Algebraic geometry Birational geometry Problem

Lüroth problem

Builds on Algebraic variety Unirational variety Rational variety Algebraically closed field

Problem

Let $k$ be an algebraically closed field and let $X$ be an integral algebraic variety over $k$ . Suppose that $X$ is unirational. Must $X$ be rational?

Remarks

In dimension $1$ the answer is yes, by Lüroth's theorem. In dimension $2$ over an algebraically closed field of characteristic $0$ , the answer is yes, by Castelnuovo's rationality criterion. In dimension $2$ over an algebraically closed field of characteristic $p > 0$ , the answer is no: Zariski found unirational surfaces that are not rational. In dimension $3$ over $\C$ , the answer is no. Clemens and Griffiths showed that a smooth cubic threefold in $\mathbb{P}^4$ is unirational but not rational. Iskovskikh and Manin showed that a smooth quartic threefold in $\mathbb{P}^4$ is not rational, and some of these are unirational. Artin and Mumford gave another unirational threefold that is not rational, detected by torsion in its third cohomology group. So the problem is settled in the negative in general, and the open questions now concern specific families of varieties.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`WGGZEKF224`
URI	`m2://WGGZEKF224`
Kind	problem
Writing	W0 stub
MSC	14E08, 14M20
Status	This copy of the site has no build data. Run `m2 status WGGZEKF224`.
About	Unirational variety, Rational variety
Uses	Algebraic variety, Unirational variety, Rational variety, Algebraically closed field

Lüroth problem

Problem

Remarks

Reading path

Further reading

What this page needs for W1

What links here