# A boundedness theorem for morphisms between threefolds

Ekatarina Amerik; Marat Rovinsky; Antonius Van de Ven

Annales de l'institut Fourier (1999)

- Volume: 49, Issue: 2, page 405-415
- ISSN: 0373-0956

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topAmerik, Ekatarina, Rovinsky, Marat, and Van de Ven, Antonius. "A boundedness theorem for morphisms between threefolds." Annales de l'institut Fourier 49.2 (1999): 405-415. <http://eudml.org/doc/75343>.

@article{Amerik1999,

abstract = {The main result of this paper is as follows: let $X,Y$ be smooth projective threefolds (over a field of characteristic zero) such that $b_2(X)=b_2(Y)=1$. If $Y$ is not a projective space, then the degree of a morphism $f:X\rightarrow Y$ is bounded in terms of discrete invariants of $X$ and $Y$. Moreover, suppose that $X$ and $Y$ are smooth projective $n$-dimensional with cyclic Néron-Severi groups. If $c_1(Y)=0$, then the degree of $f$ is bounded iff $Y$ is not a flat variety. In particular, to prove our main theorem we show the non-existence of a flat 3-dimensional projective variety with $b_2=1$.},

author = {Amerik, Ekatarina, Rovinsky, Marat, Van de Ven, Antonius},

journal = {Annales de l'institut Fourier},

keywords = {second Betti class; finite morphism; flat variety; Chern class; three-folds; degree of a morphism; Néron-Severi groups},

language = {eng},

number = {2},

pages = {405-415},

publisher = {Association des Annales de l'Institut Fourier},

title = {A boundedness theorem for morphisms between threefolds},

url = {http://eudml.org/doc/75343},

volume = {49},

year = {1999},

}

TY - JOUR

AU - Amerik, Ekatarina

AU - Rovinsky, Marat

AU - Van de Ven, Antonius

TI - A boundedness theorem for morphisms between threefolds

JO - Annales de l'institut Fourier

PY - 1999

PB - Association des Annales de l'Institut Fourier

VL - 49

IS - 2

SP - 405

EP - 415

AB - The main result of this paper is as follows: let $X,Y$ be smooth projective threefolds (over a field of characteristic zero) such that $b_2(X)=b_2(Y)=1$. If $Y$ is not a projective space, then the degree of a morphism $f:X\rightarrow Y$ is bounded in terms of discrete invariants of $X$ and $Y$. Moreover, suppose that $X$ and $Y$ are smooth projective $n$-dimensional with cyclic Néron-Severi groups. If $c_1(Y)=0$, then the degree of $f$ is bounded iff $Y$ is not a flat variety. In particular, to prove our main theorem we show the non-existence of a flat 3-dimensional projective variety with $b_2=1$.

LA - eng

KW - second Betti class; finite morphism; flat variety; Chern class; three-folds; degree of a morphism; Néron-Severi groups

UR - http://eudml.org/doc/75343

ER -

## References

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