A curiosity: “supersmooth” varieties
I want to share a curious condition on varieties for which I have found no use. Let be a field and let be a locally finite type -scheme. Recall that is said to be smooth if, for every Artin local -algebra with a proper ideal and every -morphism , there exists a -morphism extending it, i.e., the map is surjective. (It takes real work to show that this is equivalent to other more concrete notions.) It is an amusing exercise to check that this is equivalent to the following statement: for every complete noetherian local -algebra and every ideal such that is Artin, the map is surjective. Motivated by this, we will say that a locally finite type -scheme is supersmooth if for any noetherian local -algebra and any ideal such that is Artin, the map is surjective. Of course, by the above discussion every supersmooth -scheme is smooth. One could define a relative notion of supersmoothness, but it is not a “good” condition because it cannot be checked etale-locally, see below.
Supersmoothness will turn out to be a rather rigid condition; to show that our results are not vacuous, let us first give some examples.
Examples. (1) Affine -space is supersmooth: if and are as in the definition, then a morphism is equivalent to the data of elements of the ring . Since is surjective, it is clear that is surjective.
(2) Because the rings involved in the definition are local, any open subscheme of a supersmooth -scheme is supersmooth, and supersmoothness may be checked Zariski-locally on any locally finite type -scheme. Moreover, supersmoothness is preserved by field extension (although it may not be checked after field extension, see Example (5)). The product of supersmooth varieties is clearly supersmooth.
(3) If is a split reductive group, then using the existence of the open cell and the fact that its translates by rational points cover , it follows from (1) and (2) that is supersmooth. One sees furthermore that is supersmooth whenever is a parabolic -subgroup of . For example, , flag varieties, and Grassmannians (in particular, projective spaces) are all supersmooth.
(4) Let be a finite separable field extension, and suppose that is a quasi-projective -scheme satisfying the condition for supersmoothness whenever is merely a semi-local noetherian local ring (e.g., ). Then the Weil restriction is supersmooth, as the reader familiar with Weil restrictions may easily check.
(5) If for some finite field extension , then is supersmooth if and only if . Since in particular is smooth over , the extension must be separable and thus for some by the primitive element theorem. Let denote the minimal polynomial of over , so that . Thus by supersmoothness there is a -homomorphism , so that because the fraction field of is , which does not contain any nontrivial finite extension of . This example shows already that supersmoothness is not a “geometric” condition, in the sense that it may not be checked after extending : note that if is separable then for , and this is evidently supersmooth over .
(6) If is a supersmooth integral -scheme of dimension , then admits a nonconstant rational map from . Indeed, since is smooth, , so there is some closed point whose residue field is finite separable over , say as in the previous example. Let be the minimal polynomial of over , and let , . Since , there is a surjective homomorphism of -algebras , and by supersmoothness this extends to an -point of . Since the map on cotangent spaces is nontrivial, the generic point of maps to a nonclosed point of , and thus there is indeed a nonconstant rational map from . In particular, any integral -scheme of dimension admitting no nonconstant rational maps from is not supersmooth: for example, the only supersmooth integral curves are open in , and abelian varieties are never supersmooth.
Note that all of the examples we have given are rational. The last two examples in particular suggest that supersmoothness is a rather restrictive condition, and that it is somewhat similar to rationality. (We will see below that there exist non-rational examples.) Let be a locally finite type -scheme. We call nearly rational if the following condition holds: for every point , there is some open subscheme of an affine space over and a smooth -morphism such that lies in the image of and there exists a section .
Proposition. If is nearly rational, then it is supersmooth. If is of characteristic , then the converse is true.
Proof. First, suppose is nearly rational. Let be a noetherian local -algebra and let be an ideal such that is Artin. Suppose given a -morphism , and let denote the unique point in the image of this morphism. Let be some smooth neighborhood of as in the statement of the Proposition, and let be a -rational point mapping to . If denotes the maximal ideal of , then this means that the morphism lifts to morphism with image . Since is smooth, it follows from the infinitesimal criterion for smoothness that lifts to a morphism . By Examples (1) and (2) above, is supersmooth and thus there is an extension of this latter morphism to some map . The composition of this map with is then the desired extension of .
For the other direction, we need two Lemmas.
Lemma 1. Let be a finitely generated extension of fields. There exists a positive integer and a point such that .
Proof of Lemma 1. Let be a finite type domain over with fraction field . There is some surjective map , and if is the kernel of then .
Lemma 2. Suppose that is a smooth -algebra. If is a prime ideal such that is separable over , then there is a natural split short exact sequence
Proof of Lemma 2. By separability of over , note that is generically smooth over , so by localizing (using the good behavior of with respect to localization of ), Neron Models, 2.2, Prop. 7 shows that there is a natural split short exact sequence
Localizing at the generic point then gives the desired result.
Now we finish the proof of the Proposition. Suppose that is supersmooth over the field of characteristic , and let . Let , so that there exist and as in Lemma 1. By increasing as necessary, we may assume that there is an injective homomorphism inducing an isomorphism on residue fields. By supersmoothness, the induced map extends to a map . Since is locally finite type, this is the localization of some map , where is an open subscheme of containing . Note that this map induces an isomorphism of residue fields and moreover the map is injective. Since is of characteristic (and thus all extension fields of are perfect), it follows from Lemma 2 that the map is injective. Since and are smooth over , the map is therefore smooth at by Neron Models, 2.2, Prop. 8. By openness of the smooth locus we may shrink around to assume that in fact is smooth. This satisfies all the conditions in the Proposition, and we have completed the proof.
Note that this implies that any supersmooth -scheme is unirational, and hence over an infinite field it has a dense set of rational points.
This Proposition can be used to give more examples of supersmooth varieties. For example, we will use it to show that any stably rational variety is generically supersmooth (i.e., that it admits a dense open subscheme which is supersmooth). Recall that an integral scheme locally of finite type over is called stably rational if is rational for some . The fact that non-rational stably rational varieties exist was first proven in Variétés stablement rationnelles non rationnelles by Beauville–Colliot-Thélène–Sansuc–Swinnerton-Dyer. A field-theoretic formulation of this condition is the following: if is the function field of , then there exists some such that the purely transcendental extension is -isomorphic to for some .
In general, suppose given an integral scheme of finite type over with function field and a finitely generated separable extension such that is purely transcendental. If is finite (i.e., ) then clearly and is supersmooth, so assume . We can spread out these extensions to obtain an open subscheme of some affine space over and a smooth -morphism whose generic fiber has function field . If the generic fiber admits a -point, then we may spread this out to a section of over some dense open of , and this is supersmooth by the Proposition. Since , is infinite and thus the generic fiber admits a rational point whenever is purely transcendental (since the -points are dense in ). Such exists when is stably rational, so indeed stably rational varieties are supersmooth.
I don’t know whether the second statement in the Proposition is true when the characteristic of is . It would be interesting to know whether there are any counterexamples.