I also have an excuse for my absence… I’ve also been working on grant applications, job applications, my thesis, etc! I’ll take a little time now to write more about rational varieties. In general, it can be quite difficult to decide if a given variety is rational or not. Charles mentioned in a comment that he would like to see the proof that cubic threefolds are not rational. This is perhaps a good topic for the future, but maybe it requires more work to explain than I have time for right now. In any case, in this post I’ll talk about one way to verify that a variety is not rational. We won’t be distinguishing any unirational varieties from rational ones today though.
Suppose is a smooth projective variety defined over a field . Let denote the sheaf of differential forms. The main theorem uses the existence of a generically finite map from projective space to the given variety to force numerical constraints.
Theorem: If (as above) is rational, then it has no non-trivial global one forms. In general, the group for all positive integers m. In case the ground field has characteristic zero, the same is true for unirational varieties.
Proof: Suppose that there is a generically finite, dominant map . Let be an open set where this map is defined. Because is projective, the complement of this open set may be assumed to have codimension two. The claim is that non-zero differential forms on pull back to be non-zero on , or in other words, that there is an inclusion . When the map is birational, this is obvious. When the map is only finite though, this is only true in characteristic zero! Indeed, the pull back map on differential forms can be zero, even for finite maps (consider Frobenius). Now, because the complement has codimension two, differential forms extend from to all of . The theorem will be complete if we can prove that . To see this, note that injects into and this sheaf has no global sections so that there can be no global one forms on projective space. By taking m-th powers, we get the same result for all differential forms.
One sees that in positive characteristic the notion of being unirational is not well behaved. So we make the definition that a variety is separably unirational if there is a dominant, generically etale map from projective space to our variety. This means that the induced inclusion of function fields is separable and exactly implies that the pullback on differentials is injective. Using the same argument, there are no global differential forms on a smooth projective separably unirational variety in any characteristic.
As a corollary, we have that all the plurigenera for a separably unirational variety (which includes rational varieties) must be zero. Sometimes these are easier to calculate because they just involve working with invertible sheaves. For a smooth variety , recall that the m-th plurigenus of is the dimension of the space of global sections of the invertible sheaf .
Proof: There is an injection . Since these latter have no global setions, neither will . Taking m-th powers again gives the result that all the plurigenera vanish.
In fact, we don’t even need the strength of our assumptions to prove that these numbers vanish. All we need is that our variety is separably uniruled. A variety is (separably uni)ruled, if there exists a variety and a birational (resp. generically finite seperable map) . Loosely, this means that the variety has many rational curves on it. Ruled varieties are separably uniruled, and unirational varieties are also uniruled. These implications are strict. In any case, we can prove that
Theorem 2: The plurigenera are zero for any smooth projective separably uniruled variety .
Proof: Let be the separable uniruling. As in the first theorem, we will prove that . To do this, we again get an injection from the pullback of the differential forms into those of . So it will be enough to prove that . However, where is the projection map. So finally, . The last term though is zero (when m is positive) because which has global sections.
Finally, we can give an example of smooth varieties which are rational. Suppose is a smooth hypersurface in of degree . By adjunction, the canonical class of is . The first term is where is the hyperplane class on projetive space restricted to . The second term is the normal bundle of in projective space; in this case it is just a line bundle and it is where d is the degree. Putting this all together then, we see that . Since we get that all powers of have global sections, and so cannot be separably uniruled, and in particular, not rational! The same argument works for smooth complete intersections in projetive space where the degrees of the defining equations add up to more than the dimension of the projective space.
This is really the only numerical (that is, easy!) conclusion we can make in the case of smooth hypersurfaces. Low degree hypersurfaces (the Fano ones) are harder to analyze, but not impossible! Perhaps I can talk about it more in the future.