Dihedral Galois covers of algebraic varieties and the simple cases
aa r X i v : . [ m a t h . AG ] N ov DIHEDRAL GALOIS COVERS OF ALGEBRAICVARIETIES AND THE SIMPLE CASES
FABRIZIO CATANESE, FABIO PERRONI
Dedicated to Ugo Bruzzo on the occasion of his 60-th birthday.
Abstract.
In this article we investigate the algebra and geometryof dihedral covers of smooth algebraic varieties. To this aim wefirst describe the Weil divisors and the Picard group of divisorialsheaves on normal double covers. Then we provide a structuretheorem for dihedral covers, that is, given a smooth variety Y , wedescribe the algebraic “building data” on Y which are equivalentto the existence of such covers π : X → Y . We introduce thentwo special very explicit classes of dihedral covers: the simple andthe almost simple dihedral covers, and we determine their basicinvariants. For the simple dihedral covers we also determine theirnatural deformations. In the last section we give an application tofundamental groups. Contents
1. Introduction 22. Direct images of sheaves under Galois covers 42.1. The non-Galois case 93. Line bundles and divisorial sheaves on flat double covers 104. Line bundles on hyperelliptic curves 134.1. Powers of divisorial sheaves on double covers 175. Dihedral field extensions and generalities on dihedral covers 185.1. Dihedral field extensions 195.2. Structure of D n -covers 215.3. D -covers and triple covers. 306. Simple and almost simple dihedral covers and theirinvariants 326.1. Invariants of simple dihedral covers 356.2. Almost simple dihedral covers 367. Deformations of simple dihedral covers 40 Keywords : Galois covers of algebraic varieties; classification theory of algebraicvarieties.The present work took place in the framework of the DFG Forschergruppe 790‘Classification of algebraic surfaces and compact complex manifolds’ and of theERC-2013-Advanced Grant - 340258- TADMICAMT. The second author was alsosupported by the grant FRA 2015 of the University of Trieste, PRIN “Geometriadelle Variet`a Algebriche” and INDAM.
8. Examples and applications 418.1. Simple dihedral covers of projective spaces 418.2. An application to fundamental groups 43References 441.
Introduction
A main issue in the classification theory of algebraic varieties is theconstruction of interesting and illuminating examples. For instancein the book of Enriques [Enr49] one can see that a recurrent methodis the one of considering the minimal resolution S of double covers f : X → Y ; these are called ‘piani doppi’, double planes, when Y = P .Burniat [Bur66] considered more general bidouble covers, i.e., Galoiscovers f : X → Y with group G = ( Z / , and some work of thefirst author [Cat84], [Cat98] was focused on looking at the invariantsand the deformations of bidouble covers, deriving basic results for themoduli spaces of surfaces.Comessatti [Com30] was the first to study Galois coverings withabelian group G , and their relations to topology, while Pardini [Par91]described neatly the algebraic structure of such coverings, their invari-ants and deformations.Just to give a flavour of the result: when G is a cyclic group of order n and Y is factorial, normal G -coverings correspond to building data( L, D , . . . , D n − ) consisting of reduced effective divisors D , . . . , D n − without common components, and of the isomorphism class of a divisor L such that nL ≡ P i iD i ( ≡ is the classical notation for linear equiv-alence). The theorem is the scheme counterpart of the field theoreticdescription C ( X ) = C ( Y )[ z ] / ( z n − Π i δ ii ) . Here D i = { δ i = 0 } and the branch locus B f is the union of the divisors D i .In Pardini’s theorem one can replace the field C by any algebraicallyclosed field of characteristic coprime to n , but over the complex num-bers we have a more general result, namely, the extension of the Rie-mann existence theorem due to Grauert and Remmert: normal schemeswith a finite covering f : X → Y , and with branch locus contained ina divisor B correspond to conjugacy classes of monodromy homomor-phisms µ : π ( Y \B ) → S n . The scheme is a variety (i.e., irreducible) iffIm( µ ) is a transitive subgroup, and Galois if it is simply transitive (sim-ilarly for Abelian coverings in [BaCa08] the criterion for irreducibilityof X was explicitly given).On the other hand, the goal of finding explicit algebraic equations forthe case of Galois coverings with non Abelian group G is like the questfor the Holy Graal, and in this paper, widely extending previous results IHEDRAL GALOIS COVERS 3 of Tokunaga [Tok94], we give a full characterization of Galois coverings π : X → Y with Y smooth and with Galois group the dihedral group D n of order 2 n .The underlying idea is of course the same and is easy to explain ingeneral terms: we can factor π : X → Y as the composition of a cycliccovering of order n , p : X → Z := X/H , (here H ⊂ D n is the groupof rotations) and a (singular) double covering q : Z → Y . A main newtechnical tool, developed in sections 3 and 4, consists in describing Weildivisors and the Picard group of divisorial sheaves on a normal doublecover Z . To make the results more appealing, we describe in detail thevery special case of the Picard group of hyperelliptic curves, especiallythe algebraic determination of torsion line bundles; this is an extensionof a previous partial result by Mumford [Mum84], it boils down todeterminantal equations for triples of polynomials in one variable, andbears interesting similarities with the resultant of two polynomials.In section 5 we describe the general theorem, whose application inconcrete cases is however not so straightforward. For this reason weconcentrate in the next sections on two special very explicit classes ofdihedral covers of algebraic varieties: the simple and the almost simpledihedral covers.The underlying idea of simple covers is the one of giving a schematicequation which looks exactly as the equation describing the field exten-sion. The first well known instance is the one of a simple cyclic cover: X is given by an equation z n = F, where z is a fibre variable on the line bundle L associated to a divisor L on Y , and F is a section in H ( O Y ( nL )).In [Cat89] the analysis of everywhere nonreduced moduli spaces wasbased also on the technical notion of an almost simple cyclic cover,given as the locus X inside the P -bundle P ( L ⊕ L ∞ ), associated to apair of two line bundles L , L ∞ , defined by the equation z n a ∞ = z n ∞ a , a ∈ H ( O Y ( nL )) , a ∞ ∈ H ( O Y ( nL ∞ )) , and where the divisors E = { a = 0 } and E ∞ = { a ∞ = 0 } are disjoint(else the covering is not finite).Let us now explicitly describe the equations of a simple dihedralcovering: X is contained in a rank two split vector bundle of the form L ⊕ L over Y , and is defined by equations ( u n + v n = 2 a, a ∈ H ( O Y ( nL )) ,uv = F, F ∈ H ( O Y (2 L )) . The dihedral action is generated by an element σ of order n such that u ζ u, v ζ − v, FABRIZIO CATANESE, FABIO PERRONI where ζ is a primitive n -th root of 1, and by an involution τ exchanging u with v .The branch locus is here the divisor B = { a − F n = 0 } , Z is definedby z = a − F n , and the cyclic covering X is defined by u n = z + a : it issmooth if the two divisors { a = 0 } and { F = 0 } intersect transversallyand B is smooth outside of F = 0.In particular, as it is well known from the work of Zariski [Zar29] andmany other followers, the fundamental group π ( Y \ B ) is non Abelianand admits a surjection to the group D n (irreducibility is granted ifthere are points where { a = 0 } and { F = 0 } , since there the coveringis totally ramified).In the case of simple dihedral coverings, the covering X → Z is onlyramified in the A n − singularities of Z , the points where z = a = F = 0.In order to get a covering with more ramification we introduce then thealmost simple dihedral covers, defined this time on the fibre product oftwo P -bundles. X is the subset of P ( C ⊕ L ) × Y P ( C ⊕ L ) defined bythe following two equations: ( u v − u v F = 0 a ∞ v n u n − a v n u n + a ∞ v n u n = 0 , where, v , u are fibre coordinates on L , v , u are fibre coordinateson C (so that ([ u : u ] , [ v : v ]) are projective fibre coordinates on P ( C ⊕ L ) × Y P ( C ⊕ L ) ).As we said earlier, we want to find the invariants and study thedeformations of the varieties that we construct in this way. We firstdo this in full generality for any Galois covering in section 2; laterwe write more explicit formulae for simple and almost simple dihedralcoverings. We apply these in the case of easy examples, deferring thestudy of more complicated cases to the future.One interesting application, to real forms of curves with cyclic sym-metry, and related loci in the moduli space of real curves, shall be givenin a forthcoming joint work with Michael L¨onne [CLP].Finally, in this paper we consider also the non Galois case: namely,coverings of degree n whose Galois group is D n . In doing so, we estab-lish connections with the theory of triple coverings of algebraic varietiesdescribed by Miranda [Mir85] (if the triple covering is not Galois, thenits Galois group is D !).2. Direct images of sheaves under Galois covers
We consider (algebraic) varieties which are defined over the fieldof complex numbers C , though many results remain valid for complexmanifolds and for algebraic varieties defined over an algebraically closedfield of characteristic p not dividing 2 n . IHEDRAL GALOIS COVERS 5
Definition 2.1.
Let Y be a variety and let G be a finite group. A Galois cover of Y with group G , shortly a G -cover of Y , is a finitemorphism π : X → Y , where X is a variety with an effective action by G , such that π is G -invariant and induces an isomorphism X/G ∼ = Y .A dihedral cover of Y is a D n -cover of Y , where D n is the dihedralgroup of order n .If π : X → Y is a G -cover, the ramification locus of π is thelocus R := R π ⊂ X of points with non-trivial stabilizer; the (reduced)branch locus of π is B π := π ( R ) ⊂ Y .In the case where X, Y are smooth, R is the reduced divisor definedby the Jacobian determinant of π , and also B π is a divisor (purity ofthe branch locus). In the following, if there is no danger of confusion, we will denote B π by B .Let π : X → Y be a G -cover. Since π : X → Y is finite, π ∗ O X is acoherent sheaf of O Y -algebras. The O Y -algebra structure on π ∗ O X cor-responds to a morphism m : π ∗ O X ⊗ O Y π ∗ O X → π ∗ O X of O Y -modulesthat gives a commutative and associative product on π ∗ O X . Thesedata determine X as Spec π ∗ O X . Furthermore, the G -action on X cor-responds to a G -action on π ∗ O X which is the identity on O Y . The twostructures on π ∗ O X , of an algebra over O Y and of a G -module, imposerestrictions that sometimes allow one to determine certain “buildingdata” for the G -cover π . One of our aims here is the determination ofsuch building data when G = D n (Theorem 5.6).We will restrict ourselves to the case where π is flat, or equivalently π ∗ O X is locally free. We recall in the next proposition a useful charac-terization of flat G -covers. Proposition 2.2.
