Essential dimension of moduli of curves and other algebraic stacks

Patrick Brosnan; Zinovy Reichstein; Angelo Vistoli

Journal of the European Mathematical Society (2011)

  • Volume: 013, Issue: 4, page 1079-1112
  • ISSN: 1435-9855

Abstract

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In this paper we consider questions of the following type. Let k be a base field and K / k be a field extension. Given a geometric object X over a field K (e.g. a smooth curve of genus g ), what is the least transcendence degree of a field of definition of X over the base field k ? In other words, how many independent parameters are needed to define X ? To study these questions we introduce a notion of essential dimension for an algebraic stack. Using the resulting theory, we give a complete answer to the question above when the geometric objects X are smooth, stable or hyperelliptic curves. The appendix, written by Najmuddin Fakhruddin, answers this question in the case of abelian varieties.

How to cite

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Brosnan, Patrick, Reichstein, Zinovy, and Vistoli, Angelo. "Essential dimension of moduli of curves and other algebraic stacks." Journal of the European Mathematical Society 013.4 (2011): 1079-1112. <http://eudml.org/doc/277591>.

@article{Brosnan2011,
abstract = {In this paper we consider questions of the following type. Let $k$ be a base field and $K/k$ be a field extension. Given a geometric object $X$ over a field $K$ (e.g. a smooth curve of genus $g$), what is the least transcendence degree of a field of definition of $X$ over the base field $k$? In other words, how many independent parameters are needed to define $X$? To study these questions we introduce a notion of essential dimension for an algebraic stack. Using the resulting theory, we give a complete answer to the question above when the geometric objects $X$ are smooth, stable or hyperelliptic curves. The appendix, written by Najmuddin Fakhruddin, answers this question in the case of abelian varieties.},
author = {Brosnan, Patrick, Reichstein, Zinovy, Vistoli, Angelo},
journal = {Journal of the European Mathematical Society},
keywords = {essential dimension; stack; gerbe; moduli of curves; moduli of abelian varieties; essential dimension; stack; gerbe; moduli of curves; moduli of abelian varieties},
language = {eng},
number = {4},
pages = {1079-1112},
publisher = {European Mathematical Society Publishing House},
title = {Essential dimension of moduli of curves and other algebraic stacks},
url = {http://eudml.org/doc/277591},
volume = {013},
year = {2011},
}

TY - JOUR
AU - Brosnan, Patrick
AU - Reichstein, Zinovy
AU - Vistoli, Angelo
TI - Essential dimension of moduli of curves and other algebraic stacks
JO - Journal of the European Mathematical Society
PY - 2011
PB - European Mathematical Society Publishing House
VL - 013
IS - 4
SP - 1079
EP - 1112
AB - In this paper we consider questions of the following type. Let $k$ be a base field and $K/k$ be a field extension. Given a geometric object $X$ over a field $K$ (e.g. a smooth curve of genus $g$), what is the least transcendence degree of a field of definition of $X$ over the base field $k$? In other words, how many independent parameters are needed to define $X$? To study these questions we introduce a notion of essential dimension for an algebraic stack. Using the resulting theory, we give a complete answer to the question above when the geometric objects $X$ are smooth, stable or hyperelliptic curves. The appendix, written by Najmuddin Fakhruddin, answers this question in the case of abelian varieties.
LA - eng
KW - essential dimension; stack; gerbe; moduli of curves; moduli of abelian varieties; essential dimension; stack; gerbe; moduli of curves; moduli of abelian varieties
UR - http://eudml.org/doc/277591
ER -

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