About
This website records rationality results for smooth complex hypersurfaces
\[ X_d^n\subset\mathbb P^{n+1}_{\mathbb C}. \]It is maintained by Pieter Belmans.
Definitions
Let \(X\) be an integral variety of dimension \(n\) over a field \(k\).
- \(X\) is rational if it is birational to \(\mathbb P_k^n\).
- \(X\) is stably rational if \(X\times\mathbb P_k^m\) is rational for some \(m\geq0\).
- \(X\) is retract rational if there are rational maps \[ X\dashrightarrow\mathbb P_k^N\dashrightarrow X \] whose composition is the identity on a dense open subset of \(X\).
- \(X\) is unirational if there is a dominant rational map \(\mathbb P_k^N\dashrightarrow X\).
- \(X\) is rationally connected if two general geometric points of \(X\) can be joined by a rational curve.
Over \(\mathbb C\), these properties satisfy
\[ \text{rational}\Longrightarrow\text{stably rational} \Longrightarrow\text{retract rational} \Longrightarrow\text{unirational} \Longrightarrow\text{rationally connected}. \]The table keeps rationality, stable rationality, unirationality, and rational connectedness separate. Retract rationality is included in the detailed statements when the result is stronger than stable irrationality. The table also distinguishes results for every smooth hypersurface from results for a very general member.
Here very general means outside a countable union of proper closed subsets of the parameter space. It is stronger than general, which means outside a single proper closed subset.
Corrections can be submitted as a GitHub issue, pull request, or by sending me an email.