Let π : X → Y be a G -cover, with X irreducible and Y smooth. Then π is flat ⇔ X is Cohen-Macaulay.Proof. ( ⇐ ) This is a direct application of the Corollary to Theorem23.1 in [Mat89].( ⇒ ) Let x ∈ X and y = π ( x ). Since π is flat, for any regular sequence a , . . . , a r ∈ m y for O Y,y , π ∗ ( a ) , . . . , π ∗ ( a r ) is a regular sequence for O X,x . Hence, since Y is smooth and π is finite, X is Cohen-Macaulay. (cid:3) In the rest of this section we assume that X and Y are smooth. Theaim here is to relate the basic sheaves of X , namely Ω iX , Θ X , . . . , withthe corresponding ones of Y . Concretely, for example for the Ω iX ’s,there is a chain of inclusions as follows:Ω iY ⊗ π ∗ O X ֒ → π ∗ Ω iX ֒ → Ω iY (log B ) ∗∗ ⊗ π ∗ O X , where Ω iY (log B ) is the sheaf of i -forms with logarithmic poles along B and Ω iY (log B ) ∗∗ is its double dual. These three sheaves coincide on Y \ B and it is possible to describe explicitly π ∗ Ω iX inside Ω iY (log B ) ∗∗ ⊗ FABRIZIO CATANESE, FABIO PERRONI π ∗ O X at the generic points B i of B . Then, by Hartogs’ theorem, oneobtains a description of π ∗ Ω iX . We work out in detail the steps of thisprocedure for Ω X and Θ X .Let C ( X ) and C ( Y ) be the fields of rational functions on X and Y , respectively. The space of differentials of C ( X ), Ω C ( X ) , is a vectorspace of dimension d = dim C ( X ) Ω C ( X ) = dim X and a basis is given by the differentials of a transcendence basis of thefield extension C ⊂ C ( X ). The same holds true for Ω C ( Y ) and, sincedim X = dim Y , the pull-back morphism π ∗ : C ( Y ) → C ( X ) inducesan isomorphism of C ( X )-vector spaces:Ω C ( Y ) ⊗ C ( Y ) C ( X ) → Ω C ( X ) . (1)In other words, the rational differentials of X can be written as linearcombinations d X i =1 f i d y i , where f , . . . , f d ∈ C ( X ), y , . . . , y d ∈ C ( Y ) form a transcendence basisfor the extension C ⊂ C ( Y ). Since a regular differential 1-form on X is a rational differential 1-form which is regular at each point, theisomorphism (1) induces an inclusion ϕ : π ∗ Ω X ֒ → Ω C ( Y ) ⊗ C ( Y ) C ( X ) . (2)Since X and Y are smooth, by the theorem of purity of the branchlocus [Zar58] the ramification locus R and the branch locus B of π aredivisors in X and Y respectively. Let B = P ri =1 B i , where B i ⊂ Y areprime divisors. We recall, following [CHKS06], the following definition:Ω Y (log B ) := Im (cid:0) Ω Y ⊕ O ⊕ rY → Ω Y ( B ) (cid:1) , where Ω Y ֒ → Ω Y ( B ) is the natural inclusion, O ⊕ rY → Ω Y ( B ) is given by e i d b i b i , i = 1 , . . . , r , where, for any i , e i is the i -th element of the standard O Y -basis of O ⊕ rY ,and b i is a global section of O Y ( B i ) with B i = { b i = 0 } .Let Ω Y (log B ) ∗∗ be the double dual of Ω Y (log B ). It coincides withthe sheaf of germs of logarithmic 1-forms with poles along B definedin [Sai80]. Indeed both sheaves are reflexive and they coincide on thecomplement Y \ Sing( B ) of the singular locus of B . Proposition 2.3.
Let π : X → Y be a G -cover with X and Y smooth.Then there are G -equivariant inclusions of sheaves of O Y -modules asfollows: Ω Y ⊗ π ∗ O X ֒ → π ∗ Ω X ֒ → Ω Y (log B ) ∗∗ ⊗ π ∗ O X , IHEDRAL GALOIS COVERS 7 where the morphisms are isomorphisms on the complement Y \ B of thebranch divisor.Moreover, let ξ i ∈ π ∗ O X be the generator of the ideal of the re-duced divisor R i := π − ( B i ) and a maximal local root of b i ( b i = ξ m i i ),for any i = 1 , . . . , r . Then π ∗ Ω X is characterized as the subsheaf of Ω Y (log B ) ∗∗ ⊗ π ∗ O X such that at the smooth points of B i = { b i = 0 } it coincides with the subsheaf of O Y -modules generated by Ω Y ⊗ π ∗ O X and by the elements ξ ki d log( b i ) , k = 1 , . . . , m i − .Proof. The first arrow to the left is the push-forward under π ∗ of thenatural inclusion ( T π ) ∗ : π ∗ Ω Y ֒ → Ω X , where T π is the tangent map of π .Now consider the morphism ϕ in (2). We first show that (Im ϕ ) q ⊂ (Ω Y (log B ) ∗∗ ⊗ π ∗ O X ) q for any q ∈ Y ′ := Y \ Sing( B ). On the comple-ment of B , Y \ B , this follows from the projection formula, since π isnot ramified on π − ( Y \ B ) and hence π ∗ Ω X ∼ = Ω Y ⊗ π ∗ O X ∼ = Ω Y (log B ) ∗∗ ⊗ π ∗ O X on Y \ B . Let now q ∈ B ′ := B\ Sing( B ) and p ∈ π − ( q ). Choose local coordinates y , . . . , y d at q and x , . . . , x d at p , such that π has the following localexpression: y = x m , y = x , . . . , y d = x d . Hence O X,p = O Y,q [ x ] / ( x m − y ), and the stalk (Ω Y ) q is generated over O Y,q by d y , d y , . . . , d y d , while the stalk (Ω X ) p is generated over O X,p byd x , d x , . . . , d x d , with the obvious relations m · d log( x ) = d log( y ) , equivalently d x = x my d y , d x i = d y i , i ≥ . From this it follows that ( π ∗ Ω X ) q is generated as O Y,q -module by theelements of the G -orbit of x i d y j ≤ i ≤ m − , ≤ j ≤ d,x k y y , ≤ k ≤ m − y which implies the claim at the points q ∈ B ′ , since y = 0 is a definingequation for B near q . Moreover, we see that ( π ∗ Ω X ) q is generated, asan O Y -module, by (Ω Y ) q ⊗ ( π ∗ O X ) q and by the elements x k y y , ≤ k ≤ m − π ∗ Ω X ) q ⊂ (Ω Y,q (log B ) ∗∗ ⊗ ( π ∗ O X )) q is G -equivariant, if q ∈ Y \ Sing( B ). FABRIZIO CATANESE, FABIO PERRONI
To conclude the proof, observe that π ∗ Ω X is locally free and thatΩ Y (log B ) ∗∗ ⊗ π ∗ O X is reflexive, so that any section of π ∗ Ω X on Y \ Sing( B ) has a unique extension in Ω Y (log B ) ∗∗ ⊗ π ∗ O X . (cid:3) Let now Θ X := H om O X (Ω X , O X ) and Θ Y := H om O Y (Ω Y , O Y ) bethe tangent sheaves of X and Y , respectively. In the following propo-sition we relate the sheaf π ∗ Θ X with Θ Y .Define as in [Sai80] or [Cat88, Def. 9.15] the sheaf Θ Y ( − log B ) oflogarithmic vector fields on Y with respect to B = { b = 0 } asΘ Y ( − log B ) := { v | v · log( b ) ∈ O Y } = { v | v ( b ) ∈ b O Y } = (Ω Y (log B )) ∗ . Observe that the quotient sheaf Θ Y / Θ Y ( − log B ) equals the equisin-gular normal sheaf N ′B| Y of B in Y , which coincides with the normalsheaf at the points where B is smooth ([Cat88], rem. 9.16).Let us recall that the usual pairing Θ Y × Ω Y → O Y extends to aperfect pairing Θ Y ( − log B ) × Ω Y (log B ) ∗∗ → O Y . Proposition 2.4.
Let π : X → Y be a G -cover with X and Y smooth.Let B ⊂ Y be the branch divisor of π . Then the tangent map of π identifies π ∗ Θ X with a subsheaf of Θ Y ⊗ π ∗ O X . Under this identificationwe have the following inclusions of sheaves Θ Y ( − log B ) ⊗ π ∗ O X ⊂ π ∗ Θ X ⊂ Θ Y ⊗ π ∗ O X which are compatible with the G -actions. Moreover the three sheavescoincide on Y \ B and π ∗ Θ X is characterized as the subsheaf of Θ Y ⊗ π ∗ O X sending π ∗ Ω X to π ∗ O X .More concretely, it is the subsheaf such that at the smooth points of B i = { b i = 0 } coincides with the subsheaf of O Y -modules generated by Θ Y ( − log B ) ⊗ π ∗ O X and by ξ − i b i ∂∂b i , where ξ i ∈ π ∗ O X is the generatorof the ideal of the reduced divisor R i := π − ( B i ) (and a maximal localroot of b i , b i = ξ m i i ), while b i ∂∂b i is a local generator of N ′B| Y .Proof. The tangent morphism
T π : Θ X → π ∗ Θ Y gives a G -equivariantmorphism of sheaves(3) π ∗ Θ X → Θ Y ⊗ π ∗ O X . We first prove that (3) is injective and that its image contains Θ Y ( − log B ) ⊗ π ∗ O X . By Hartogs’ theorem and the definition of Θ Y ( − log B ) it suf-fices to prove this on the complement of Sing( B ).On Y \ B , π is not ramified, hence (3) is an isomorphism. Let now q ∈ B \ Sing( B ) and let p ∈ π − ( q ). Choose as before coordinates y , . . . , y d at q and x , . . . , x d at p , such that π has the following localexpression: y = x m , y = x , . . . , y d = x d . IHEDRAL GALOIS COVERS 9
Then ( π ∗ Θ X ) q is generated by the G -orbit of x k (cid:18) ∂∂x + mx m − ∂∂y (cid:19) , x k ∂∂y , . . . , x k ∂∂y d , ≤ k ≤ m − , as O Y,q -module. The image of these generators under (3) is the G -orbitof m ∂∂y ⊗ x m − k , ∂∂y ⊗ x k , . . . , ∂∂y d ⊗ x k , ≤ k ≤ m − . From this it follows that (3) is injective and that its image contains thesheaf Θ Y ( − log B ) ⊗ π ∗ O X , which is generated by the G -orbit of y ∂∂y ⊗ x k = ∂∂y ⊗ x m + k , ∂∂y ⊗ x k , . . . , ∂∂y d ⊗ x k , ≤ k ≤ m − , as O Y,q -module. The only missing term to get from Θ Y ( − log B ) ⊗ π ∗ O X the full direct image is then y ∂∂y ⊗ x − . We observe that y ∂∂y ⊗ x − = x y ∂∂y and the vector field y ∂∂y is the local generator of N ′B| Y =Θ Y / Θ Y ( − log B ). (cid:3) Remark 2.5.
Notice that the previous results are valid for more gen-eral Galois covers of complex manifolds.2.1.
The non-Galois case.
We extend propositions 2.3 and 2.4 tobranched covers π : X → Y which are not necessarily Galois. Noticethat also in this case we have an inclusion of the sheaf π ∗ Ω X in theconstant sheaf Ω C ( Y ) ⊗ C ( Y ) C ( X ) as in (2). Proposition 2.6.
Let π : X → Y be a branched cover with X and Y smooth. Let B ⊂ Y be the branch divisor of π . Then we have thefollowing inclusions of O Y -modules: Ω Y ⊗ π ∗ O X ֒ → π ∗ Ω X ֒ → Ω Y (log B ) ∗∗ ⊗ π ∗ O X , where the morphism on the left is induced by the pull-back morphism ( T π ) ∗ : π ∗ Ω Y → Ω X and the one on the right is induced by (2) .Proof. Since π : X \ R → Y \ B is non-ramified, ( T π ) ∗ : π ∗ Ω Y → Ω X isinjective. Applying the push-forward functor π ∗ we obtain an injectivehomomorphism Ω Y ⊗ π ∗ O X ֒ → π ∗ Ω X .We now prove that the morphism (2) has image in Ω Y (log B ) ∗∗ ⊗ π ∗ O X . Notice that by the same argument as in the proof of Proposition2.3, it suffices to prove this on the complement of the singular locus of B . So we assume that B is smooth. The assertion is true on Y \ B ,because here π is unramified. Let now q ∈ B and let V ⊂ Y be anopen neighbourhood (in the complex topology) of q , such that: π − ( V ) = U ⊔ . . . ⊔ U s , with U , . . . , U s ⊂ X disjoint subsets; for any i = 1 , . . . , s there arecoordinates ( x i, , . . . , x i,d ) on U i and ( y , . . . , y d ) on V , such that the restriction of π on U i has the form( x i, , . . . , x i,d ) ( y = x e i i, , y = x i, , . . . , y d = x i,d ) ,e , . . . , e s are integers ≥ π ∗ Ω X ) | V isgenerated as O Y ( V )-module by: ( x k i i, d x i, = x ki +1 i, e i d y y , k i = 0 , . . . , e i − , i = 1 , . . . , sx k i i, d x i,j = x k i i, d y j , j = 2 , . . . , d , i = 1 , . . . , s , k i = 0 , . . . , e − . Since y = 0 is a local equation for B , it follows that the image of π ∗ Ω X under (2) is contained in Ω Y (log B ) ∗∗ ⊗ π ∗ O X . (cid:3) Analogously, for the tangent sheaves we have the following
Proposition 2.7.
