The Tate-Oort group scheme TOp and Godeaux surfaces

This website supports the papers

[TOp] Miles Reid, The Tate-Oort group scheme TOp, Proceedings of the
Steklov Institute of Mathematics, 307 (2019), 267-290.

[KR] Soonyoung Kim and Miles Reid, The Tate-Oort group scheme TOp
and Godeaux surfaces (in progress)

Abstract: We construct Godeaux surfaces and Calabi-Yau 3-folds whose Pic^tau contains
5-torsion, in mixed characteristic at 5. The same ideas give Campedelli surfaces and
Calabi-Yau 3-folds with 7-torsion, and (eventually) Godeaux surfaces with 3-torsion. The
aim is to put varieties in characteristic p with action of ZZ/p, al_p and mu_p into a single
family, together with the characteristic zero case with a ZZ/p action.

Representation theory and invariant theory We know by [TOp, Section 2.1] that
TOp is a p-torsion subgroup of the 2-dimensional matrix group ( 1 0 \\ x 1+tx ) in GL(2,B).
(Here B is any base ring and t in B any element.) The corresponding free module of rank 2
is the given representation Vgiv. The representations that occur in our arguments are almost
always associated representation of this 2-dimensional representation. Notably, the regular
representation is Vreg = Sym^{p-1}(Vgiv): indeed, the regular representation of a group
scheme is the action on its own coordinate ring, and the coordinate ring of TOp is
B[TOp] = B[x]/(F) with F a monic polynomial of degree p, so is the free module based by

The invariant theory currently depends on unwieldy computer algebra. Even when the
answer can be cleaned up to be short and simple (comparable to the case of Fermat
hypersurfaces), the derivation involves long and tricky calculation. This website
gives Magma routines to carry out these derivations. Almost everything works in the
online Magma calculator.

First case: plane cubic E3 in PP^2 The plane cubic E3 in PP^2 with a free action by
a group scheme of order 3 is treated in [TOp, Section 5]. It corresponds over CC to the
Hesse pencil. Over a field, a group action is translation by a group subscheme in E of
order 3. In char 3 this may be ZZ/3 or al_3 or mu_3. As described in [TOp, Section 4],
there are two different derivations of the 4 invariants. The first starts from the reductive
case with t invertible, then shaves off successive powers of t. The second works
directly with Linear algebra in the rank 10 free module of cubics in u0,u1,u2. This
Magma code does all the necessary calculations. It is a warm-up for TO5-invariant
quintics that we need in [KR] for the 5-torsion Godeaux surfaces and CY 3-folds.

Quintic elliptic curve E5 in PP^4 The elliptic quintic E5 in PP^4 is defined by the
4x4 Pfaffians of a 5x5 skew matrix with linear entries. It is an interesting exercise, and
a useful warm-up for our 7-torsion Campedelli surfaces and CY 3-folds. Take the linear
forms on PP^4 to be the regular representation Vreg of TO5 with basis u0,u1,u2,u3,u4.
The space of skew bilinear forms on PP^4 is wedge^2 of this; however, what we need are
skew forms on PP^4 with entries linear in the ui, that is, TO5-invariant homomorphisms
N: Vreg -> wedge^2 Vreg. We write N for such a matrix (and ignore the distinction
between Vreg and its dual for the present). The matrix Du that represents the action of
TO5 on Vreg is given in [TOp, Section 2.3, (2.5)]. Then TO5-invariance turns out to be
expressed by the condition that Du*N*Transpose(Du) = D(N) (the matrix obtained by
applying D to each linear entry of N). This condition is linear in N, and it turns out that
there are 10 basic solutions corresponding to skew matrices Nij with ijth entry = u0 + l.o.t.
We take a sufficiently general linear combination and reduce modulo (S,t,p), and verify
that this defines a nonsingular elliptic curve E5 in PP^4 over FF5 with a free action
of al_5. This Magma code verifies everything. One of several derivations of the 10 skew
matrices Nij is here (only roughly documented). See also draft chapter.

