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Numerical Godeaux surfaces with an involution in positive characteristic.

1. Introduction. Godeaux surfaces are surfaces over C of general type with the smallest invariants [p.sub.g] = q = 0 and [K.sup.2.sub.X] = 1. Information on the torsion groups of numerical Godeaux surfaces was obtained by Bombieri, Miyaoka, and Reid. It is known that Tors X has order at most 5 and Z/2Z [direct sum] Z/2Z is impossible [1,17,19]. Simply connected examples were first constructed by Barlow in 1982, and Lee and Park [13] gave a more recent construction. Godeaux surfaces with an involution over C were studied by Keum and Lee [9], and subsequently Calabri, Ciliberto, and Mendes Lopes [4] classified the possibilities for the quotient space of a Godeaux surface by its involution, proving that it is either rational or birational to an Enriques surface.

Lang [11] showed Godeaux surfaces exist in every characteristic. In his treatment Pic X is reduced, [Pic.sup.[tau]] X = Z/5Z, and X is a quotient of a quintic hypersurface Y in [P.sup.3] by an action of the multiplicative group scheme [[mu].sub.5]. A minimal surface X of general type over C with [K.sup.2.sub.X] = 1 and x([O.sub.X]) = 1 has [P.sub.g](X) = [h.sup.1]([O.sub.X]) = 0, but [p.sub.g](X) = [h.sup.1]([O.sub.X]) = 1 can also happen in characteristic p = 2, 3 and 5 [15]. These Godeaux surfaces are called nonclassical, and have nonreduced Pic X.

Miranda [16] constructed a Godeaux surface with nonreduced Picard scheme in characteristic 5 via a Godeaux-like construction. In a similar way, Liedtke constructed an action of the additive group scheme [[alpha].sub.5] on a quintic in characteristic 5 [15] by a nowhere zero additive vector field.

In these three cases, [Pic.sup.[tau]] X is isomorphic to Z/5Z, [[mu].sub.5] and [[alpha].sub.5] respectively, and [Pic.sup.[tau]] determines a finite flat morphism [phi]: Y [right arrow] X which is a torsor over X under the group scheme [([Pic.sup.[tau]] X).sup.[disjunction]], where [G.sup.[disjunction]] denotes the Cartier dual group scheme of G. We obtain the same bound [absolute value of (Tors X)] [less than or equal to] 5 as in characteristic 0, and we show that the quotient X/[sigma] of X by its involution is rational, or is birational to an Enriques surface. We study the three families in characteristic 5 due to Lang [11], Miranda [16], and Liedtke [15] with [Pic.sup.[tau]] X.

We show explicit examples of quintic hypersurface Y having symmetry by Aut G [congruent to] Z/4Z which is the holomorph [H.sub.20] = Hol G = G [varies] Z/4Z of G to give an involution on examples in each family in characteristic 5.

2. Godeaux surfaces in positive characteristic.

2.1. Notation and basic results. We work over an algebraically closed field k of characteristic p [not equal to] 2. Recall the following definitions.

[chi]([O.sub.X]) := [[summation].sup.n.sub.i=0][(-1).sup.i][h.sup.i]([O.sub.X])

[b.sup.et.sub.i] := dim [H.sup.i.sub.et] (X, [Q.sub.l])

e(X) := [[chi].sub.top](X) := [[summation].sup.n.sub.i=0] [(-1).sup.i] [b.sup.et.sub.i] (X)

[w.sub.X] := dualizing sheaf of X

[p.sub.g] := [h.sup.2] (X, [O.sub.X]) = dim [H.sup.0] (X, [w.sub.X])

q := dim Alb X

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[Pic.sup.[tau]] X := subscheme of Pic X of numerically trivial Cartier divisors

[W.sub.2](k) := ring of second Witt vectors of k.

Proposition 2.1 (Proposition 1, [15]). Let X be a minimal surface of general type with [K.sup.2.sub.X] = 1. Then the following equalities and inequalities hold:

1 [less than or equal to] [chi]([O.sub.X]) [less than or equal to] 3, [P.sub.g](X) [less than or equal to] 2, [h.sup.1]([O.sub.x]) [less than or equal to] 1, [b.sub.1](X) = 0, [absolute value of ([[pi].sup.et.sub.1] (X))] [less than or equal to] 6.

