In algebraic geometry, a quartic threefold is a degree 4 hypersurface of dimension 3 in 4-dimensional projective space.
showed that all non-singular quartic threefolds are irrational, though some of them are
unirational In mathematics, a rational variety is an algebraic variety, over a given field ''K'', which is birationally equivalent to a projective space of some dimension over ''K''. This means that its function field is isomorphic to
:K(U_1, \dots , U_d), ...
.
Examples
*
Burkhardt quartic In mathematics, the Burkhardt quartic is a quartic threefold in 4-dimensional projective space studied by ,
with the maximum possible number of 45 nodes.
Definition
The equations defining the Burkhardt quartic become simpler if it is embedded in ' ...
*
Igusa quartic
References
*{{Citation , last1=Iskovskih , first1=V. A. , last2=Manin , first2=Ju. I. , title=Three-dimensional quartics and counterexamples to the Lüroth problem , doi= 10.1070/SM1971v015n01ABEH001536 , mr=0291172 , year=1971 , journal=
Matematicheskii Sbornik
''Matematicheskii Sbornik'' (russian: Математический сборник, abbreviated ''Mat. Sb.'') is a peer reviewed Russian mathematical journal founded by the Moscow Mathematical Society in 1866. It is the oldest successful Russian mat ...
, series=Novaya Seriya , volume=86 , pages=140–166
3-folds