Let π : X → Y be a branched cover, with X and Y smooth. Let B ⊂ Y be the branch locus. Then the tangent map of π induces an injective morphism of sheaves π ∗ Θ X ֒ → Θ Y ⊗ π ∗ O X whose image contains the sheaf Θ Y ( − log B ) ⊗ π ∗ O X . Remark 2.8.
We can obtain more precise results by taking the Galoisclosure p : W → Y , which is a normal variety, and writing X = W/H where H is a suitable subgroup of the Galois group G . The only prob-lem is that W may not be smooth, and we denote then by W ′ itssmooth locus. Then we can write π ∗ Ω X as essentially the submoduleof H -invariants inside p ∗ Ω W : for instance we have π ∗ Ω X = (( p ∗ Ω W ) H ) ∗∗ , since the regular 1-forms on X ′ = W ′ /H are just the H -invariant 1-forms on W ′ .3. Line bundles and divisorial sheaves on flat doublecovers
In this section we consider the following general situation. We havea flat finite double cover, where Y is smooth: q : Z → Y, Z = Spec( q ∗ O Z ) , R := q ∗ O Z = ( O Y ⊕ z O Y ( − L ) / ( z − F )) , F ∈ H ( O Y (2 L )) . Our goal is to have a description of divisorial sheaves, that is, rank 1reflexive sheaves L on the Gorenstein variety Z in terms of their directimage q ∗ ( L ) =: N . By flatness of L over Y , which we assume throughout this section, N is a rank two vector bundle on Y (a locally free sheaf of rank 2),and its R - module structure is fully encoded in an endomorphism N : N ( − L ) → N IHEDRAL GALOIS COVERS 11 such that N = F · Id N : N ( − L ) → N . In particular, observe that Tr( N ) = 0 , det( N ) = − F , so that, on anyopen set where N and the divisor L are trivialized, N is given by amatrix of the form a bc − a , a + bc = F. Conversely, any such pair ( N , N ) as above determines a divisorial sheaf L on Z . Lemma 3.1.
Given a pair ( N , N ) where N : N ( − L ) → N satisfies N = F · Id N : N ( − L ) → N , it determines a saturated rank 1 torsionfree R -module which is locally free exactly in the points of Y where theendomorphism N does not vanish. In particular, a divisorial sheaf L on Z .Proof. View locally N as O Y : then a local section ( x, y ) is a local R generator at a point P if and only if ( x, y ) and N ( x, y ) are a local basis.This means, in terms of the matrix N , that ( x, y ) and ( ax + by, cx − ay )are linearly independent; equivalently, the determinant q ( x, y ) = cx − axy − by = 0 . Hence N is an invertible R -module if and only if the quadratic formobtained by evaluating a, b, c at P is not identically zero, which amountsto the condition that a, b, c do not vanish simultaneously. (cid:3) Remark 3.2.
Of course, the points where N vanishes yield singularpoints of the branch locus B = { F = 0 } , and singular points of Z .In the case where Z is normal, then L is determined by its restrictionon the smooth locus Z of Z since L = i ∗ ( L| Z ) (we are denoting by i : Z → Z the inclusion map), hence we are just dealing with thePicard group Pic( Z ). We want now to spell out in detail the groupstructure of Pic( Z ) in terms of the direct image rank 2 vector bundles. Proposition 3.3.
Let Z be normal, let L , L be invertible sheaveson Z , and let N j = ( q ◦ i ) ∗ ( L j ) , j = 1 , . Then the tensor productoperation in
Pic( Z ) , associating to L , L their tensor product L ⊗ O Z L gives the following pair:1) the vector bundle ( q ◦ i ) ∗ ( L ⊗ O Z L ) equals the O Y -double dualof N ⊗ R N , which is the cokernel of the following exact sequence: ψ : ( N ⊗ O Y N )( − L ) → N ⊗ O Y N → N ⊗ R N → , where ψ = N ⊗ O Y Id N − Id N ⊗ O Y N .
2) To the inverse invertible sheaf L − corresponds the R -moduleassociated to the pair ( N ∗ ( − L ) , t N ( − L )) , where N ∗ := H om ( N , O Y ) .3) Two invertible sheaves on Z , L , L , are isomorphic if and only ifthere is an isomorphism Ψ : N → N such that Ψ ◦ N = N ◦ Ψ( − L ) .4) Effective Weil divisors D on Z correspond to points [ δ ] of someprojective space P ( H ( N )) , for some isomorphism class of a pair ( N , N ) as above. The sum D + D corresponds to the image of δ ⊗ R δ inthe double dual ( N ⊗ R N ) ∗∗ .Proof.
1) First of all we have that ( q ◦ i ) ∗ ( L ⊗ O Z L ) is the saturationof the sheaf equal to N ⊗ R N on the open set Y = Y \ Sing( B )(letting j : Y → Y be the inclusion, the saturation of F is here for us j ∗ ( F | Y )).Moreover, by definition, N ⊗ R N is the quotient of N ⊗ O Y N bythe submodule generated by the elements ( z · n ) ⊗ O Y n − n ⊗ O Y ( z · n ),i.e., the submodule image of N ⊗ O Y Id N − Id N ⊗ O Y N : N ⊗ O Y N ( − L ) → N ⊗ O Y N . Since the above is an antisymmetric map of vector bundles, its rank atone point is either 2, or 0: but the latter happens, as it is easy to verify,exactly when both N and N vanish, i.e., in a locus contained in thesingular locus of the branch locus B = { F = 0 } , which has codimension2 since we assume Z to be normal.Hence we have that N ⊗ R N is a rank 2 bundle at y ∈ Y if either N or N do not vanish at y ; in the contrary case, the sheaf N ⊗ R N has rank 4 at the point y , and one needs to take ( N ⊗ R N ) ∗∗ , wherewe set F ∗ := H om O Y ( F , O Y ).We observe moreover that we have an infinite complex . . . → ( N ⊗ O Y N )( − L ) → ( N ⊗ O Y N )( − L ) → N ⊗ O Y N → N ⊗ R N → , where the maps are given by ψ ′ = N ⊗ O Y Id N + Id N ⊗ O Y N , respectively by the above ψ . The complex is exact on the right andalso at the points where either N or N are not vanishing.2) Given the module H om R ( N , R ), its elements consist of elements φ = φ + zφ , where φ ∈ N ∗ , and φ : N → O Y ( − L ). R -linearity isequivalent to φ ( N n ) = zφ ( n ), hence to φ ( N n ) = φ ( n ) , φ ( N n ) =
F φ ( n ) . The first equation implies the second, hence φ determines φ ; moreover,given any φ : N → O Y ( − L ), and defining φ ( n ) := φ ( N n ), we getan R -linear homomorphism by the above formulae.Clearly, multiplication by z acts on φ by sending φ = φ + zφ F φ + zφ , hence it sends φ : N → O Y ( − L ) to φ ◦ N , and it is givenby t N ( − L ) : N ∗ ( − L )( − L ) → N ∗ ( − L ). IHEDRAL GALOIS COVERS 13
3) Is easy to show, applying ( q ◦ i ) ∗ to the given isomorphism.4) A section of L determines, again by applying ( q ◦ i ) ∗ , a section δ ∈ H ( N ), which represents the image of 1 ∈ O Y and which in turndetermines the image of an element α + zβ ∈ R : since we must have α + zβ δ ( α ) + N δ ( β ). (cid:3) Line bundles on hyperelliptic curves
We apply the theory developed in the previous section to discuss theparticular, but very interesting case, of line bundles on hyperellipticcurves. Here Y = P , and Z = C is a hyperelliptic curve of genus g ≥ z = F ( x , x ) , where F is a homogeneous polynomial of degree 2 g +2 without multipleroots.The results of this section are also very relevant in order to showthe complexity of the equations defining dihedral covers. In fact, a D n -covering of P , X → P , factors through a hyperelliptic curve C .In the case where X → C is ´etale, X is determined by the choice of aline bundle L ∈ Pic ( C ) which is an element of n -torsion. This is thereason why we dedicate special attention to the description of n -torsionline bundles on hyperelliptic curves.For shorthand notation we write O ( d ) := O P ( d ) and we observethat every locally free sheaf on P is a direct sum of some invertiblesheaves O ( d j ).Our first goal here is to parametrize the Picard group of C . Thefirst preliminary remark is that it suffices to parametrize the subsetsPic ( C ) , Pic − ( C ), since for any line bundle L we can choose an integer d such that deg( L ⊗ O C q ∗ ( O ( d )) ∈ { , − } . Let us then consider a line bundle L of degree zero or −
1. In thecase L = O C we know that the corresponding pair is the vector bundle O ⊕ z O ( − g − , N ( α + zβ ) = F β + zα, and the corresponding matrix is N = F whose determinant yields a trivial factorization F = 1 · F .If instead we assume that L is nontrivial, i.e. it has no sections, wecan write N := q ∗ ( L ) = O ( − a ) ⊕O ( − b ) , < a ≤ b, a + b = g +1 − d, d = deg( L ) . The integer a is equal tomin { m | H ( L ⊗ q ∗ O ( m )) = 0 } , and it determines a Zariski locally closed stratification of the Picardgroup. We fix a and d ∈ { , } for the time being, and observe thatthen the vector bundle N is uniquely determined, whereas the matrix N (determining the R -structure) has the form N = P fq − P , − det( N ) = ( P + qf ) = F. Here,deg( P ) = g + 1 , deg( f ) = g + 1 − a + b, deg( q ) = g + 1 + a − b. Recall in fact that, in terms of fibre variables ( u, v ), we have zu = P u + qv, zv = f u − P v.
We see the matrix N as providing a factorization F − P = qf. It follows that N is fully determined by the degree g + 1 polyno-mial P and by a partial factorization of the polynomial F − P =Π g +21 l i ( x , x ) (here the l i are linear forms), as the product f q of twopolynomials of respective degrees deg( f ) = g +1 − a + b ≥ g +1 , deg( q ) = g + 1 + a − b ≤ g + 1. Observe that the choice for the polynomial q (hence of f ) is only unique up to a constant λ ∈ C ∗ , once the partialfactorization is fixed.Since for P general the linear forms l i are distinct, we have in thiscase exactly (2 g +2)!( g +1 − a + b )!( g +1+ a − b )! such possible factorizations.Denote by V ( a, b ) the variety of such matrices: V ( a, b ) = { ( P, f, q ) | P + qf = F } . The preceding discussion shows that the variety V ( a, b )parametrizing such matrices N is a C ∗ bundle over a finite coveringof degree (2 g +2)!( g +1 − a + b )!( g +1+ a − b )! of the affine space C g +2 parametrizingour polynomials P . Hence V ( a, b ) is an affine variety of dimension g + 3. However, in order to get isomorphism classes of line bun-dles (elements of the Picard group) we must divide by the adjointaction of the group G of the automorphisms of the vector bundle N := q ∗ ( L ) = O ( − a ) ⊕ O ( − b ) which have determinant 1.For a = b we get G = SL (2 , C ), whereas for a < b we get a triangulargroup of matrices B = λ β λ − , where β is any homogeneous polynomial of degree b − a . IHEDRAL GALOIS COVERS 15
We also observe that the stabilizer of a matrix N corresponds toan isomorphism of L , hence to a scalar µ ∈ C ∗ whose square equalsdet( B ) = 1, hence µ = ± V ( a, b ) yields an open set of dimension g in the casewhere a = b , else it gives a stratum of dimension g + 1 − ( b − a ).Recalling that a + b = g + 1 − d , for d = 0 the open set correspondsto: the case a = b = g +12 for g odd, and the case a = g , b = g + 1 for g even; similarly for d = − L on the hyperelliptic curve C , with two motivations:the first one is to try to get useful results towards the description ofdihedral coverings of the projective line, the second in order to describetorsion line bundles on hyperelliptic curves. Proposition 4.1.