In particular, if [h.sup.1]([O.sub.X]) [not equal to] 0, then X has nonreduced Picard scheme, which can happen only in positive characteristic.

Definition 2.2. A numerical Godeaux surface is a minimal surface X of general type over an algebraically closed field with [K.sup.2.sub.X] = 1 and [chi][O.sub.X] = 1. In this paper we abbreviate numerical Godeaux surface to Godeaux surface.

Theorem 2.3 (Corollary 1, [15]). Nonclassical Godeaux surfaces can exist only in characteristic 2 [less than or equal to] p [less than or equal to] 5.

2.2. Tors X. Let G be a subgroup scheme of order n in [Pic.sup.[tau]] X. Then there is a finite morphism [phi]: Y [right arrow] X that is a nontrivial [G.sup.[disjunction]]-torsor. If the cover is purely inseparable, Y may be singular, but is still an irreducible Gorenstein surface [6] .If [phi]: Y [right arrow] X is a [[mu].sub.p]-torsor then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [L.sub.0] = [O.sub.X], [L.sub.1] [member of] Pic X is a line bundle with [L.sup.[cross product]p.sub.1] = [O.sub.X], and [L.sub.i] [congruent to] [L.sup.[cross product]i.sub.1].

If [phi]: Y [right arrow] X is a [[alpha].sub.p]-torsor then [[phi].sub.*][O.sub.Y] is a successive extension of sheaves isomorphic to [O.sub.X] [6, Proposition I.1.7]. And the equalities

(2.1) [chi]([O.sub.Y]) = [P.sub.[chi]]([O.sub.X]) and [K.sup.2.sub.Y] = p[K.sup.2.sub.X] hold as for a finite degree n Galois etale cover [15].

Proposition 2.4. Let X be a minimal surface of general type over an algebraically closed field k. Suppose characteristic p [greater than or equal to] 5. If [K.sup.2.sub.X] = 1 and [chi][O.sub.X] = 1, then [absolute value of ([Pic.sup.[tau]] X)] [less than or equal to] 5.

Proof. The proof is similar to Reid [18]. Let [phi]: Y [right arrow] X be the [G.sup.[disjunction]]-torsor associated to G = [Pic.sup.[tau]] X of order n. Since char k [not equal to] 2, 3, the Noether inequality [K.sup.2] [greater than or equal to] 2[p.sub.g] - 4 and (2.1) imply [absolute value of ([Pic.sup.[tau]] X)] [less than or equal to] 6.

Suppose [absolute value of ([Pic.sup.[tau]] X)] = 6. There is 6-to-1 etale cover [phi]: Y [right arrow] X with [p.sub.g](Y) = 5, [K.sup.2.sub.Y] = 6. Then Y is a Horikawa surface with [h.sup.1]([O.sub.Y]) = 0. The canonical map is a double cover [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and restricts to a [g.sup.3.sub.6] on a general C [member of] [absolute value of ([K.sub.Y])]. The classical Clifford theorem on an irreducible Gorenstein curve says that C is hyperelliptic [5]. The canonical image Z is an irreducible surface of degree 3 spanning [P.sup.4] [6, Proposition 0.1.2 (iii)], [14, Theorem 2.3], and Z is either [F.sub.1] embedded in [P.sup.4] as the cubic scroll or the cone over a rational normal curve of degree 3 in [P.sup.4] [14, Theorem 3.3]. In either case, the Horikawa double cover induces a biregular involution, and the composite p [phi] := f (where p is the projection p: F [right arrow] [P.sup.1]) is a canonically defined pencil of curves f: Y [right arrow] [P.sup.1] with fibers of genus 2. The Horikawa double cover induces a biregular involution since we work in characteristic [not equal to] 2, and the surface Y has a canonically defined pencil of curves of genus 2. This contradicts the free action of Z/3Z.

Remark 2.5. Proposition 2.4 implies that if [Pic.sup.[tau]] X contains a nontrivial subgroup scheme of odd order, then [Pic.sup.[tau]] X has no 2-torsion.

