# 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

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topBrosnan, 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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