There are exact sequences → K := O ( − a − b − ( g + 1)) → S ( q ∗ L ) → q ∗ ( L ) → , → K := K ⊗ q ∗ L → S ( q ∗ L ) → q ∗ ( L ) → , → K n := K ⊗ S n − ( q ∗ L ) → S n ( q ∗ L ) → q ∗ ( L n ) → . Proof.
The local sections of N = q ∗ L can be written as pairs ( u, v ).Hence q ∗ ( L ) is generated by u , uv, v , subject to the relation u ( f u − P v ) = u ( zv ) = z ( uv ) = v ( zu ) = v ( P u + qv ) ⇔⇔ Ξ := f u − P uv − qv = 0 . That this is the only relation follows since the kernel K has rank 1and first Chern class equal to − a − b − ( g + 1).Similarly, q ∗ ( L ) is generated by the cubic monomials u , u v, uv , v ,and we can simply multiply the relation Ξ by u , respectively v .In general, we simply observe that there is a natural morphism q L : C → Proj( q ∗ L )given by evaluation, and whose image is the relative quadric Γ := { ( u, v ) | Ξ( u, v ) = 0 } .Recall that, since L is invertible, the polynomials f, P, q cannot van-ish simultaneously, in particular Γ := { Ξ( u, v ) = 0 } is irreducible.Moreover the branch locus of Γ → P is { ( x , x ) | f q + P = F = 0 } ,therefore Γ is isomorphic to C .Hence C is the hypersurface in P ′ := Proj( q ∗ L ) defined by Ξ = 0,where Ξ is a section of O P ′ (2) ⊗ q ∗ ( K ) − ; hence we obtain the generalexact sequence via the pushforward of0 → O P ′ ( n )( − C ) = O P ′ ( n − ⊗ q ∗ ( K ) → O P ′ ( n ) → O C ( n ) → . (cid:3) Corollary 4.2.
A line bundle L of n -torsion on the hyperelliptic curve C corresponds to a pair ( N , N ) , N = O ( − a ) ⊕ O ( − b ) , < a ≤ b, a + b = g + 1 ,N = P fq − P , det( N ) = − ( P + qf ) = − F. where deg( P ) = g + 1 , deg( f ) = 2 b, deg( q ) = 2 a, such that the followinglinear map H ( S n ( O ( a ) ⊕ O ( b ))( − → H ( S n − ( O ( a ) ⊕ O ( b ))(2 g )) , equal to the dual of K ⊗ S n − ( q ∗ L ) → S n ( q ∗ L ) twisted by ( − , is notsurjective.Proof. Let L be a degree zero line bundle on C , so that we have a + b = g + 1. The condition that L n is trivial is clearly equivalent to thecondition H ( q ∗ ( L n )) = 0 . In view of the exact cohomology sequence(here H ( S n ( q ∗ L )) = H ( S n ( N )) = 0 since a, b > → H ( K n ) = 0 → H ( S n ( q ∗ L )) = 0 → H ( q ∗ ( L n )) →→ H ( K n ) → H ( S n ( q ∗ L )) → H ( q ∗ ( L n )) → , the condition H ( q ∗ ( L n )) = 0 is equivalent to the non injectivity of H ( K n ) → H ( S n ( q ∗ L )), equivalently, to the non surjectivity of thehomomorphism of Serre dual vector spaces:coker( H ( S n ( N ∗ )( − → H ( S n − ( N ∗ )(2 g ))) = 0 ⇔⇔ coker( H ( S n ( O ( a ) ⊕O ( b ))( − → H ( S n − ( O ( a ) ⊕O ( b ))(2 g ))) = 0 . For n = 2 this means that, denoting by A [ m ] = H ( O ( m )) the spaceof homogeneous polynomials of degree m , the linear map A [2 a − ⊕ A [ a + b − ⊕ A [2 b − → A [2 a + 2 b − f, − P, q ) is not surjective (equivalently, the linearmap given by the matrix ( f, P, q ) is not surjective).Writing in non homogeneous coordinates f = b X i f i x i , P = a + b X i P i x i , q = a X i q i x i , IHEDRAL GALOIS COVERS 17 this means that the matrix A ( f, P, q ) = f f . . . f b − f b . . . f f . . . f b − f b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . f f . . . f b − f b P P . . . P a + b . . . P P . . . P a + b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P P . . . P a + b q q . . . q a . . . q q . . . q a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . q q . . . q a , does not have maximal rank. (cid:3) Remark 4.3.
The reader may notice the similarity of the matrix A ( f, P, q ) with the matrix giving the resultant of two homogenouspolynomials in two variables. Moreover, notice that A ( f, P, q ) is a(3 a + 3 b − × (2 a + 2 b −
1) matrix, hence the condition that its rankbe at most 2 a + 2 b − g = a + b − ( C ).4.1. Powers of divisorial sheaves on double covers.
We wouldnow like to show how the result of proposition 4.1 generalizes to anyflat double cover q : Z → Y . Let L be a divisorial sheaf on Z , and( N , N ) the associated pair, where N = q ∗ ( L ). There is a natural map q L : Z Proj( N ) given by evaluation, and which is a morphism onthe smooth locus of Z .Set P ′ := Proj( N ), and let Γ be the image of q L . Clearly Z isbirational to Γ, which is a Gorenstein variety, since it is a divisor in P ′ ,and q L is bijective outside of the inverse image of the branch locus B q .Moreover q factors through q L and the natural projection π : P ′ → Y . At the points where L is invertible, then L is isomorphic with O Z ,with an isomorphism compatible with the projection q , hence we con-clude that q L is an embedding on the smooth locus of Z .At a point P where L is not invertible, we use the local descriptionof the pair ( N , N ) as ( O Y , N ), where N is the matrix sending twolocal generators x, y to ax + cy, respectively to bx − ay . Since L is notinvertible, by lemma 3.1 we get that a, b, c vanish at P . We use nowthe relations xz = ax + cy ⇔ x ( z − a ) = cy, yz = bx − ay ⇔ y ( z + a ) = bx to obtain ( x : y ) = ( c : z − a ) = ( z + a : b ) . These formulae (observe that ( c : z − a ) = ( z + a : b ) since z − a = bc )show that, at the points P where a, b, c (hence also z ) vanish, therational map q L blows up the point q − ( P ) to the whole fibre P lyingover P . Hence Γ is a small resolution of Z , and we have that the inverseof q L is obtained blowing down these P ’ s lying over such points P .Since we have an isomorphism Z ∼ = Γ , we see that the line bundle O Γ (1) restricts to L on Z , hence the sections of L n on Z correspondto sections of O Γ ( n ) on Γ .Notice that Pic( P ′ ) is generated by Pic( Y ) and by O P ′ (1), hence thereis an invertible sheaf K on Y such that Γ is the zero set of a sectionof O P ′ (1) ⊗ π ∗ ( K ) − .Consider now the exact sequence0 → O P ′ ( n )( − Γ) = O P ′ ( n − ⊗ π ∗ ( K ) → O P ′ ( n ) → O Γ ( n ) → . Taking the direct image, and observing that π ∗ O Γ ( n ) = q ∗ ( L n ), weobtain the following Proposition 4.4.
For each divisorial sheaf L on Z there is an exactsequence → K n := K ⊗ S n − ( q ∗ L ) → S n ( q ∗ L ) → q ∗ ( L n ) → . Dihedral field extensions and generalities on dihedralcovers
In this section we describe dihedral field extensions C ( Y ) ⊂ C ( X )in the case where Y is factorial and X is normal, thus we obtaina birational classification of D n -covers. Recall that for any G -cover π : X → Y of normal varieties the field extension C ( Y ) ⊂ C ( X ) isGalois with group G (a G -extension); on the other hand any such fieldextension determines π as the normalization of Y in C ( X ).In the second part of this section we determine the geometric build-ing data that make the above normalization process explicit. This isimportant, for instance to calculate invariants of X , to determine thedirect images of basic sheaves on X (e.g. π ∗ O X , π ∗ Ω X , π ∗ Θ X , etc.) andto provide explicitly families of such covers. IHEDRAL GALOIS COVERS 19
Dihedral field extensions.