Godeaux quintic hypersurface Y5 in PP^4 and F5 in PP^3 Godeaux surfaces
in characteristic 5 with Pic^0 a group scheme of order 5 (that is, ZZ/5 or al_5 or
mu_5) were constructed separately by Bill Lang, Rick Miranda and Christian Liedtke.
Soonyoung KIM's 2014 Seogang Univ. PhD thesis under Yongnam LEE constructed
these surfaces with additional ZZ/4 = FF5^x symmetry, and also clarified the argument
for nonsingularity in the inseparable cases (see below). Here we put all these varieties
into one family. The task is clear: take the regular representation Vreg of TO5, and
calculate the TO5 invariant quintic forms in Sym^5(Vreg). In fact, as with the calculations
over CC, it is better to calculate the whole graded ring of invariant forms: we find
u0 in degree 1, B1,B2 in degree 2, new C1..C4 in degree 3, new D1..D4 in degree 4, and
E1..E4 in degree 5. This Magma file carries this out. It also writes out a linear combination
of the invariants in degree 5 that after reducing modulo (S,t,p) defines an invariant quintic
surface F5 in PP^3 with al_5 action, whose quotient S = F5/al_5 is a nonsingular surface.
For the quintic 3-fold Y5 in PP^4, the current version of the nonsingularity calculation
does not work in the projective context, but works instantly in each affine piece.

Nonsingularity When we try to construct a variety X as a quotient of Y by a nonreduced
group al_p acting freely (or mu_p), we usually can't hope to have X nonsingular. In this
respect, the above elliptic curves are quite unrepresentative. Instead, the criterion we use
is that X should have a number n of points of type A_{p-1}, locally analytically xy = z^p,
with the singular subscheme (x=y=z^p) an orbit of the group; this is a sufficient condition
for the quotient X to be nonsingular. This is discussed in Soonyoung Kim's thesis, and in
her paper

[KSY] KIM Soonyoung, Numerical Godeaux surfaces with an involution in positive
characteristic, Proc. Japan Acad. Ser. A Math. Sci. 90:8 (2014) 113-118

The group action is given by a vector field having no zeros. Thus a surface Y with such
an action must have etale Euler number 0. If its singular subscheme is 0-dimensional, it
is a disjoint union of al_p orbits. For example, in the Godeaux case, if a quintic surface
Y in PP^3 has isolated singular locus and has an al_5 (or mu_5) action then its singular
subscheme always has degree 55 (the c2 of a general nonsingular quintic), and the
quotient S is nonsingular if this consists of 11 distinct al_5 orbits. A quintic 3-fold
with an al_5 action has singular subscheme of degree 200 (the c3 or Euler number of a
general quintic 3-fold), and the quotient X is nonsingular if this consists of 40
distinct al_5 orbits. Then Y has 40 points that are analytically x^2+y^2+z^2+t^5=0.

Campedelli varieties with 7-torsion Over CC, we construct Campedelli surfaces with
ZZ/7 in Pic S as the mu_7 quotient of F14 in PP^5 defined by the 6x6 Pfaffians of a
7x7 skew matrix. Our construction for the T07 version is quite similar to the above
exercise with an elliptic curve of degree 5. As there, we view the defining skew matrix
with linear entries as a map Vreg -> wedge2 Vreg. We write the conditions for it to be
TO7-invariant as 147 linear equations in the entries of a 7x21 matrix. It turns out there
are 21 basic solutions Nij having ijth entry u0 + l.o.t. The current calculation of
invariants is here. This is a temporary draft, only roughly documented inside the Magma
code. The nonsingularity calculation is still in progress. It works for a dense open set
of choices -- once is enough. However, for a random linear combination of the 21
basic matrices with coefficients in [1..6], it only works with probability about 60%.
We have settled on one solution that works, but we hope to find a more elegant one.

Next step: Godeaux with 3-torsion Still to do.

Part II. The T-nonsplit form

The paper is still in preparation. This Magma code works to give the T-nonsplit
form as a Hopf algebra.