Some Elementary Examples of Unirational Varieties Which are Not Rational 论文

摘要

An outstanding problem in the algebraic geometry of varieties of dimension n ^ 3 over an algebraically closed field k has been whether there exist unirational varieties which are not rational. Here V is unirational if it has the equivalent properties: (a) there exists a rational surjective map /: V n-> V, or, there exists an embedding k(V) < = k(X1}...,-XJ; while V rational means equivalently: (b) there exists a birational map /: P 71-> V, or, there exists an isomorphism k(V) = k(Xv...,1^). For n—\\, these are equivalent (Liiroth's theorem). For n = 2 they are equivalent in characteristic 0 (Castelnuovo's theorem) or if the map / in (a) is assumed separable (Zariski's extension of Castelnuovo's theorem). In 1959 ([13]), Serre clarified classical work on this problem for n = 3. It has been generally accepted since then that none of the examples proposed by Fano or Roth had been correctly proved irrational. In the past year, two solutions of this problem have been found: Clemens and Griffiths ([6]) showed that all non-singular cubic hypersurfaces in P 4 are irrational, and Iskovskikh and Manin ([16]) showed that all non-singular quartic hypersurfaces in P 4 are irrational. Some are unirational (Segre ([11])). Both of these solutions are quite deep and it seems worth while to have an elementary example as well, even if our method applies to a very special kind of variety. Ramanujam suggested using torsion in H s and this led us to the examples presented here. We construct varieties, of all dimensions n ^ 3 and all characteristics £> # 2, which are unirational and which have 2-torsion in H 3. With the present state of resolution of singularities, we can show that such a V cannot be rational if the characteristic is 0 or if the characteristic is not 2 and n = 3.