Let us fix the following presentationfor the dihedral groups: D n = h σ, τ | σ n = τ = ( στ ) = 1 i . The split exact sequence0 → h σ i = Z /n Z → D n → Z / Z → D n -cover π : X → Y as the compositionof two intermediate cyclic covers: π = q ◦ p , where p : X → Z is a Z /n Z -cover, q : Z → Y is a Z / Z -cover, Z := X/ h σ i . If X is normaland irreducible, then also Z and Y are so, and we have the followingchain of field extensions: C ( Y ) ⊂ C ( Z ) ⊂ C ( X ) , where the field of rational functions on Z is the field C ( X ) h σ i of invari-ant functions under σ .Let us recall, following [Cat10], the structure of cyclic extensions C ( W ) ⊂ C ( V ), where W, V are normal varieties. Here the Galoisgroup G is cyclic of order m , G ∼ = Z /m Z ; later we use this descriptionwith m = 2 and with m = n to study D n -field extensions. Let σ ∈ G be a generator and let ζ ∈ C be a primitive m -th root of the unity.Then there exists v ∈ C ( V ) and f ∈ C ( W ), such that(4) ( C ( V ) = C ( W )( v ) , v m = f ,σ ( v ) = ζ v . Concretely, v can be chosen to be any non-zero element of the form v = m − X i =0 ζ m − i σ i (˜ v ) , ˜ v ∈ C ( V ) . Furthermore, it is possible to choose v and f , such that (4) holds and(5) f = m − Y i =1 ( δ i ) i , where δ , . . . , δ m − are regular sections of invertible sheaves on W \ Sing( W ). To see this, let ˆ v and ˆ f satisfying (4), and consider the Weildivisor associated to ˆ f , ( ˆ f ) = P U v U ( ˆ f ) U , where U ⊂ W varies amongthe prime divisors of W and v U is the valuation of U . On the non-singular locus W \ Sing( W ), U ∩ ( W \ Sing( W )) is an effective Cartierdivisor, hence U ∩ ( W \ Sing( W )) = { δ U = 0 } , where δ U is a regularsection of an invertible sheaf on W \ Sing( W ), henceˆ f = Y U δ v U ( ˆ f ) U . Let now v U ( ˆ f ) = mq U ( ˆ f )+ r U ( ˆ f ), where q U ( ˆ f ) , r U ( ˆ f ) ∈ Z , 0 ≤ r U ( ˆ f ) Let C ( Y ) ⊂ C ( X ) be a D n -extension. Then thereexist a, F ∈ C ( Y ) and x ∈ C ( X ) , such that C ( X ) = C ( Y )( x ) , x n − ax n + F n = 0 , with D n -action given as follows: σ ( x ) = ζ x , τ ( x ) = F/x , where ζ ∈ C is a primitive n -th root of .Conversely, given a, F ∈ C ( Y ) , such that x n − ax n + F n ∈ C ( Y )[ x ] is irreducible, then C ( Y )[ x ]( x n − ax n + F n ) is a D n -field extension of C ( Y ) with D n -action given as before. Hence the normalization of Y in C ( Y )[ x ]( x n − ax n + F n ) is a D n -covering of Y .Proof. Consider the quotient Z := X/ h σ i of X by the cyclic subgroup h σ i ∼ = Z /n Z and let q : Z → Y be the induced double cover. From theprevious description of cyclic field extensions, we have that ( C ( Z ) = C ( Y )( z ) , z = f ∈ C ( Y )¯ τ ( z ) = − z where ¯ τ ∈ D n / h σ i is the class of τ . Since Y is factorial we can assumethat f is a regular section of an invertible sheaf on Y .Now consider the extension C ( Z ) ⊂ C ( X ). Let x ∈ C ( X ), g ∈ C ( Z ),such that ( C ( X ) = C ( Z )( x ) , x n = g ,σ ( x ) = ζ x , IHEDRAL GALOIS COVERS 21 where ζ ∈ C is a primitive n -th root of 1, and notice that g ∈ C ( Z )can be written as g = a + zb , with a, b ∈ C ( Y ) . Without loss of generality b = 1 in the previous formula, otherwise wereplace f with b f and z with bz . Moreover, xτ ( x ) is invariant underthe action of D n , therefore xτ ( x ) = F ∈ C ( Y ) , equivalently τ ( x ) = F/x . Finally, F n = x n τ ( x n ) = g ¯ τ ( g ) = ( a + z )( a − z ) = a − f , hence f = a − F n . To conclude, we observe that z ∈ C ( Y )( x ) because x n = a + z , so C ( X ) = C ( Y )( x ) and by construction it follows that x n − ax n + F n = 0 . For the last statement, notice that the field extension C ( Y ) ⊂ C ( Y )[ x ]( x n − ax n + F n )is Galois with group D n , indeed the conjugates of x , namely ζ i x and ζ i F/x , for i = 0 , . . . , n − 1, belong to C ( Y )[ x ]( x n − ax n + F n ) . (cid:3) Structure of D n -covers. Let π : X → Y be a flat D n -cover with Y smooth. Then π ∗ O X is a locally free sheaf of O Y -modules, i.e. a vec-tor bundle on Y . Recall that the sheaf π ∗ O X carries a natural structureof O Y -algebras, which is given by the product of regular functions on X ,(7) m : π ∗ O X ⊗ O Y π ∗ O X → π ∗ O X ;the D n -action on X gives to π ∗ O X the structure of a D n -sheaf (seebelow). On the other hand, the variety X is completely determined by π ∗ O X and m , since X = Spec( π ∗ O X ) ([Hart77, II, 5.17]); the D n -actionon X is given by the structure of D n -sheaf on π ∗ O X .Recall that, for any finite group G the regular representation of G , C [ G ], is the vector space with a basis { e g } g ∈ G indexed by the elementsof G ; for any h ∈ G , an endomorphism of C [ G ] is defined by e g e hg ,for all g ∈ G . In a similar way one defines a sheaf of O Y -algebras O Y [ G ], for any variety Y . A sheaf F of O Y -modules is a G -sheaf , if ithas a structure of sheaf of O Y [ G ]-modules. If moreover F is a vectorbundle, then its fibers carry a linear G -action and so we can see F asa family of representations of G parametrized by Y .For any representation ρ : G → GL( V ) of G , its canonical decompo-sition (see [Serre77, § V = V ⊕ . . . ⊕ V N defined as follows. Let W , . . . , W N be the different irreducible repre-sentations of G . Then each V i is the direct sum of all the irreduciblerepresentations of G in V that are isomorphic to W i . If χ , . . . , χ N arethe characters of W , . . . , W N , and n i = dim W i , then(8) p i = n i | G | X g ∈ G χ i ( g ) ρ ( g ) ∈ End( V )is the projection of V onto V i , for any i = 1 , . . . , N , where χ i ( g ) is thecomplex-conjugate of χ i ( g ).Let now F be a locally free G -sheaf on Y with action ρ : G → GL O Y ( F ). Via (8) we define an endomorphism p i ∈ End O Y ( F ), forany i = 1 , . . . , N . Setting F i := Im( p i ), we have the following decom-position: F = F ⊕ . . . ⊕ F N . For any i = 1 , . . . , N , F i is the eigensheaf of F corresponding to the (ir-reducible) representation with character χ i . Notice that F i is a vectorsub-bundle of F , for all i . In particular, when F = π ∗ O X , π : X → Y is a flat G -cover, we have that π ∗ O X is a locally free sheaf of O Y [ G ]-modules of rank one, i.e. the fibres of π ∗ O X are isomorphic to C [ G ]as G -representations. Indeed, the previous procedure gives the decom-position π ∗ O X = ( π ∗ O X ) ⊕ . . . ⊕ ( π ∗ O X ) N , with ( π ∗ O X ) i ⊂ π ∗ O X asub-bundle, for any i . By construction, the fibres of ( π ∗ O X ) i are iso-morphic to each other as G -representations. So, it is enough to considerthe restriction of π ∗ O X on the complement Y \ B of the branch divisor B , where the assertion follows easily.In order to describe the sheaves ( π ∗ O X ) i , i = 1 , . . . , N , when π : X → Y is a flat D n -cover, let us briefly recall the representation theory ofthe dihedral groups. They depend on the parity of n . n odd. There are two irreducible representations of degree 1 with char-acters χ and χ , σ k σ k τχ χ − , and n − irreducible representations of degree 2, ρ ℓ ( σ k ) = (cid:18) ζ kℓ ζ − kℓ (cid:19) , ρ ℓ ( σ k τ ) = (cid:18) ζ kℓ ζ − kℓ (cid:19) , (9)where ζ ∈ C ∗ is a primitive n -th root of 1, 1 ≤ ℓ ≤ n − , k = 0 , . . . , n − IHEDRAL GALOIS COVERS 23 n even. In this case there are 4 representations of degree 1 with char-acters χ , χ , χ and χ , σ k σ k τχ χ − χ ( − k ( − k χ ( − k ( − k +1 , and, for any 1 ≤ ℓ ≤ n − 1, the irreducible representation ρ ℓ definedby (9).As a consequence we have that π ∗ O X = O Y ⊕ L ⊕ n − ℓ =1 ( π ∗ O X ) ℓ , if n is odd , and π ∗ O X = O Y ⊕ L ⊕ M ⊕ N ⊕ n − ℓ =1 ( π ∗ O X ) ℓ , if n is even , where L , M , N are the line bundles corresponding to the 1-dimensionalrepresentations with characters respectively χ , χ , χ .Notice that the sections of L are invariant under the action of thesubgroup h σ i ∼ = Z /n Z ≤ D n , hence they are regular functions on Z .Moreover we have:(10) q ∗ O Z = O Y ⊕ L , L ⊗ ∼ = O Y ( −B q ) , where B q ⊂ Y is the branch divisor of q .If n is even, the line bundles M and N have a similar interpretation,they arise from the ( Z / Z × Z / Z )-cover X/ h σ , τ i → Y .In order to get further information on the rank 4 vector bundles( π ∗ O X ) ℓ , we assume that X is normal and use the factorization π = q ◦ p and the theory of cyclic covers. Let us denote with Z the smoothlocus of Z , X = p − ( Z ) and p : X → Z be the restriction of p . From the structure theorem for cyclic covers ([Par91], [BaCa08],[Cat10]) it follows that p is determined by divisor classes L , . . . , L n − and reduced effective divisors D , . . . , D n − ⊂ Z without commoncomponents, such that(11) L i + L j ≡ L i + j − n − X k =1 ε ki,j D k , where i + j ∈ { , . . . , n − } , i + j = i + j (mod n ), L := O Z .Let us briefly recall the geometric interpretation of the previous data.For any i = 0 , . . . , n − 1, the line bundle O ( L i ) is the subsheaf of( p ) ∗ O X consisting of the regular functions f on X such that σ ∗ f =exp (cid:16) π √− n i (cid:17) f . For any k = 1 , . . . , n − 1, the divisor D k ⊂ Z isthe union of the components ∆ of the branch divisor B p of p suchthat, for any component T ⊂ ( p ) − (∆), the stabilizer of the generic point of T is the cyclic subgroup of h σ i generated by σ k and there is auniformizing parameter x ∈ O X ,T such that ( σ k ) ∗ x = exp (cid:16) π √− |h σ k i| (cid:17) x ,where |h σ k i| is the order of σ k . For every i, j = 0 , . . . , n − ε ki,j isdefined as follows: let ı i ( k ) , ı j ( k ) ∈ { , . . . , |h σ k i| − } be such thatexp (cid:18) π √− n ik (cid:19) = exp (cid:18) π √− |h σ k i| (cid:19) ı i ( k ) , exp (cid:18) π √− n jk (cid:19) = exp (cid:18) π √− |h σ k i| (cid:19) ı j ( k ) respectively, then(12) ε ki,j = ( , if ı i ( k ) + ı j ( k ) ≥ |h σ k i| , otherwise . Proposition 5.3. Let Y be a smooth variety and let π : X → Y be aflat D n -cover with X normal. Let p : X → Z and q : Z → Y be