Toric varieties in some sense a "canonical rational variety" in that one can construct them from purely combinatorial data and this combinatorial data makes it possible to turn many typically intractable geometric problems into ones that may be computed by combinatorial means.
I wonder if it might be possible to make a similar construction for unirational varieties as follows. If an $ n $-dimensional variety $ Z $ is unirational, then there is a birational map $ \phi: \mathbb{P}^{n}_{k} \to Z $ such that if $ K(Z) $ is the field of fractions of $ Z $, then $ \operatorname{Gal}(K(Z)/K(\mathbb{P}^{n}_{k})) $ is a finite group $ G $. The torus of a toric variety $ X_{\Sigma} $ is isomorphic to $ \operatorname{Spec}(k[M]) $. If one started with an action of a finite group $ G $ on $ M_{\mathbb{R}} $ which stabilized $ M $, then one might be able to construct an Etale cover from $ X_{\Sigma} $ to a unirational variety by creating the underlying unirational variety via gluings of $ M_{\mathbb{R}} $ and some sort of fan-like structure.
Choose a finite group $ G $, and assume that there exists a representation $ \beta: G \to \operatorname{GL}(n,\mathbb{Z}) $. Via the representation $ \beta $, the group $ G $ acts on $ M $. The torus of a toric variety $ X_{\Sigma} $ is isomorphic to $ \operatorname{Spec}(k[M]) $, and so $ G $ acts on the torus of $ X_{\Sigma} $.
There is a morphism of topological spaces $ \pi: M_{\mathbb{R}} \to M_{\mathbb{R}}//G $. For $ \sigma \in \Sigma $, let $ \sigma^{?} $ be the image of $ \sigma^{\vee} $ under $ \pi $ and let $ M^{?} $ be the image of $ M $. I believe that it is possible to impose a semigroup structure on $ M^{?} $. As a result, there should be a ring homomorphism $ \Phi^{\sharp}: k[\sigma^{?} \cap M^{?}] \to k[\sigma^{\vee} \cap M] $, and that it should be possible to prove that $ k[\sigma^{?}\cap M^{?}] \cong k[\sigma^{\vee} \cap M]^{G} $.
I believe that the map $ \{\cdot,\cdot\} \operatorname{Hom}_{\mathbb{R}}(M_{\mathbb{R}}//G,\mathbb{R})\times M_{\mathbb{R}}//G \to \mathbb{R} $ obtained from $ \{f,[x]\} = f([x]) $ should have the following form. If $ \pi^{\vee}: \operatorname{Hom}_{\mathbb{R}}(M_{\mathbb{R}}//G, \mathbb{R}) \to N_{\mathbb{R}} $ sends $ f $ to $ u $ and $ \pi(x)=[x] $, then $ \{f,[x]\} $ should equal $ \left(\sum_{g \in G} \langle g \cdot x, g \cdot u \rangle\right)/\operatorname{ord}(G) $.
So $ \{\cdot,\cdot\} $ should be a $ G $-invariant, perfect pairing between $ \operatorname{Hom}_{\mathbb{R}}(M_{\mathbb{R}}//G, \mathbb{R}) $ and $ M_{\mathbb{R}}//G $. If $ N^{?} $ is equal to $ \operatorname{Hom}_{\mathbb{R}}(M^{?},\mathbb{R}) $, then $ \{\cdot,\cdot\} $ should identify $ N^{?} $ and $ M^{?} $ as dual lattices. This should allow one to make some sort of structure analagous to a fan (call it $ \Sigma^{?} $), and to obtain a variety $ X_{\Sigma^{?},\beta} $ along with an Etale cover $ \Phi: X_{\Sigma} \to X_{\Sigma^{?},\beta} $.
If $ K(X_{\Sigma})^{G} \not \cong K(\mathbb{G}_{m}^{n}) $, then $ X_{\Sigma^{?},\beta} $ is unirational and not rational.
This is obviously a very loose idea. Has anyone worked on something of this sort, and if not would anyone want to collaborate and work on this idea? I believe that it might allow one to relate a topological space obtained by gluing to a variety and thus turn topological problems into algebro-geometric ones.
