A class of non-rational surface singularities with bijective Nash map

Camille Plénat; Patrick Popescu-Pampu

Bulletin de la Société Mathématique de France (2006)

  • Volume: 134, Issue: 3, page 383-394
  • ISSN: 0037-9484

Abstract

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Let ( 𝒮 , 0 ) be a germ of complex analytic normal surface. On its minimal resolution, we consider the reduced exceptional divisor E and its irreducible components E i , i ∈ I . The Nash map associates to each irreducible component C k of the space of arcs through 0 on 𝒮 the unique component of E cut by the strict transform of the generic arc in C k . Nash proved its injectivity and asked if it was bijective. As a particular case of our main theorem, we prove that this is the case if E · E i < 0 for any  i ∈ I .

How to cite

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Plénat, Camille, and Popescu-Pampu, Patrick. "A class of non-rational surface singularities with bijective Nash map." Bulletin de la Société Mathématique de France 134.3 (2006): 383-394. <http://eudml.org/doc/272440>.

@article{Plénat2006,
abstract = {Let $(\mathcal \{S\},0)$ be a germ of complex analytic normal surface. On its minimal resolution, we consider the reduced exceptional divisor $E$ and its irreducible components $E_\{i\}$, $i \in I$. The Nash map associates to each irreducible component $C_k$ of the space of arcs through $0$ on $\mathcal \{S\}$ the unique component of $E$ cut by the strict transform of the generic arc in $C_k$. Nash proved its injectivity and asked if it was bijective. As a particular case of our main theorem, we prove that this is the case if $E\cdot E_\{i\} &lt;0$ for any $\{i \in I\}$.},
author = {Plénat, Camille, Popescu-Pampu, Patrick},
journal = {Bulletin de la Société Mathématique de France},
keywords = {space of arcs; Nash map; Nash problem},
language = {eng},
number = {3},
pages = {383-394},
publisher = {Société mathématique de France},
title = {A class of non-rational surface singularities with bijective Nash map},
url = {http://eudml.org/doc/272440},
volume = {134},
year = {2006},
}

TY - JOUR
AU - Plénat, Camille
AU - Popescu-Pampu, Patrick
TI - A class of non-rational surface singularities with bijective Nash map
JO - Bulletin de la Société Mathématique de France
PY - 2006
PB - Société mathématique de France
VL - 134
IS - 3
SP - 383
EP - 394
AB - Let $(\mathcal {S},0)$ be a germ of complex analytic normal surface. On its minimal resolution, we consider the reduced exceptional divisor $E$ and its irreducible components $E_{i}$, $i \in I$. The Nash map associates to each irreducible component $C_k$ of the space of arcs through $0$ on $\mathcal {S}$ the unique component of $E$ cut by the strict transform of the generic arc in $C_k$. Nash proved its injectivity and asked if it was bijective. As a particular case of our main theorem, we prove that this is the case if $E\cdot E_{i} &lt;0$ for any ${i \in I}$.
LA - eng
KW - space of arcs; Nash map; Nash problem
UR - http://eudml.org/doc/272440
ER -

References

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