At the Seattle workshop I mentioned in the previous post I was a mentor of a group of recent PhDs consisting of Jennifer Park, Daniel Litt, Runpu (Z)Hong, Dingxin Zhang, Sam Raskin, and Francois Greer. The topic was “supersingular surfaces in positive characteristics”. Any mistakes and/or misrepresentations in this post are mine.

Let us define a smooth projective variety over a field K of characteristic p > 0 to be *supersingular* if for each i the Newton slopes on the “motive” H^i(X) are all i/2. (This is just one possible choice of definition.) If the Tate conjecture is true for H^2, then for a supersingular variety over an algebraically closed field the rank of the Picard group is equal to the second betti number (!).

Motivation for the choice of topic was the recent successes (due to my colleague Maulik, as well as Liedtke, Lieblich, and Charles) on moduli of supersingular K3 surfaces. Basically, we know that supersingular K3 surfaces satisfy the Tate conjecture, are unirational, form a family of dimension 9, and that any degeneration of a supersingular K3 has potentially good reduction.

Now, let’s discuss some things one can say for other surfaces.

**Degeneration** of supersingular surfaces. Let K = k((t)) and let X be a supersingular surface over K. The method here, following Rudakov-Zink-Shafarevich (RZS), is to note that the formal Brauer group of any special fibre X_0 has vanishing p-divisible part (they only prove this under the assumption that H^1(X_0, O_{X_0}) = 0). This imposes a strong condition on the limits. For example, if X_0 is equal to the union of two smooth surfaces S_1, S_2 glued along transversally along a nonsingular curve C, then S_1 and S_2 are forced to have slopes 1 of the Newton polygon of H^2 and the map Pic^0(S_1) x Pic^0(S_2) —> Pic^0(C) has to be surjective.

One can make examples of this kind of degeneration, by considering a family of supersingular genus 2 curves specializing to a good curve consisting of two supersingular elliptic curves glued in a point and taking the product with another supersingular elliptic curve.

Looking at quintic surfaces in P^3, we found that applying the RZS criterion to GIT-stable limits does not always give enough information, and that it is better to consider stable limits of surfaces in the sense of birational geometry. In fact, it appears likely that supersingular quintics have potential good reduction at least for large enough primes (but this may be an empty statement — see below).

Degenerations of supersingular elliptic surfaces with a section. If (X, σ) is an elliptic surface over a curve C and if all fibres are semi-stable, then this determines a morphism C —> \bar M_{1, 1}. Hence we can use Abramovich-Vistoli(+Olsson) to take a limit in \bar M_g(\bar M_{1, 1}, degree) and see that our elliptic surface degenerates to an elliptic surface X_0 over a semi-stable limit C_0 of C, right? For example if g = g(C) = 0, then C_0 is a tree of curves and at first it looks like RZS implies that C_0 has to be irreducible: namely the glueing curves are elliptic and the component surfaces are elliptic with nonconstant j-invariant hence have trivial Pic^0. Well, this is not quite the case as in this game you have to allow the base curve to become stacky at the nodes. And then the analysis of RZS still works (we think), but now you are glueing along stacky curves whose Pic^0 may be zero (or finite).

In fact, looking at 1-parameter families of supersingular elliptic K3 surfaces (which we know exist) we proved this kind of behaviour must happen, i.e., the Abramovich-Vistoli limit must produce a reducible stacky limit curve C_0. This is not a contradiction with the previously mentioned good reduction of K3 surfaces, as what (probably) happens is that one of the irreducible components X_0 is a K3 and the others are (for example) rational elliptic surfaces. Slogan: there is a difference between limits of X as an abstract surface and limits of X as an elliptic surface.

**Unirational** surfaces are supersingular. Shioda gave an example of a supersingular surface with q = p_g = 0 which is not unirational. However, he also conjectured

Let X be a simply-connected surface (i. e. without connected etale covering of degree > 1) in char p > 0. Then X is unirational if and only if it is supersingular.