3. Numerical Godeaux surfaces with an involution in odd characteristic. Let X be a smooth Godeaux surface in positive characteristic p [not equal to] 2 with an involution [sigma]. The quotient double cover [pi]: X [right arrow] T := X/[sigma] fits in the diagram

(3:1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Given an involution [sigma] on X, its fixed locus is the union of a smooth curve R and n isolated fixed points [p.sub.1], ..., [p.sub.n]. The singularities of T are canonical and the adjunction formula gives [K.sub.X] [equivalent to] [[pi].sup.*] [K.sub.T] + R. In diagram (3.1), let [epsilon] be the blowup of X at n isolated fixed points in X of the action of [sigma]. The quotient map [pi] induces a double cover [??], where [eta] is the minimal resolution of the n ordinary double points of T. We set [E.sub.i] := [[epsilon].sup.*]([p.sub.i]), [R.sub.0] := [[epsilon].sup.*] (R) on V and [C.sub.i] := [??]([E.sub.i]), [B.sub.0] := [??]([R.sub.0]) on the smooth surface [[??].sub.*]. The [C.sub.i] are n disjoint--2-curves.

The map [??] is a finite flat double cover with branch locus [??] := [B.sub.0] + [[summation].sup.n.sub.i=1] [C.sub.i]. Thus there exists a line bundle L on W for which 2L [equivalent to] [??] and [[??].sub.*][O.sub.V] = [O.sub.W] [direct sum] [L.sup.-1]. Later in Lemma 3.6, we assume in addition that Kx is ample, so that [H.sub.m] has no -2-curves other than the four [C.sub.i].

Proposition 3.1. Let X be a minimal surface of general type in odd characteristic with an involution a. Then:

(i) 2[K.sub.W] + [B.sub.0] is nef and big;

(ii) [(2[K.sub.W] + [B.sub.0]).sup.2] = 2[K.sup.2.sub.X];

(iii) [K.sub.W]([K.sub.W] + L) [less than or equal to] 0;

(iv) The Kodaira dimension [kappa](W) [less than or equal to] 0.

Proof. (i) and (ii) follow from [[??].sup.*] (2[K.sub.W] + [B.sub.0]) = [[epsilon].sup.*](2[K.sub.X]). Part (iii) is clear by formula (ii). For (iv), since the Kodaira dimension [kappa] is a birational invariant, consider [phi]: D [subset] W [right arrow] D' [subset] [W.sub.min] := W', where D [member of] [absolute value of (2[K.sub.W] + [B.sub.0])] and D' = [[phi].sub.*]D. Suppose by contradiction that [kappa](W) [greater than or equal to] 1. Then D[K.sub.W] [less than or equal to] 0, which implies D'[K.sub.W'] [less than or equal to] 0. For m [much greater than] 0 we have D'm[K.sub.W'] = D' (M + F), where M is the moving part and F the fixed part. Then D'[K.sub.W'] > 0, which contradicts D'[K.sub.W'] [less than or equal to] 0. Hence [kappa](W) [less than or equal to] 0.

3.1. Vanishing theorem for 2[K.sub.W] + L. The Kodaira vanishing theorem and its extension due to Kawamata and Viehweg may fail in positive characteristic [7]. However, under additional assumptions, notably lifting to [W.sub.2](k), the Kawamata Viehweg vanishing theorem does hold.

Assumption 3.2. We fix the notation used in Theorem 3.3. X denotes a d-dimensional projective smooth variety over a perfect field k. Let E = [[summation].sup.m.sub.j=1] [E.sub.j] be a reduced simple normal crossing divisor on X. Assume that E [subset] X has a lifting [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 3.3 (Corollary 3.8, [8]). Let X be projective over a Noetherian affine scheme and let D be an ample Q-divisor on X such that Supp(D -[D]) [subset or equal to] Supp(E). Assume that E [subset] X admits a lifting [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then, if i + j > d = dim X and if p > d, we have

(3.2) [H.sup.i](X, [[OMEGA].sup.j.sub.X] (log E)(-E -[-D])) = 0.

Proposition 3.4 ([12]). Let X be an algebraic surface with isolated normal singularities, [pi]: V [right arrow] X its minimal resolution, and E the reduced exceptional divisor. If X has a cyclic quotient singularity of type [1/n] (1, n - 1), we assume that n is coprime to p. Then we have equality [[pi].sub.*] [T.sub.V] (-log E) = [[pi].sub.*] [T.sub.V] = [T.sub.X].

We keep the notation of diagram (3.1), C := [[summation].sup.n.sub.i=1] [C.sub.i] and [H.sub.m] := [K.sub.W] + L - ([1/2] + [1/m])C. Lemma 3.5. Hm is an ample Q-divisor for m [much greater than] 0.

