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A birational construction of projective compactifications of ℂ3 with second Betti number equal to onewith second Betti number equal to one

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Abstract

We give an explicit construction of all projective compactifications of ℂ3 with second Betti number equal to one.

Abbreviations

N Y|X :

normal bundle ofY inX

c1(ℱ):

first Chern class

h0(ℒ):

dimC H 0(ℒ)

Bs|ℒ|:

base locus of the linear system |ℒ|

κ(X):

Kodaira dimension

mult A X :

multiplicity ofX at a general point ofA

K X :

canonical divisor ofX

∼:

linear equivalence

≃:

isomorphism

\(\mathbb{F}_n \) :

Hirzebruch surface of degreen

n :

smooth quadric hypersurface in ℙn+1

0 2 :

quadric cone in ℙ3

(X, Y)≃(X′, Y′):

φ:XX′ isomorphism with φ(Y)=Y′

\((X,Y)\tilde \to (X',Y')\) :

φ::XX′ birational mapping withX−YX′−Y′

References

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Dedicated to K.-N. Hatsue

Entrata in Redazione il 30 novembre 1998.

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Furushima, M. A birational construction of projective compactifications of ℂ3 with second Betti number equal to onewith second Betti number equal to one. Annali di Matematica pura ed applicata 178, 115–128 (2000). https://doi.org/10.1007/BF02505891

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Keywords

  • Exact Sequence
  • Betti Number
  • Exceptional Divisor
  • Hyperplane Section
  • Birational Mapping