“Fermat quintic threefold”的意思、由来-开放百科全书

词条

Fermat quintic threefold

释义

Rational curves
References

In mathematics, a Fermat quintic threefold is a special quintic threefold, in other words a degree 5, dimension 3 hypersurface in 4-dimensional complex projective space, given by the equation

.

This threefold, so named after Pierre de Fermat, is a Calabi–Yau manifold.

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

Rational curves

{{harvs|txt|last=Clemens|first=Herbert|authorlink=Herbert Clemens |year=1984}} conjectured that the number of rational curves of a given degree on a generic quintic threefold is finite. The Fermat quintic threefold is not generic in this sense, and {{harvs|txt| last1=Albano | first1=Alberto | last2=Katz | first2=Sheldon | authorlink2= Sheldon Katz |year=1991}} showed that its lines are contained in 50 1-dimensional families of the form

for and . There are 375 lines in more than one family, of the form

for fifth roots of unity and .

References

{{Citation | last1=Albano | first1=Alberto | last2=Katz | first2=Sheldon | authorlink2= Sheldon Katz | title=Lines on the Fermat quintic threefold and the infinitesimal generalized Hodge conjecture | jstor=2001512 | mr=1024767 | year=1991 | journal=Transactions of the American Mathematical Society | issn=0002-9947 | volume=324 | issue=1 | pages=353–368 | doi=10.2307/2001512}}
{{Citation | last=Clemens | first=Herbert |authorlink= Herbert Clemens | title=Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982) | publisher=Princeton University Press | series=Annals of Mathematics Studies | mr=756858 | year=1984 | volume=106 | chapter=Some results about Abel-Jacobi mappings | pages=289–304}}
{{Citation | last1=Cox | first1=David A. | authorlink1= David A. Cox | last2=Katz | first2=Sheldon | authorlink2= Sheldon Katz |title=Mirror symmetry and algebraic geometry | url=http://www.ams.org/bookstore-getitem/item=surv-68.s | publisher=American Mathematical Society | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-1059-0 | mr=1677117 | year=1999 | volume=68}}

2 : 3-folds|Complex manifolds

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