Proof. Let N := [K.sub.W] + L - [1/2] [[summation].sup.n.sub.i=1] [C.sub.i], then N = 1/2 (2[K.sub.W] + [B.sub.0]) in (3.1). [[??].sup.*]N = [1/2] [[epsilon].sup.*] (2[K.sub.X]) is a nef and big divisor on V. For s [much greater than] 0 the linear system [absolute value of (sN)] is basepoint free and the associated morphism is birational, and contracts exactly Ci. Hence N = [[eta].sup.*] A for some ample Q-divisor A, and then L + v([[epsilon].sup.*] [K.sub.X]) = L + v([[??].sup.*] N) is ample for v [much greater than] 0 by [10, Proposition 1.45]. Therefore [H.sub.m] is ample for m [much greater than] 0.

Consider the following sequence:

(3.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Lemma 3.6. (W,C) lifts over [W.sub.2](k).

Proof. (3.3) gives the long exact sequence:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By results of Lee and Nakayama in [12], and by Section 1 of Burns and Wahl [3], the morphism [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is surjective. If W is rational or an Enriques surface, then [H.sup.2]([T.sub.W]) = 0 holds in any characteristic except possibly 2. Hence [H.sup.2]([T.sub.W](-log C)) = 0 by the above exact sequence, so that (W,C) lifts to [W.sub.2](k) by [20, Lemma 4.1].

Lemma 3.7. Let W, L be as above. Then [H.sup.i] (W, [O.sub.W](2[K.sub.W] + L))= 0 for i > 0.

Proof. By Lemma 3.5, [H.sub.m] is ample for m [much greater than] 0. Now (W, C) lifts over [W.sub.2](k) by Lemma 3.6. Applied to Theorem 3.3, this gives the vanishing

[H.sup.i](W, [O.sub.W](2[K.sub.W] + L)) = 0 for i> 0.

By Riemann Roch on surfaces, and [[??].sub.*] [O.sub.V] = [O.sub.W] [direct sum] [L.sup.-1], we get [chi]([O.sub.V]) = 2[chi]([O.sub.W]) + 1/2 L([K.sub.W] + L), which holds in odd characteristic.

Theorem 3.8. Let X, V and W be as above, and n the number of fixed points of [sigma]. Then:

(i) [chi]([O.sub.W]) = 1, and (ii) n [greater than or equal to] 5.

Proof. By the Riemann Roch theorem, Lemma 3.7 and the standard double cover formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Suppose by contradiction that [chi]([O.sub.W]) [less than or equal to] 0. Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A contradiction, and hence [chi]([O.sub.W]) = 1. From the standard double cover formula, (i), and by Proposition 3.1 (iii), we have [([K.sub.W] + L).sup.2] [less than or equal to] -2. So that [K.sup.2.sub.V] = [[??].sup.*] [([K.sub.W] + L).sup.2] [less than or equal to] -4, hence n [greater than or equal to] 5.

Corollary 3.9. Either W is rational or its minimal model is birational to an Enriques surface.

Proof. From Theorem 3.8 (i), Proposition 3.1 (iv) and the Table of possible invariants for surfaces with k = 0 in [2], [p.sub.g](W) = [h.sup.1]([O.sub.W]) = 0. The result thus follows from the classification of surfaces.

Lemma 3.10. [H.sup.i](W, [O.sub.W] (2[K.sub.W] + L)) = 0 for i > 0, hence [h.sup.0](W, [O.sub.W](2[K.sub.W] + L)) = 0.

Proof. By Theorem 3.8 (i), [chi]([O.sub.W]) = 1 and L([K.sub.W] + L) = -2. Therefore [chi](2[K.sub.W] + L) [less than or equal to] 0. Thus [H.sup.i](2[K.sub.W] + L) = 0 if i > 0, hence [h.sup.0](W, [O.sub.W] (2[K.sub.W] + L)) = 0.

Corollary 3.11. Let [phi] be the bicanonical map of X. Then:

(i) [phi] is composed with [sigma];

(ii) ([K.sub.W] + L)[K.sub.W] = 0;

(iii) n = 5.

Proof. Lemma 3.10 gives (i). Vanishing for 2[K.sub.W] + L and (i) give [h.sup.0](2[K.sub.W] + L)= [K.sub.W] ([K.sub.W] + L) = 0, [K.sup.2.sub.V] = F[([K.sub.W] + L).sup.2] = 2[([K.sub.W] + L).sup.2], and n = [K.sup.2.sub.X] - [K.sup.2.sub.V] = 1 - 2[([K.sub.W] + L).sup.2] = 5.

