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.