theintermediate cyclic covers defined previously. Then the following holdstrue. (i) p ∗ O X = ⊕ n − i =0 F i , where F = O Z and, for any i = 1 , . . . , n − , F i is a divisorial sheaf on Z . For any i, j = 1 , . . . , n − theproduct (7) induces an isomorphism as follows: (13) ( F i ⊗ O Z F j ) ∗∗ ∼ = F i + j ⊗ O Z O Z − n − X k =1 ε ki,j D k !! ∗∗ , where i + j ∈ { , . . . , n − } , i + j = i + j (mod n ) , for any k = 1 , . . . , n − , D k = D k is the closure of the divisor D k in (11) , and () ∗∗ is the double dual in the category of O Z -modules. (ii) For any i = 1 , . . . , n − , U i := q ∗ ( F i ) is a vector bundle of rank on Y , in particular F i is flat over Y . Furthermore, ( π ∗ O X ) ℓ = U ℓ ⊕ U n − ℓ , where ℓ = 1 , . . . , n − , if n is odd, and ℓ = 1 , . . . , n − if n iseven; in the case where n is even, U n = M ⊕ N .Proof. Consider the following Cartesian diagram:(14) X ı −−−→ X p y y p Z −−−→ Z where Z is the smooth locus of Z , : Z → Z is the inclusion, X = p − ( Z ), ı : X → X is the inclusion, and p = p | X . Notice that underour hypotheses Z is normal. IHEDRAL GALOIS COVERS 25 Since O X = ı ∗ O X and O Z = ∗ O Z (see, e.g. [ReidC3f]), p ∗ O X = p ∗ ı ∗ O X = ( p ◦ ı ) ∗ O X == ∗ p ∗ O X = ∗ (cid:0) O Z ⊕ n − i =1 O Z ( L i ) (cid:1) =(15) = O Z ⊕ n − i =1 ∗ O Z ( L i ) . Let us define F i := ∗ O Z ( L i ), for any i = 1 , . . . , n − 1. The product(7) gives morphisms m ij : F i ⊗ F j → F i + j . Since F i + j is reflexive, m ij is determined by its restriction on the smooth locus Z . On Z , by(11), m ij gives an isomorphism O Z ( L i + L j ) ∼ = O Z ( L i + j − P n − k =1 D k ).This implies (13).To prove (ii), apply q ∗ to (15) and use the following equalities: π = q ◦ p , q ∗ O Z = O Y ⊕ L . Notice that, q ∗ F i is locally free for any i =0 , . . . , n − 1, since π ∗ O X is locally free and π ∗ O X = ⊕ n − i =0 q ∗ F i . Theequality ( π ∗ O X ) ℓ = U ℓ ⊕ U n − ℓ follows directly from the definition of( π ∗ O X ) ℓ ⊂ π ∗ O X as the eigensheaf corresponding to the representation ρ ℓ . Finally, in the case where n is even, τ ∗ acts on U n ; M is the τ ∗ -invariant subsheaf, while N is the τ ∗ -anti-invariant one. (cid:3) Notice that (13) implies the following relation between F and theWeil divisors D , . . . , D n − , (cid:0) F ⊗ n (cid:1) ∗∗ ∼ = O Z ( − n − X k =1 kD k ) , which is the analogous of Remark 5.1 and equation (6) when the baseof the cover is normal.For later use we observe that the branch divisor B p = P n − k =1 D k of p : X → Z is invariant under the involution τ : Z → Z induced by τ , that is τ ∗ B p = B p , since στ = τ σ − . So, there exists an effectiveCartier divisor ∆ p ⊂ Y , such that ( q ) ∗ ∆ p = B p , where q := q | Z .Let us define(16) ∆ p := ∆ p ∈ Div( Y ) . Remark 5.4. Notice that, for any i = 1 , . . . , n − τ ( D i ) = D n − i , so τ ( D i ) does not have any common component with D i for any i < n/ n is even, τ ( D n ) = D n .In the remaining of the section we determine building data for D n -covers of Y . To this aim we first derive some properties of the vectorbundles U , . . . , U n − ( – below). For any i = 1 , . . . , n − 1, the involution τ : X → X induces anisomorphism τ ∗ : U i → U n − i , such that τ ∗ ◦ τ ∗ is the identity. Proof. For any open V ⊂ Y , U i ( V ) := F i ( q − ( V )) consists of the reg-ular functions f ∈ O X ( π − ( V )) such that σ ∗ f = exp( π √− n i ) f . From the relation στ = τ σ − it follows that f τ ∗ f gives a morphism τ ∗ : U i ( V ) → U n − i ( V ) of O Y -modules. Since τ = 1, τ ∗ ◦ τ ∗ = Id and τ ∗ is an isomorphism. (cid:3) For any i = 1 , . . . , n − U i has a structure of R -module givenby the restriction m i : L ⊗ U i → U i of the product (7), where R = O Y ⊕ L = q ∗ O Z (see Section 3.). In particular m i = δ q · Id U i , where δ q ∈ H ( Y, O Y ( B q )) is such that B q = { δ q = 0 } ( δ q = F of Section3). Furthermore, the isomorphism τ ∗ : U i → U n − i allows us to identify m n − i with − m i , as it follows from the commutativity of the followingdiagram and the fact that m i ◦ ( τ ∗ ⊗ Id U i ) = − m i (17) L ⊗ U i m i ◦ ( τ ∗ ⊗ Id Ui ) −−−−−−−−→ U i Id L ⊗ τ ∗ y y τ ∗ L ⊗ U n − i m n − i −−−→ U n − i For any i, j = 1 , . . . , n − 1, the product (7) induces, by restriction,a morphism m ij : U i ⊗ U j → U i + j , where i + j ∈ { , . . . , n − } , i + j = i + j (mod n ), and by definition U = q ∗ O Z . In particular, for any i = 1 , . . . , n − 1, there is a morphism m i,n − i : U i ⊗ U n − i → O Y ⊕ L . In the following, we will denote again with m i,n − i the previous mor-phism under the identification τ ∗ : U n − i → U i , hence m i,n − i : U i ⊗ U i → O Y ⊕ L . Let m + i,n − i : U i ⊗ U i → O Y and m − i,n − i : U i ⊗ U i → L be the com-positions of m i,n − i with the projections onto O Y and L respectively.Notice that, with respect to the involution on U i ⊗ U i that exchangesthe factors, m + i,n − i is symmetric, while m − i,n − i is antisymmetric, hencethey can be seen as morphisms m + i,n − i : Sym ( U i ) → O Y , m − i,n − i : ∧ ( U i ) → L , (18)or equivalently as sections m + i,n − i ∈ H ( Y, Sym ( U ∨ i )) , m − i,n − i ∈ H ( Y, ∧ ( U ∨ i ) ⊗ L ) . (19) Proposition 5.5. For any i = 1 , . . . , n − the following statementshold true. (i) m + i,n − i is determined by m i and m − i,n − i . (ii) The divisor of zeros of m − i,n − i coincides with the divisor ∆ p de-fined in (16) . In particular m − i,n − i yields an isomorphism be-tween ∧ U i and L ⊗ O Y ( − ∆ p ) . IHEDRAL GALOIS COVERS 27 Proof. (i) Let y ∈ Y , let ( U i ) y be the stalk of U i over y and let s , s ∈ ( U i ) y . Then m i,n − i ( s ⊗ s ) = s τ ∗ ( s ) , hence m + i,n − i ( s ⊗ s ) = 12 ( s τ ∗ ( s ) + τ ∗ ( s ) s ) ,m − i,n − i ( s ⊗ s ) = 12 ( s τ ∗ ( s ) − τ ∗ ( s ) s ) ;where the product s τ ∗ ( s ) (respectively τ ∗ ( s ) s ) is the usual onebetween stalks of regular functions defined in some neighborhood of π − ( y ). For any r ∈ ( L ) y , the associativity of the multiplication im-plies that m i,n − i ( m i ( r ⊗ s ) ⊗ s ) = m ( r ⊗ m i,n − i ( s ⊗ s ))and hence m ± i,n − i ( m i ( r ⊗ s ) ⊗ s ) = m (cid:0) r ⊗ m ∓ i,n − i ( s ⊗ s ) (cid:1) . This implies that, under the natural identification O Y,y ∼ = End(( L ) y ), m + i,n − i ( s ⊗ s ) = m − i,n − i ( m i (( ) ⊗ s ) ⊗ s ), and hence the claim follows.(ii) Without loss of generality we assume that Z is smooth. Indeed,since Z is normal, its singular locus Sing( Z ) has codimension ≥ 2, so,for Z := Z \ Sing( Z ) and Y := q ( Z ), the divisor of zeros of m − i,n − i and ∆ p are determined by their restrictions to Y .For any y ∈ Y , let us consider the stalk ( m − i,n − i ) y of m − i,n − i at y . Let u i , v i be a basis of ( U i ) y as O Y,y -module, then u i ⊗ v i − v i ⊗ u i is a basisof ( ∧ U i ) y , viewed as the submodule of U i ⊗ U i . We have: m − i,n − i ( u i ⊗ v i − v i ⊗ u i ) = u i τ ∗ ( v i ) − v i τ ∗ ( u i )= u i τ ∗ ( v i ) − τ ∗ ( u i τ ∗ ( v i )) , (20)where we consider u i and v i as stalks of regular functions definedon some open neighbourhood of π − ( y ) in X , such that σ ∗ ( u i ) =exp( π √− n i ) u i and σ ∗ ( v i ) = exp( π √− n i ) v i .Let us choose local analytic coordinates ( y , . . . , y d ) for Y at y , and w on Z , such that Z is given locally by the equation w = y . Further-more, for any i = 1 , . . . , n − 1, let e i be a basis of F i as (cid:16) C [ y ,...,y d ,w ]( w − y ) (cid:17) -module. Then let(21) u i := 1 · e i , v i := w · e i . Notice that we can choose the e i ’s in such a way that τ ∗ ( e i ) = e n − i , forany i = 1 , . . . , n − 1, since τ ∗ identifies F i with F n − i . So, substituting(21) in (20) and using the equations τ ∗ ( w ) = − w and τ ∗ ( e i ) = e n − i , we obtain: m − i,n − i ( u i ⊗ v i − v i ⊗ u i ) = u i τ ∗ ( v i ) − τ ∗ ( u i τ ∗ ( v i ))= e i τ ∗ ( we i ) − τ ∗ ( e i τ ∗ ( we i ))= e i ( − we n − i ) − τ ∗ ( e i ( − we n − i ))= − we i e n − i − we i e n − i = − we i e n − i = − wb p , (22)where we have used the fact that e i e n − i = b p , b p is a local equation for B p (this follows from (11)). Finally, let δ p be a local equation for ∆ p such that q ∗ ( δ p ) = b p , then the previous computation shows that m − i,n − i ( u i ⊗ v i − v i ⊗ u i ) = − wq ∗ ( δ p ) , from which the claim follows. (cid:3) The following result is a converse to Proposition 5.3. Theorem 5.6 (Structure of D n -covers) . Let Y be a smooth variety and n be a positive integer. Then, to the following data a), b) and c), wecan associate a D n -cover π : X → Y in a natural way. a) A line bundle L and an effective reduced divisor B q on Y , suchthat L ⊗ ∼ = O Y ( −B q ) . b) Reduced effective Weil divisors D , . . . , D ⌊ n ⌋ on Z := Spec( O Y ⊕L ) without common components, such that τ ( D ∪ . . . ∪ D ⌊ n − ⌋ ) doesn’t have common components with D ∪ . . . ∪ D ⌊ n − ⌋ , and,in the case where n is even, τ ( D n ) = D n ; where ⌊ a ⌋ denotesthe integer part of a number a ∈ Q , and τ is the involution ofthe double cover q : Z → Y . c) Divisorial sheaves F , . . . , F ⌊ n ⌋ on Z flat over O Y , such that,for any i, j = 1 , . . . , n − , (13) holds, and if n is even F n = τ ∗ ( F n ) ; where for ⌊ n ⌋ < ℓ, k ≤ n − , F ℓ := τ ∗ ( F n − ℓ ) and D k := τ ( D n − k ) , the coefficients ε kij are defined in (12) .The variety X so constructed is normal if and only if, setting κ :=gcd { k = 1 , . . . , n − | D k = 0 } , then either κ = 1 , or n κ F − P n − k =1 k κ D k has order precisely κ in the group of divisorial sheaves of Z . In thiscase π ∗ O X = O Y ⊕ L ⊕ ⌊ n/ ⌋ i =1 q ∗ F i , in particular π is flat. Before giving the proof of the previous theorem, two remarks are inorder. Remark 5.7. 