Quintic threefold

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In mathematics, a quintic threefold is a degree 5 dimension 3 hypersurface in 4-dimensional projective space. Non-singular quintic threefolds are Calabi–Yau manifolds.

The Hodge diamond of a non-singular quintic 3-fold is

1
0 0
0 1 0
1 101 101 1
0 1 0
0 0
1

Rational curves[edit]

Clemens (1984) conjectured that the number of rational curves of a given degree on a generic quintic threefold is finite. (Some smooth but non-generic quintic threefolds have infinite families of lines on them.) This was verified for degrees up to 7 by Katz (1986) who also calculated the number 609250 of degree 2 rational curves. Philip Candelas, Xenia C. de la Ossa, and Paul S. Green et al. (1991) conjectured a general formula for the number of rational curves of any degree, which was proved by Givental (1996) The number of rational curves of various degrees on a generic quintic threefold is given by

5, 2875, 609250, 317206375, 242467530000, ...(sequence A076912 in OEIS).

Since the generic quintic threefold is a Calabi–Yau threefold and the moduli space of rational curves of a given degree is a discrete, finite set (hence compact), these have well-defined Donaldson–Thomas invariants (the "virtual number of points"); at least for degree 1 and 2, these agree with the actual number of points.

Examples[edit]

References[edit]