers. Besides [E2], the recent work by L. Mainò on enriched structures in her Ph.D. thesis =-=[M]-=- was important for us. Actually, as we deal here only with limits of canonical systems, we used only a fraction of what is available in [M]. There are further recent works on (generalizations of) limi... |

3 |
Limit linear series and families of equisingular plane curves
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(Show Context)
Citation Context ...degenerations of linear systems to a stable curve with two components. It’s particularly interesting the conditions he gives for such a curve to be a limit of smooth plane quintics. Finally, there is =-=[B2]-=-, where applications to (variants of) a problem of Severi’s are given.LIMIT CANONICAL SYSTEMS ON CURVES WITH TWO COMPONENTS 3 We thank A. Bruno, M. Coppens, D. Eisenbud, J. Harris, S. Kleiman and spe... |

3 | The irreducibility of the space of curves with given genus, Publ - Deligne, Mumford - 1969 |

2 |
Limit Weierstrass schemes on stable curves with 2 irreducible components, preprint
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(Show Context)
Citation Context ...with two elliptic components meeting at two points in general position. In fact, it was only very recently that partial progress was made by M. Coppens and L. Gatto towards answering the question. In =-=[CG]-=- Coppens and Gatto consider a stable curve C with just two irreducible components meeting at δ points in general position, and show that all the points in each collection of δ−1 smooth points of C are... |

2 |
Limits of Weierstrass points in regular smoothings of curves with two components
- Esteves, Medeiros
(Show Context)
Citation Context ... Introduction 1.1. Goal. In [EH2] D. Eisenbud and J. Harris asked: “What are the limits of Weierstrass points in families of curves degenerating to stable curves not of compact type?” As announced in =-=[EM]-=-, in the present article we answer their question for stable curves with just two irreducible components meeting at points in general position. For a glimpse of our results, consider the following par... |

2 |
Degenerations of linear systems
- Ran
- 1985
(Show Context)
Citation Context ... the topic generated in the 80’s, the theory of limit linear series could not be extended to stable curves of more general type. There was unpublished work of Z. Ran on degenerations of linear series =-=[R]-=-, but the crucial relationship with degenerations of ramification points was yet to be established. It was only recently that this relationship was established by the first author in [E2], building up... |

1 |
A Geometric Invariant Theory approach to admissible covers and generalized linear series. A rough draft
- Pandharipande
- 1995
(Show Context)
Citation Context ... There are further recent works on (generalizations of) limit linear series worth noticing, even though they are not completely related to what we do here. First, there is a draft by R. Pandharipande =-=[P]-=- who takes a Geometric Invariant Theory approach to proving the existence of what he calls generalized linear series for certain degenerations. Second, there is A. Bruno’s thesis [B1], where he studie... |

1 |
Infinitesimal deformations of singularities
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- 1964
(Show Context)
Citation Context ...mation theory. Let C be a nodal curve defined over a field k. Since C is one– dimensional, generically smooth, and a local complete intersection, Ext 2 C (Ω1 C , OC) = 0 by [DM, Lemma 1.3, p. 79]. By =-=[S]-=-, there exists a versal formal deformation V of C over the base scheme M := Spec(k[[t1, . . ., te]]), where e := dimk Ext 1 C (Ω1C , OC). Since C is a curve, H 2 (C, OC) = 0. By [S], the formal scheme... |