1. Using the results of Section 3, the divisorial sheavesin c) correspond to pairs ( U , m ) , . . . , ( U ⌊ n ⌋ , m ⌊ n ⌋ ) consisting of rank2 vector bundles on Y , U , . . . , U ⌊ n ⌋ , and morphisms m i : U i ⊗ L → U i such that m i = δ q Id U i , where B q = { δ q = 0 } . With this notation, τ ∗ F i corresponds to ( U i , − m i ), for any i = 1 , . . . , n − 1. Furthermore,using Proposition 3.3, the relation (13) can be written in terms of the( U i , m i )’s and the D i ’s. IHEDRAL GALOIS COVERS 29 A similar criterion for the existence of D n -covers π : X → Y hasbeen given in [Tok94], where X is constructed as the normalizationof Y in a certain dihedral field extension of C ( Y ). Here, using thestructure theorem for cyclic covers, we construct π : X → Y explicitly,we hope that this procedure could be useful for further investigationsof Galois covers. Proof. The data a) determines a flat double cover q : Z → Y in theusual way, Z := Spec ( O Y ⊕ L ) and q is given by the inclusion O Y ֒ →O Y ⊕ L as the first summand. Notice that Z is normal since B q isreduced.We define X := Spec ( O Z ⊕ F ⊕ . . . ⊕ F n − ), where O Z ⊕ F ⊕ . . . ⊕ F n − is the sheaf of O Z -algebras with algebra structure given bythe isomorphisms (13) in the following way. Since F i + j is a divisorialsheaf, for i, j = 1 , . . . , n − 1, a morphism F i ⊗ F j → F i + j is uniquelydetermined by its restriction on the smooth locus Z of Z . Let us fixfirstly sections δ k ∈ H ( Z, O Z ( D k )), such that D k = { δ k = 0 } on Z ,for any k = 1 , . . . , n − 1. By Lemma 3.1, on Z the sheaves F , . . . , F n − are locally free, so, locally where they are trivial, we choose generators e , . . . , e n − such that e n − i = τ ∗ ( e i ), for any i . The algebra structure onthe restriction of O Z ⊕ F ⊕ . . . ⊕ F n − on such open subsets is definedas usual by the equations e i e j = e i + j n − Y k =1 δ ε kij k , for any i, j = 1 , . . . , n − . Notice that, if we choose different generators, ˜ e , . . . , ˜ e n − satisfying thesame conditions ˜ e n − i = τ ∗ (˜ e i ), then we obtain an algebra canonicallyisomorphic to the previous one. Hence the construction globalizes and O Z ⊕ F ⊕ . . . ⊕ F n − becomes a sheaf of O Z -algebras. The morphism π : X → Y is defined as usual.From the previous construction and from the fact that D n − k = τ ( D k ), it follows that τ ∗ : O Z ⊕ F ⊕ . . . ⊕ F n − → O Z ⊕ F ⊕ . . . ⊕ F n − is a morphism of O Z -algebras. This defines an involution τ : X → X .By construction we have that σ ∗ ◦ τ ∗ = τ ∗ ◦ ( σ − ) ∗ , hence X carries anaction of D n such that π : X → Y is a D n -cover.The condition for X to be normal follows from Theorem 1.1 in[Cat10]. (cid:3) Remark 5.8. Similarly as for cyclic covers, X in Theorem 5.6 is de-termined by the data a), b) and F (or ( U , m )), such that( F ⊗ n ) ∗∗ ∼ = O Z − n − X k =1 kD k ! . Furthermore, by Proposition 4.4, ( F ⊗ n ) ∗∗ = (Sym n ( U )) / K n . D -covers and triple covers. In this section we restrict our-selves to D -covers and relate the results that we obtained so far withthe structure theorem for flat finite morphisms of degree 3 of algebraicvarieties ([Mir85]), also called triple covers . This relation originatesfrom the following fact: for every D -cover π : X → Y , any quotient W := X/ h σ i τ i has a natural structure of triple cover of Y induced by π ; conversely, if W → Y is a triple cover which is not Galois, thenits Galois closure is a D -cover. Since the elements σ i τ are pairwiseconjugate, the corresponding triple covers are isomorphic, hence in thefollowing we consider only X/ h τ i . Notice that the structure of D -covers has been studied also in [East11].Let π : X → Y be a D -cover. We have seen in the previous sec-tion that π is determined by the locally free sheaves L , U , U and amorphism of O Y -modules m : ( O Y ⊕ L ⊕ U ⊕ U ) ⊗ → O Y ⊕ L ⊕ U ⊕ U which gives an associative commutative product. The involution τ yields an isomorphism τ ∗ : U → U as explained in the previous sec-tion. Under this identification one easily see that π : X → Y is deter-mined by O Y , L , U , m , m and m (see the previous section fortheir definitions). In particular, O X = O Y ⊕ L ⊕ U ⊕ U and the involution τ ∗ acts on the element ( a, b, c, d ) ∈ O Y ⊕ L ⊕ U ⊕ U as follows: τ ∗ ( a, b, c, d ) = ( a, − b, d, c ) . The subsheaf of τ ∗ -invariants, ( O X ) τ ∗ ⊂ O X , consists of the elementsof the form ( a, , c, c ) ∈ O Y ⊕ L ⊕ U ⊕ U , so the sheaf of regularfunctions on W := X/ h τ i is(23) O W ∼ = O Y ⊕ U with inclusion in O Y ⊕ L ⊕ U ⊕ U given by ( a, c ) ( a, , c, c ). Themorphism π : X → Y descends to a morphism f : W → Y , which is atriple cover. We refer to [Mir85] for definitions, notations and resultsconcerning triple covers. Proposition 5.9. Under the identification (23) , U coincides with theTschirnhausen module of O W over O Y , that is U consists of the el-ements of O W whose minimal cubic polynomial has no square term( U = E in the notation of [Mir85] ). Furthermore, the tensor φ in [Mir85] coincides with m : U ⊗ U → U , which has been defined inthe previous section.Proof. For (0 , c ) ∈ O Y ⊕ U , we calculate its cubic power (0 , c ) . Tothis aim, we consider its image (0 , , c, c ) ∈ O Y ⊕ L ⊕ U ⊕ U anduse the product m of O X . In order to simplify the notation, we write IHEDRAL GALOIS COVERS 31 (0 , , c, c ) = c + τ ∗ ( c ) ∈ O X and denote the product m simply by · .Then we have the following expression:( c + τ ∗ ( c )) = c + τ ∗ ( c ) + 2 cτ ∗ ( c ) ( c + τ ∗ ( c )) . Since c + τ ∗ ( c ) , cτ ∗ ( c ) ∈ O Y , the minimal cubic polynomial of c + τ ∗ ( c ) has no square term and so it belongs to the Tschirnhausen moduleof O W over O Y . This proves the first claim.For the second claim, recall that by definition φ is the compositionof the product in O W followed by the projection onto U . For any c, c ′ ∈ U , the product between c + τ ∗ ( c ) , c ′ + τ ∗ ( c ′ ) ∈ O W has thefollowing expression:( c + τ ∗ ( c )) ( c ′ + τ ∗ ( c ′ )) = cc ′ + τ ∗ ( cc ′ ) + cτ ∗ ( c ′ ) + τ ∗ ( cτ ∗ ( c ′ )) . Notice that cτ ∗ ( c ′ )+ τ ∗ ( cτ ∗ ( c ′ )) ∈ O Y and that cc ′ + τ ∗ ( cc ′ ) correspondsto (0 , τ ∗ ( cc ′ )) ∈ U under the identification (23). So φ coincides with m under the identification τ ∗ : U → U . (cid:3) Let f : W → Y be a flat finite map of degree 3, let E be the Tschirn-hausen module of O W over O Y and let φ : Sym ( E ) → E be theassociated triple cover homomorphism, as defined in [Mir85]. Underthe standard identification E ∼ = E ∨ ⊗ ∧ ( E ) = Hom( E, ∧ ( E )), u ( v u ∧ v ), we can view φ as a morphism φ : Sym ( E ) ⊗ E → ∧ ( E ).By [Mir85, Prop. 3.5], φ is symmetric, hence it induces a morphismΦ : Sym ( E ) → ∧ ( E ). The structure theorem for triple covers statesthat conversely f : W → Y is determined by a rank 2 locally free sheaf E of O Y -modules and an O Y -morphism Φ : Sym ( E ) → ∧ ( E ) ([Mir85,Thm. 3.6]). The proof of the fact that φ is symmetric given in [Mir85]is done by a direct calculation using local coordinates and is a con-sequence of the fact that E is the Tschirnhausen module of O W over O Y .We extend part of these results to D n -covers, n ≥ 2, and give adifferent proof of [Mir85, Prop. 3.5] quoted above. For any D n -cover π : X → Y , there is a O Y -morphism φ n − : Sym n − ( U ) → U defined as follows: for any u ⊗ . . . ⊗ u n − ∈ Sym n − ( U ), their product m ( u ⊗ . . . ⊗ u n − ) is in U n − , so applying τ ∗ : U n − → U we obtainan element in U , this is by definition φ n − ( u ⊗ . . . ⊗ u n − ). Underthe identification U ∼ = U ∨ ⊗ ∧ ( U ) as before, we can see φ n − as amorphism φ n − : Sym n − ( U ) ⊗ U → ∧ ( U ) . Proposition 5.10. The morphism φ n − is symmetric, hence inducesa morphism of O Y -modules Φ n − : Sym n ( U ) → ∧ ( U ) . Proof. By the commutativity of m , Φ n − is symmetric in the first n − n − ( u ⊗ . . . ⊗ u n − ⊗ v ) = Φ n − ( u ⊗ . . . ⊗ u n − ⊗ v ⊗ u n − ) , for any u , . . . , u n − , v ∈ U . To this aim, let us consider the morphism m − ,n − : ∧ ( U ) → L defined in (18). By Proposition 5.5, m − ,n − identifies ∧ ( U ) with L ( − ∆ p ), hence (24) is equivalent to m − ,n − (Φ n − ( u ⊗ . . . ⊗ u n − ⊗ v )) == m − ,n − (Φ n − ( u ⊗ . . . ⊗ u n − ⊗ v ⊗ u n − )) . (25)From the explicit form of the isomorphism U ∼ = U ∨ ⊗∧ ( U ) recalledbefore, we have thatΦ n − ( u ⊗ . . . ⊗ u n − ⊗ v ) = τ ∗ ( m ( u ⊗ . . . ⊗ u n − )) ∧ v . Furthermore, by definition, m − ,n − ( τ ∗ ( m ( u ⊗ . . . ⊗ u n − )) ∧ v ) is theprojection onto L of m ( τ ∗ ( m ( u ⊗ . . . ⊗ u n − )) ⊗ τ ∗ ( v )) ∈ O Y ⊕ L .Since τ ∗ commutes with m , m ( τ ∗ ( m ( u ⊗ . . . ⊗ u n − )) ⊗ τ ∗ ( v )) = τ ∗ m ( m ( u ⊗ . . . ⊗ u n − )) ⊗ v ) . Using the associativity and the commutativity of m , we deduce that τ ∗ m ( m ( u ⊗ . . . ⊗ u n − )) ⊗ v ) = τ ∗ m ( m ( u ⊗ . . . ⊗ u n − ⊗ v )) ⊗ u n − ) . Projecting both sides of this equation onto L we obtain (25), fromwhich the statement follows. (cid:3) Remark 5.11. From Theorem 5.6 it follows that, for a D n -cover π : X → Y , the intermediate degree n cover f : W := X/ h τ i → Y embeds naturally in O Y ⊕ U ⊕ . . . ⊕ U n − if n is odd, respectively in O Y ⊕ U ⊕ . . . ⊕ U n − ⊕ M when n is even. Then the symmetric mor-phism Φ n − : Sym n ( U ) → ∧ ( U ) determines f : W → Y birationally.6. Simple and almost simple dihedral covers and theirinvariants In this section we define the simple dihedral covers and the almostsimple dihedral covers , and we investigate some of their properties.Throughout the section we assume that Y is a smooth variety.Recall that, from Proposition 5.2, a D n -field extension C ( Y ) ⊂ E isgiven by E = C ( Y )( x ), where x satisfies an irreducible equation of thefollowing form: x n − ax n + F n = 0 , with a, F ∈ C ( Y ). In order to construct varieties having E as fieldof rational functions, one could proceed in the following way. Con-sider a geometric line bundle L → Y , sections a ∈ H ( Y, L ⊗ n ), F ∈ IHEDRAL GALOIS COVERS 33 H ( Y, L ⊗ ), and define X ′ := { v ∈ L | v n − av n + F n = 0 } ⊂ L , where v is a fibre coordinate of L . The singular locus of X ′ containsthe locus v = F = 0, so it has codimension 