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On Minimal Compactifications of ℂ2

✍ Scribed by Mnkio Furushima

Publisher
John Wiley and Sons
Year
2009
Tongue
English
Weight
555 KB
Volume
186
Category
Article
ISSN
0025-584X

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✦ Synopsis

Let (X,

Y

) be a minimal compactifiurtion of C 2 . In this paper, we determine the structure of such a (X,Y) in the case where X is a normal hypersurface of degree d 5 4 i n P3.

1. Introduction

A projective normal Gorenstein surface X over C is called a compactification of C if there is a closed subvariety Y such that X -Y is biholomorphic to C 2 . We shall denote simply the compactification by the pair ( X , Y ) . The Hartogs theorem shows that the' boundary Y is of pure codimension one, that is, Y is a Weil divisor on X .

A compactification (X, Y ) of Q: is said to be minimal if the second Betti number b 2 ( X ) = dimH2(X, R) = 1 (what is equivalent to say that Y is irreducible).

gives an example of a rational Z -homology plane, that is, a normal rational surface X with H i ( X ; Z ) = Hi(P2;Z) for i 2 0.

Conversely, we have

A minimal compactification of C Question 1.1. Is there a rational +-homology plane which is not a compactification of (c2 ?

In this paper, we shall mainly study the following Problem 1.2. Determine the minimal compactifications of C which are hypersurfaces in the projective 3 space P3.

Our main result is the following Theorem 1.3. Let (X,j,Y,j) be a minimal compactification of C 2 which is a hypersurface of degree d 2 2 in P3. Assume that d L 4.

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