Lemma 3.12. Let f: X [right arrow] E be a double cover with X a nonsingular surface of general type and E birational to an Enriques surface in characteristic p [not equal to] 2. Then [f.sup.*][K.sub.E] is a nontrivial torsion element in Pic X. Equivalently, if K [right arrow] E is the K3 double cover, then the fiber product Y = X [x.sub.e] K is irreducible.

Proof. Consider

(3.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We can assume that X [right arrow] E is the quotient by an involution, so E has only 1/2 (1,1) singularities. The ramification locus of X [right arrow] E is a nonzero divisor De. For otherwise [K.sub.X] = [f.sup.*][K.sub.E] is numerically zero, which contradicts X of general type. Consider the fiber product Y = X [x.sub.E] K as the composite Y [right arrow] K [right arrow] E. Here [phi]: K [right arrow] E is the K3 double cover. Also Y [right arrow] K is ramified in the divisor [D.sub.K] = [[phi].sup.*][D.sub.E] by fiber product. Now [D.sub.K] > 0 and therefore Y is an irreducible surface. By base change, Y [right arrow] X is the double cover corresponding to [f.sup.*] [K.sub.E], so that [f.sup.*] [K.sub.E] [not equal to] 0 in Pic X if and only if Y is irreducible.

Corollary 3.13. Let X be a Godeaux surface with 5-torsion in characteristic 5. Then the birational type of the quotient space of X by an involution cannot be an Enriques surface.

Proof. Assume that the quotient of X is an Enriques surface W. We may assume W is minimal. An Enriques surface w in characteristic [not equal to] 2 has [K.sub.W] a 2-torsion class. Therefore the algebraic fundamental group [[pi].sup.et.sub.1](W) is isomorphic to Z/2Z. The fundamental group is a birational invariant of surfaces with at worst rational singularities. There is an etale 2-to-1 cover f: S [right arrow] [W.sub.min] with S a K3 surface in any characteristic [2]

(3.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If the quotient of X by its involution is birational to an Enriques surface W, the pullback of the 2-torsion class [K.sub.W] defines a nontrivial 2-torsion class on X by Lemma 3.12, so [absolute value of ([Pic.sup.[tau]] X)] must have even order. This contradicts Remark 2.5.

4. Godeaux surfaces in characteristic 5 with an involution. The Godeaux surfaces in characteristic 5 due to Lang [11], Miranda [16], and Liedtke [15] are constructed as quotients X = Y/G of a quintic hypersurface Y [subset] [P.sup.3] by a group scheme G of order 5 action freely. Here if G = Z/5Z, free means that G acts without fixed points. In the inseparable cases [[mu].sub.5] or [[alpha].sub.5], it means that G acts by a nowhere zero vector field. They prove the non-singularity of X by using Bertini's theorem for a very ample linear system on [P.sup.3]/G. Instead, in each case, we give an explicit example of Y having symmetry by Aut G [congruent to] Z/4Z, hence by the holomorph [H.sub.20] = Hol G = G [varies] Z/4Z of G. For the two inseparable cases, the nonsingularity of X involves a nonclassical calculation: as we show in 4.4, Y has exactly 11 singular points of type [A.sub.4].

4.1. The case G = Z/5Z. Miranda [16] takes the linear map a given by the matrix:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

He constructs a quintic surface Y invariant under <[sigma]> using the subspace V of quintic forms generated by norms N(l) = [[PI].sup.4.sub.i=0] [[sigma].sup.i](l) of linear forms l; these forms define an embedding [P.sup.3]/<[sigma]> [subset] P(V), and his X is a hyperplane section.

Lemma 4.1. The linear automorphisms

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

generate an action of Hol G = Z/5Z [varies] Z/4Z on [P.sup.3]. Clearly A, B2 generate an action of [D.sub.10].

Proof. One checks directly that [A.sup.5] = 1, [B.sup.4] = 1 and [BAB.sup.-1] = [A.sup.2].

Proposition 4.2. There exists a nonsingular hypersurface Y in [P.sup.3] invariant under Hol G action.

