# On covering and quasi-unsplit families of curves

Laurent Bonavero; Cinzia Casagrande; Stéphane Druel

Journal of the European Mathematical Society (2007)

- Volume: 009, Issue: 1, page 45-57
- ISSN: 1435-9855

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topBonavero, Laurent, Casagrande, Cinzia, and Druel, Stéphane. "On covering and quasi-unsplit families of curves." Journal of the European Mathematical Society 009.1 (2007): 45-57. <http://eudml.org/doc/277773>.

@article{Bonavero2007,

abstract = {Given a covering family $V$ of effective 1-cycles on a complex projective variety $X$, we find conditions allowing one to construct a geometric quotient $q:X\rightarrow Y$, with $q$ regular on the whole of $X$, such that every fiber of $q$ is an equivalence class for the equivalence relation naturally defined by $V$. Among other results, we show that on a normal and $\mathbb \{Q\}$-factorial projective variety $X$ with canonical singularities and $\operatorname\{dim\}X\le 4$, every covering and quasi-unsplit family $V$ of rational
curves generates a geometric extremal ray of the Mori cone $\overline\{\text\{NE\}\}(X)$ of classes of effective 1-cycles and that the associated Mori contraction yields a geometric quotient for $V$.},

author = {Bonavero, Laurent, Casagrande, Cinzia, Druel, Stéphane},

journal = {Journal of the European Mathematical Society},

keywords = {covering families of curves; extremal curves; quotient; Covering families of curves; extremal curves; quotient},

language = {eng},

number = {1},

pages = {45-57},

publisher = {European Mathematical Society Publishing House},

title = {On covering and quasi-unsplit families of curves},

url = {http://eudml.org/doc/277773},

volume = {009},

year = {2007},

}

TY - JOUR

AU - Bonavero, Laurent

AU - Casagrande, Cinzia

AU - Druel, Stéphane

TI - On covering and quasi-unsplit families of curves

JO - Journal of the European Mathematical Society

PY - 2007

PB - European Mathematical Society Publishing House

VL - 009

IS - 1

SP - 45

EP - 57

AB - Given a covering family $V$ of effective 1-cycles on a complex projective variety $X$, we find conditions allowing one to construct a geometric quotient $q:X\rightarrow Y$, with $q$ regular on the whole of $X$, such that every fiber of $q$ is an equivalence class for the equivalence relation naturally defined by $V$. Among other results, we show that on a normal and $\mathbb {Q}$-factorial projective variety $X$ with canonical singularities and $\operatorname{dim}X\le 4$, every covering and quasi-unsplit family $V$ of rational
curves generates a geometric extremal ray of the Mori cone $\overline{\text{NE}}(X)$ of classes of effective 1-cycles and that the associated Mori contraction yields a geometric quotient for $V$.

LA - eng

KW - covering families of curves; extremal curves; quotient; Covering families of curves; extremal curves; quotient

UR - http://eudml.org/doc/277773

ER -

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