| 概要 | Let Y be a quartic hypersurface in CP^4 with mild singularities, e.
g. no worse than ordinary double points.
If Y contains a surface that is not a hyperplane section, Y is not
Q-factorial and the divisor class group of Y, Cl Y, contains divisors that
are not Cartier. However, the rank of Cl Y is bounded.
In this talk, I will show that in most cases, it is possible to describe
explicitly the divisor class group Cl Y by running a Minimal Model Program
(MMP) on X, a small Q-factorialisation of Y. In this case, the generators
of Cl Y/ Pic Y are ``topological traces " of K-negative extremal
contractions on X.
This has surprising consequences: it is possible to conclude that a
number of families of non-factorial quartic 3-folds are rational.
In particular, I give some examples of rational quartic hypersurfaces
Y_4\subset CP^4 with rk Cl Y=2 and show that when the divisor class group
of Y has sufficiently high rank, Y is always rational.
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