Proof. Set

f := [x.sup.5] + 3[x.sup.3]yw + 2[x.sup.3][z.sup.2] + 3 [x.sup.2][y.sup.2]z + 2[x.sup.2]z[w.sup.2] - x[y.sup.4] - x[y.sup.2][w.sup.2] + 2 x[z.sup.4] + 3 x[w.sup.4] + 2 [y.sup.3] zw + 3[y.sup.2][z.sup.3] + yz[w.sup.3].

One checks that f is invariant under A and B, and the quintic surface Y [subset] [P.sup.3] defined by f = 0 is nonsingular. (This is easy by computer algebra, but it can also be done by hand.)

Now the quotient Y [right arrow] X is an etale Z/5Z cover of a Godeaux surface X with [p.sub.g] (X) = [h.sup.1]([O.sub.X]) = 1 and [[pi].sub.1] = Z/5Z, and the Hol G action on Y descends to a Z/4Z action on X, so in particular an involution.

4.2. The case G = [[mu].sub.5]. Lang's Godeaux surfaces [11] satisfy [p.sub.g](X) = [h.sup.1]([O.sub.X]) = 0, and work in all characteristics. The group scheme [[mu].sub.5] acts on [P.sup.3] by [epsilon]([x.sub.i]) [right arrow] [[epsilon].sup.i][x.sub.i] and [P.sup.3]/[[mu].sub.5] is nonsingular except at the 4 coordinate points. If Y does not pass through these points, the [[mu].sub.5] action on [P.sup.3] restricts to a free action on Y. The general hyperplane X = Y/[[mu].sub.5] is a nonsingular Godeaux surface.

Lemma 4.3. Let A = [[mu].sub.5] act on [P.sup.3] with coordinates x, y, z, w by [1/5] (1, 2, 4, 3). The permutation B = (x, y, z, w) of [S.sub.4] defines a linear map of P3 that normalizes the [[mu].sub.5] action, and generates an action of the semidirect product group scheme Hol [[mu].sub.5] = [[mu].sub.5] [varies] Z/4Z. Then <A, [B.sup.2]> is a dihedral group scheme [D.sub.10].

Proof. The 4-cycle B = (x, y, z, w) corresponds to the generator [epsilon] [??] [[epsilon].sup.2] of Aut [[mu].sub.5] [congruent to] Z/4Z. One checks that [BAB.sup.-1] = [A.sup.2].

Proposition 4.4. There exists a hypersurface [Y.sub.5] [subset] [P.sup.3] invariant under Hol [[mu].sub.5] such that the quotient X = Y/[[mu].sub.5] is a nonsingular Godeaux surface.

Proof. Set

f = [x.sup.5] + [y.sup.5] + [z.sup.5] + [w.sup.5] + 2([x.sup.3] zw + x[y.sup.3]w + xy[z.sup.3] + yz[w.sup.3]) + 3([x.sup.2][y.sup.2]z + [x.sup.2]y[w.sup.2] + x[z.sup.2][w.sup.2] + [y.sup.2][z.sup.2]w).

Clearly Y is invariant under Hol [[mu].sub.5] and [D.sub.10]. See 4.4 for the nonsingularity of X.

4.3. The case G = [[alpha].sub.5]. Liedtke [15] uses the vector field [delta] := y [[partial derivative]/[partial derivative]x] + z [[partial derivative]/[partial derivative]y] + w [[partial derivative]/[partial derivative]z] to generate an [[alpha].sub.5] action on [P.sup.3]. Let V be the vector space of elements of degree 5 in the fixed ring of [delta]. The morphism [phi]: [P.sup.3] [right arrow] P(V) can be identified with the quotient map [P.sup.3] [right arrow] [P.sup.3]/[[alpha].sub.5], at least outside [1 : 0 : 0 : 0]. Its general hyperplane is a nonsingular Godeaux surface X = Y/[[alpha].sub.5] quotient of a S-invariant quintic [Y.sub.5] [subset] [P.sup.3].

Lemma 4.5. The following matrices generate an action of Hol [[alpha].sub.5] = [[alpha].sub.5] [varies] Z/4Z on [P.sup.3]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

such that (A) [congruent to] [[alpha].sub.5] with [t.sup.5] = 0, (B) [congruent to] Z/4Z. Moreover A, [B.sup.2] generate a dihedral group scheme [D.sub.10].

