Zariski problem on field extensions

The Zariski problem has its motivation in a geometric question. For example, one could ask the following: Given two curves $C_1$ and $C_2$, make two surfaces by crossing a line with each curve. If the resulting surfaces are isomorphic, must the original curves also be isomorphic?

In general, one starts with two affine varieties, $V_1$ and $V_2$, of dimension $n$ (cf. also Affine variety) and crosses each with a line. Associated to each $V_i$ is its coordinate ring $c[V_i]$, and from an algebraic point of view, one wants to know if the polynomial rings $c[V_1][x_1]$ and $c[V_2][x_2]$ being isomorphic forces the coordinate rings to be isomorphic (cf. also Isomorphism). For $n$ larger than two, this is an open problem (as of 2000). However, also associated to each $V_i$ is its function field, $c(V_i)$, and one wants to know if isomorphism of the rational function fields in one variable over the function fields forces the function fields to be isomorphic. This is the so-called Zariski problem.

The problem has an affirmative answer for varieties of dimension one. This result appears in [a3], but uses ideas from [a5] and in an essential way depends on Amitsur's results about function fields of genus zero [a1]. Using a wide range of ideas from algebraic geometry, [a2] provides a family of counterexamples to the problem. In particular, there exist a field $K$ and extension fields $L$ of transcendence degree two over $K$ that are not rational and yet $L(x_1,x_2,x_3)$ is a pure transcendental extension of $K$ in five variables. Finally, in [a4] it is shown that the problem does have an affirmative answer most of the time, i.e., if the original varieties are of general type. Again, this result uses [a6] and, in an essential way, the results from [a7].

References

[a1]	S. Amitsur, "Generic splitting fields for central simple algebras" Ann. of Math. , 2 : 62 (1955) pp. 8–43 MR70624
[a2]	A. Beauville, J.-L. Colliot-Thélène, J.-J. Sansuc, P. Swinnerton-Dyer, "Varietes stablement rationnelles non rationnelles" Ann. of Math. , 121 (1985) pp. 283–318 MR0786350 Zbl 0589.14042
[a3]	J. Deveney, "Ruled function fields" Proc. Amer. Math. Soc. , 86 (1982) pp. 213–215 MR0667276 Zbl 0529.12016
[a4]	J. Deveney, "The cancellation problem for function fields" Proc. Amer. Math. Soc. , 103 (1988) pp. 363–364 MR0943046 Zbl 0652.12013
[a5]	M. Nagata, "A theorem on valuation rings and its applications" Nagoya Math. J. , 29 (1967) pp. 85–91 MR0207688 Zbl 0146.26302
[a6]	J. Ohm, "The ruled residue theorem for simple transcendental extensions of valued fields" Proc. Amer. Math. Soc. , 89 (1983) pp. 16–18 MR0706500 Zbl 0523.12021
[a7]	P. Roquette, "Isomorphisms of generic splitting fields of simple algebras" J. Reine Angew. Math. , 214/5 (1964) pp. 207–226 MR0166215 Zbl 0219.16023