[TOp] Miles Reid, The Tate-Oort group scheme TOp,
to appear in Proceedings of the

Steklov Institute of
Mathematics, **307** (2019).

[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

1,x,..x^{p-1}.

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
LLLLLinear 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.

step: Godeaux with 3-torsion Still to do.