Proof. The matrix A with [t.sup.5] = 0 defines a group scheme [[alpha].sub.5]. One sees that [BAB.sup.-1] = [A.sup.2] as in Lemma 4.1. To prove [D.sub.10] = (A, [B.sup.2]) is clear.

Proposition 4.6. There exists a hypersurface [Y.sub.5] [subset] [P.sup.3] invariant under Hol [[mu].sub.5] with quotient Y/[[alpha].sub.5] a nonsingular Godeaux surface.

Proof. Set

f := [x.sup.5] + 2x[y.sup.2][w.sup.2] + xy[z.sup.2]w + 2x[z.sup.4] + 2x[w.sup.4] - [y.sup.3]zw + [y.sup.2][z.sup.3] - yz[w.sup.3] - [z.sup.3][w.sup.2].

Y is invariant under [[alpha].sub.5], Z/4Z and hence Hol G. As in the [[mu].sub.5] case, we show in 4.4 that X is nonsingular.

4.4. Nonsingularity of the quotient X. A quintic Y with an inseparable group action as in Proposition 4.4 and 4.6 must be singular. In fact a nonsingular quintic Y has e(Y) = [c.sub.2](Y) = 55. But a nonsingular surface with an everywhere nonzero vector field has [c.sub.2](Y) = 0. Alternatively, if Y is an inseparable cover of a nonsingular Godeaux surface X, then X and Y are homeomorphic in the etale topology, so e(Y) = e(X) = 11. We define the singular subscheme of Y by V(J) [subset] Y, where J(f) := (f, [[partial derivative]f/[partial derivative]x], [[partial derivative]f/[partial derivative]y], [[partial derivative]f/[partial derivative]z], [[partial derivative]f/[partial derivative]w]) is the Jacobian ideal.

Lemma 4.7. Let f be either of the invariant quintic polynomials of Proposition 4.4 and 4.6. Then dim V (J) = 0, deg V (J)= 55 and deg V [(J).sub.red] = 11.

Proof. Computer algebra. (A Magma script is available on request.)

Corollary 4.8. Y has 11 singularities of type [A.sub.4] (of analytic type xy = [z.sup.5]), and X is nonsingular.

Proof. Lemma 4.7 says that V(J) is supported at 11 distinct singular points of Y. [[mu].sub.5] or [[alpha].sub.5] act freely on [P.sup.3] except at the fixed coordinate points and V(J) is invariant under these G actions. Define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to be the sheaf of ideals generated by J, and [J.sub.i] its stalks at the 11 singular points. Then J = [[intersection].sup.11.sub.i=1] [J.sub.i]. Each [J.sub.i] is G-invariant, so that V([J.sub.i]) contains an orbit of G. From Lemma 4.7, each V([J.sub.i]) coincides with the G-orbit of [P.sub.i], which is a subscheme of length 5. Hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We choose local regular coordinates x, y, z in the local ring [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus x, y [member of] [J.sub.i], and after a coordinate change f = xy - [z.sup.5] + higher order terms.

The group scheme G acts by an everywhere nonzero p-closed vector field D, and D(f) = 0. It follows that D = [a.sub.0] (x [[partial derivative]/[partial derivative]x] - y [[partial derivative]/[partial derivative]y]) + b [[partial derivative]/[partial derivative]z], where [a.sub.0] [member of] k[[x, y, z]] and b is unit. We want to arrange that Dx = Dy = 0 after coordinate change. Set

[xi] = x[(1 + [[alpha].sub.1]z + ... + [[alpha].sub.4][z.sub.4]), [eta] = y(1 + ... + [[alpha].sub.4][z.sup.4]).sup.-1].

We take [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This coordinate change gives D = a[z.sup.4](x [[partial derivative]/[partial derivative]x] - y [[partial derivative]/[partial derivative]y]) + b [[partial derivative]/[partial derivative]z]. Dx = ax[z.sup.4], so that [D.sup.5]x = 4![b.sup.4](ax) + ... = [alpha] x ax[z.sup.4]. [D.sup.5](x) = [D.sup.4](a[z.sup.4]x) includes the term a x 4! x [b.sup.4]x and other terms in a x [m.sup.2], where m is a maximal ideal. But [D.sup.5](x) = cD(x) = ca[z.sup.4]x with c = 0 or 1, a is divisible by z. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is UFD, hence a = 0. Similarly for D(y), then the vector field acting on z only by z [right arrow] [alpha]z + [beta] with [[alpha].sup.5] = 1, [[beta].sup.5] = 0. Thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are regular functions on the quotient X = Y/G and the ideal they generate in [O.sub.Y,P] is J[O.sub.Y]. Therefore x, y generate the maximal ideal of [O.sub.X,Q], which implies X is nonsingular.

doi: 10.3792/pjaa.90.113

Acknowledgements. I thank Prof. Lee deeply for being my supervisor and for all his support and patience. I also thank a lot Prof. Miles Reid for useful tutorials about Godeaux surfaces, including computer algebra calculations for nonsingularity of X, and Prof. Christian Liedtke for useful comments. I would like to thank the referee for careful advice.