As far as I can tell this conjecture is still open (please let me know if this is no longer the case). To try and disprove this, we can try to find “new” examples of supersinguar surfaces. E.g., we can look for whole families of them as in the next paragraph. But, we can also look for “sporadic” surfaces (like Fermat surfaces for which the conjecture is known). In fact, I don’t know if for every p > 5 there is a supersingular quintic surface we could try the conjecture on (again, please let me know if there are examples). I also tried to find new ones by computation which I will report on in the next blog post.

In fact, I **can** point out some examples for which I don’t know if the conjecture holds. Let C and C’ be supersingular hyperelliptic curves in char p > 2 and let i, i’ denote the hyperelliptic involutions. Note: for p > 2 there exists a 1-parameter family of supersingular hyperelliptic curves for genus 2 and for genus 3 too (Oort). Then let X be the resolution of singularities of (C x C’)/<(i, i')>. It seems to me that \pi_1(X) = 0 and of course X is supersingular. But I don’t know how to prove that X is unirational, do you? Countably many cases of this are discussed by Shioda and others (eg when the curves C and C’ are related to Fermat curves); also if C and C’ are elliptic curves, then X is a Kummer surface and X was proved to be unirational by Shioda (a beautiful alternative non-computational proof of this was found by Katsura).

**Moduli** of supersingular surfaces. Suppose given a family of surfaces over a base B, for example the universal family of quintic surfaces in P^3. What one can try is look at the lower bound on the codimension of the supersingular stratum in B using the fact that Newton polygons jump in codimension 1 (if they do jump). We tried this on the first day and it turns out that for quintic surfaces in P^3 and for elliptic surfaces which are not K3 and not rational, the lower bound you get is higher than the dimension of B, i.e., it is useless. In fact, I would like to know

Are there infinitely many primes p such that there is a 1-parameter family of supersingular quintic surfaces?

In fact, I don’t know a 1-parameter family of supersingular quintic surfaces except for p = 5. Of course, we can ask the displayed question for surfaces of any given degree > 4 in P^3. Also, we can ask this for elliptic surfaces of given height > 2 (i.e., not rational or K3).

But, looking around the literature, families of supersingular surfaces seem to be hard to come by and the ones I’ve found are always families of unirational surfaces (so useless from the point of trying to address Shioda’s conjecture). Please let me know if you know of examples where unirationality (currently) isn’t known.

To finish the discussion let me mention two examples of families.

For surfaces in P^3 we can consider the Zariski surfaces

X : T_0^p = F(T_1, T_2, T_3)

where F is a general homogeneous form of degree p. Such a surface has a large number of A_{p – 1} singularities and the desingularization X is a unirational surface. If I understand well, then all the new algebraic cycles come from the resolution of the singularities.

Let q be a power of p. In Shioda’s paper one finds the family of surfaces

T_0^q T_2 + T_1^q T_3 + T_0 F(T_2, T_3) + T_1 G(T_2, T_3) = O

where F, G are degree q homogeneous without common factors. These are smooth and unirational and have 2q – 2 moduli.

How could we ever disprove Shioda’s conjecture? Being (inseparably) unirational has consequences for the weights of Frobenius (e.g., Esnault’s theorem) and for the algebraic fundamental group (Ekedahl’s theorem, cf. Chambert-Loir’s wonderful expose). But Shioda’s hypotheses rule out either of those results as a way to disprove the conjecture. So the problem is, what further consequences are there of unirationality that could be used to disprove Shioda’s conjecture? In fact, we have a decomposition of the diagonal, at least after tensoring with $\mathbb{Q}$. Are there other “motivic” consequences of this that we could use? Could a simply connected, supersingular surface have nonvanishing unramified cohomology?

Yes, very well said indeed! In fact, Shioda may have been thinking a similar thought as he says in his paper “it would be very hard to disprove this conjecture, if not true.”