References

[1] E. Bombieri, Canonical models of surfaces of general type, Inst. Hautes Etudes Sci. Publ. Math. 42 (1973), 171-219.

[2] E. Bombieri and D. Mumford, Enriques' classification of surfaces in char. p. II, in Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1974, pp. 23-42.

[3] D. M. Burns, Jr. and J. M. Wahl, Local contributions to global deformations of surfaces, Invent. Math. 26 (1974), 67-88.

[4] A. Calabri, C. Ciliberto and M. Mendes Lopes, Numerical Godeaux surfaces with an involution, Trans. Amer. Math. Soc. 359 (2007), no. 4, 1605-1632 (electronic).

[5] F. Catanese, Pluricanonical-Gorenstein-curves, in Enumerative geometry and classical algebraic geometry (Nice, 1981), 51 95, Progr. Math., 24, Birkhauser Boston, Boston, MA, 1982.

[6] T. Ekedahl, Canonical models of surfaces of general type in positive characteristic, Inst. Hautes Etudes Sci. Publ. Math. 67 (1988), 97-144.

[7] H. Esnault and E. Viehweg, Lectures on vanishing theorems, DMV Seminar, 20, Birkhauser, Basel, 1992.

[8] N. Hara, A characterization of rational singularities in terms of injectivity of Frobenius maps, Amer. J. Math. 120 (1998), no. 5, 981-996.

[9] J. Keum and Y. Lee, Fixed locus of an involution acting on a Godeaux surface, Math. Proc. Cambridge Philos. Soc. 129 (2000), no. 2, 205-216.

[10] J. Kollar and S. Mori, Birational geometry of algebraic varieties, translated from the 1998 Japanese original, Cambridge Tracts in Mathematics, 134, Cambridge Univ. Press, Cambridge, 1998.

[11] W. E. Lang, Classical Godeaux surface in characteristic P, Math. Ann. 256 (1981), no. 4, 419-427.

[12] Y. Lee and N. Nakayama, Simply connected surfaces of general type in positive characteristic via deformation theory, Proc. Lond. Math. Soc. (3) 106 (2013), no. 2, 225-286.

[13] Y. Lee and J. Park, A simply connected surface of general type with [p.sub.g] = 0 and [K.sup.2] = 2, Invent. Math. 170 (2007), no. 3, 483-505.

[14] C. Liedtke, Algebraic surfaces of general type with small [c.sup.2.sub.1] in positive characteristic, Nagoya Math. J. 191 (2008), 111-134.

[15] C. Liedtke, Non-classical Godeaux surfaces, Math. Ann. 343 (2009), no. 3, 623-637.

[16] R. Miranda, Nonclassical Godeaux surfaces in characteristic five, Proc. Amer. Math. Soc. 91 (1984), no. 1, 9-11.

[17] Y. Miyaoka, Tricanonical maps of numerical Godeaux surfaces, Invent. Math. 34 (1976), no. 2, 99-111.

[18] M. Reid, Godeaux and Campedelli surfaces. (http://homepages.Warwick.ac.uk/~masda/ surf/ Godeaux.pdf).

[19] M. Reid, Surfaces with [p.sub.g] = 0, [K.sup.2] = 1, J. Fac. Sci. Univ. Tokyo Sect. iA Math. 25 (1978), no. 1, 75-92.

[20] Q. Xie, Kawamata-Viehweg vanishing on rational surfaces in positive characteristic, Math. Z. 266 (2010), no. 3, 561-570.

By Soonyoung KIM

Department of Mathematics, Sogang University, Sinsu-dong, Mapo-gu, Seoul 121-742, Republic of Korea

(Communicated by Shigefumi MORI, M.J.A., Sept. 12, 2014)
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