1 and X ′ is not normal.We slightly modify this construction in the following way. Let L → Y be a geometric line bundle, a ∈ H ( Y, L ⊗ n ) and F ∈ H ( Y, L ⊗ ).Define X ⊂ L ⊕ L to be the set of ( u, v ) ∈ L ⊕ L such that the followingequations are satisfied: ( uv = Fu n − a + v n = 0 , (26)where u (resp. v ) is a fibre coordinate of the first (resp. second) copy of L in L ⊕ L . The dihedral group D n acts on X via σ ( u, v ) = ( ζ u, ζ − v ), τ ( u, v ) = ( v, u ), where ζ ∈ C ∗ is a primitive root of 1. Theorem 6.1. Let Y be a smooth variety. Let L → Y be a geometricline bundle, a ∈ H ( Y, L ⊗ n ) and F ∈ H ( Y, L ⊗ ) , such that: (i) the zero locus of a − F n ∈ H ( Y, L ⊗ n ) is smooth in the openset F = 0 ; (ii) the divisors { a = 0 } and { F = 0 } intersect each other trans-versely.Then the variety X defined by the equations (26) is smooth and therestriction to X of the fibre bundle projection L ⊕ L → Y is a D n -cover, π : X → Y , with branch divisor B π = { F n − a = 0 } . Furthermore, if { a = 0 } ∩ { F = 0 } 6 = ∅ , then X is irreducible. Definition 6.2. A simple D n -cover of Y is the D n -cover π : X → Y given as in Theorem 6.1 by the restriction to X of the fibre bundleprojection L ⊕ L → Y .Proof. (Of Theorem 6.1.) Let us define Φ := uv − F and Φ := u n − a + v n , so that the equations (26) become ( Φ = 0Φ = 0 . Taking partial derivatives along the fibre coordinates we have: ∂ Φ ∂u ∂ Φ ∂v∂ Φ ∂u ∂ Φ ∂v = v unu n − nv n − , hence the ramification divisor of π is R = { v n − u n = 0 } . Notice that v n − u n is h σ ∗ i -invariant, while τ ∗ ( v n − u n ) = − ( v n − u n ). Therefore the branch locus B π is defined by the following equation:( v n − u n )( u n − v n ) = − ( v n − u n ) = − ( v n + u n ) + 4( uv ) n = − a + 4 F n . Let us consider first the restriction of π over the points where F = 0.In this locus, we have v = F/u and so the equation Φ = 0 is equivalentto Φ = 0 , where Φ := u n − au n + F n . It follows that π is finite over the open set F = 0. To show the flatnessof π over F = 0, by Prop. 2.2 it suffices to prove that X is smooththere. To this aim consider ∂ Φ ∂u = 2 nu n − ( u n − a ) , which can vanish only if u n − a = 0, since F = 0 ⇒ u = 0. On theother hand, over the locus u n − a = 0, ∂ Φ ∂y = − u n ∂a∂y + ∂F n ∂y = ∂ ( F n − a ) ∂y , where y is any coordinate function on Y . Since F n − a = 0 is thebranch divisor and, by hypothesis (i), it is smooth in F = 0, it followsthat X is smooth there.We consider now the restriction of π over the locus F = 0 and noticethat here uv = 0. If u = 0, then as before we see that π is finite.The smoothness of X at ( u = 0 , v = 0) follows from the fact that( u = 0 , v = 0) R . The same argument applies if v = 0. It remainsthe case where u = v = 0, which implies that a = F = 0. By hypothesis(ii), a and F are part of a local coordinate system near the points where a = F = 0, in the analytic topology. Writing the equations for X inthese coordinates, we see that X is smooth at these points. Thereremains to show that π is finite also over the points a = F = 0. Here uv and u n + v n are precisely the D n -invariants, and O X is a free O Y -module generated by 1 , u n − v n , u, u , . . . , u n − , v, v , . . . , v n − .Let us now assume that { a = 0 } ∩ { F = 0 } 6 = ∅ . The irreducibilityof X is equivalent to the surjectivity of the monodromy of the cover, µ : π ( Y \ B π ) → D n . Since the branch divisor of the intermediatecover q : Z → Y coincides with B π , it is reduced, so Z is irreducibleand hence Im( µ ) contains a reflection σ i τ . To see that also σ ∈ Im( µ ),let us choose local coordinates ( a, F, y , . . . , y dim( Y ) ) for Y at a point in { a = 0 } ∩ { F = 0 } and consider the path in Xγ ( t ) = (cid:18) exp (cid:18) π √− n t (cid:19) u , exp (cid:18) − π √− n t (cid:19) v , F, a, y , . . . , y dim( Y ) (cid:19) , IHEDRAL GALOIS COVERS 35 where, ( u , v , F, a, y , . . . , y dim( Y ) ) ∈ X . Then, γ (1) = σγ (0), and π ◦ γ is a loop contained in Y \ B π , if and only ifexp(2 π √− t ) u n − exp( − π √− t ) v n = 0 , ∀ t . Since this condition can be easily achieved, e.g. by choosing u , v with | u | 6 = | v | , the claim follows. (cid:3) Notice that the hypothesis (i) in the previous theorem is general, asit follows from the next proposition. Proposition 6.3. Under the same notation as before, for a generalchoice of a ∈ H ( Y, L ⊗ n ) and F ∈ H ( Y, L ⊗ ) , F n − a = 0 is smoothover F = 0 .Proof. This follows directly from Bertini’s theorem (see e.g. [GH]). (cid:3) Invariants of simple dihedral covers. We first determine theeigensheaves decomposition of π ∗ O X , where π : X → Y is a simple D n -cover. Let L , F and a be as in the statement of Thm. 6.1, and letus denote with O Y ( L ) the sheaf of sections of L . Then, over an opensubset S ⊂ Y where L is trivial, from (26) we deduce that π ∗ O X | S ∼ = O S [ λ, µ ]( λµ − F, λ n − a + µ n ) ∼ = O S ⊕ O S ( λ n − µ n ) ⊕ n − i =1 (cid:0) O S λ i ⊕ O S µ n − i (cid:1) , where λ and µ are fibre coordinates on the dual L ∨ of L . This localdescription globalizes and we have:(27) π ∗ O X ∼ = ⊕ n − i =0 (cid:2) O Y ( − iL ) ⊕ O Y ( − ( n − i ) L ) (cid:3) . Notice that, for the sheaves L and U i introduced in Section 5.2, wehave: L = O Y ( − nL ) , U i = O Y ( − iL ) ⊕ O Y ( − ( n − i ) L ) . To determine the canonical bundle of X , we use the canonical bundleformula for branched covers: ω X = π ∗ ω Y ⊗ O X ( R ) , where R is the ramification divisor of π . If π : X → Y is a simple D n -cover, then R = { u n − v n = 0 } (see the proof of Thm. 6.1). Since u n − v n is a generator of the eigensheaf corresponding to the irreduciblerepresentation of D n with character χ , O X ( R ) = π ∗ L ∨ = π ∗ O Y ( nL ).Hence ω X = π ∗ ( ω Y ( nL )) . (28)In particular, if dim( X ) = 2, then the self-intersection of a canonicaldivisor of X is:(29) K X = 2 n ( K Y + nL ) . To compute the Euler characteristic χ ( O X ), by the finiteness of π we have that χ ( O X ) = χ ( π ∗ O X ). In particular, if dim( X ) = 2, theRiemann-Roch theorem for surfaces yields the following formula:(30) χ ( O X ) = 2 nχ ( O Y ) + 16 n (2 n + 1) L · L + 12 n L · K Y . Almost simple dihedral covers. In this section we define thealmost simple dihedral covers, which can be seen as projectivizationsof the simple dihedral covers. The construction follows closely the oneof almost simple cyclic covers, introduced and studied in [Cat89].The covering space X of an almost simple dihedral cover π : X → Y ,over the smooth variety Y , is defined as a complete intersection in thefibre product P × Y P → Y of two P -bundles over Y in the followingway. Let L → Y be a geometric line bundle and C = Y × C → Y bethe trivial geometric line bundle. For each i = 1 , 2, let P i := P ( C ⊕ L )be the P -bundle associated to C ⊕ L → Y , and let p i : P i → Y be thecorresponding projection. Here, each fibre of p i is the projective spaceof 1-dimensional subspaces in the corresponding fibre of C ⊕ L , hence,using the notation in [Hart77], P i = Proj ( O Y ⊕ O Y ( − L )) and in par-ticular ( p i ) ∗ O P i (1) = O Y ⊕O Y ( − L ), where L is a divisor in Y such that O Y ( L ) is the sheaf of sections of L (hence L = Spec(Sym( O Y ( − L )))).On each P i there are projective fibre coordinates: [ u : u ] for P ,[ v : v ] for P . u , u are defined as follows ( v , v are defined inthe same way): let U ⊂ p ∗ ( C ⊕ L ) be the universal sub-bundle, then u : U → p ∗ ( C ) is the composition of the inclusion with the projection;similarly for u : U → p ∗ L . Notice that, since U = Spec(Sym O P (1)),then u ∈ H ( P , O P (1)) and u ∈ H ( P , O P (1) ⊗ p ∗ O Y ( L )).Let F ∈ H ( Y, L ⊗ ) and let A , A ∞ be effective divisors in Y suchthat A ≡ nL + A ∞ . Let a ∞ ∈ H ( Y, O Y ( A ∞ )) and a ∈ H ( Y, O Y ( A )) be such that A ∞ = { a ∞ = 0 } and A = { a = 0 } .Consider the subvariety X ⊂ P × Y P = P ( C ⊕ L ) × Y P ( C ⊕ L )defined by the following equations: ( Φ = 0Φ = 0 , (31)whereΦ := u v − u v F , Φ := a ∞ v n u n − a v n u n + a ∞ v n u n ,u , u , v , v are defined above and ([ u : u ] , [ v : v ]) are projectivefibre coordinates on P ( C ⊕ L ) × Y P ( C ⊕ L ).Notice that the dihedral group D n acts on X in the following way: σ ([ u : u ] , [ v : v ]) = ([ u : ζ u ] , [ v : ζ − v ]) ,τ ([ u : u ] , [ v : v ]) = ([ v : v ] , [ u : u ]) , IHEDRAL GALOIS COVERS 37 where ζ ∈ C ∗ is a primitive n -th root of 1. Then we have the followingresult. Theorem 6.4. Let Y be a smooth variety, L → Y be a geometric linebundle, C = Y × C → Y be the trivial geometric line bundle, and F ∈ H ( Y, L ⊗ ) . Let A , A ∞ be effective divisors in Y such that A ≡ nL + A ∞ , where L is a divisor in Y with O Y ( L ) being the sheaf of sections of L . Let a ∞ ∈ H ( Y, O Y ( A ∞ )) and a ∈ H ( Y, O Y ( A )) be such that A ∞ = { a ∞ = 0 } and A = { a = 0 } .Assume that the following conditions are satisfied: (i) A intersects { F = 0 } transversely, A ∩ A ∞ = ∅ and A ∞ issmooth; (ii) the locus { a − F n a ∞ = 0 } is smooth on the open set F = 0 .Then X , defined by (31) , is smooth and the restriction of the projection P ( C ⊕ L ) × Y P ( C ⊕ L ) → Y to X is a D n -cover π : X → Y with branchdivisor B π = { a ∞ ( a − a ∞ F n ) = 0 } . Furthermore, if A ∩ { F = 0 } 6 = ∅ ,then X is irreducible. Definition 6.5. An almost simple D n -cover of Y is the D n -cover π : X → Y given as in Theorem 6.4 by the restriction to X of the fibrebundle projection P ( C ⊕ L ) × Y P ( C ⊕ L ) → Y .Proof. We first prove that the intersection of X with each one of thestandard open subsets v u = 0, v u = 0 and v u = 0 is a smoothvariety. This suffices because τ ( { v u = 0 } ) = { v u = 0 } .To this aim, observe that the restriction of π to the locus where u v = 0 is a simple D n -cover. Indeed, setting u = u /u and v = v /v ,the equations (31) reduce to ( uv − F = 0 a ∞ v n − a + a ∞ u n = 0 . Since A ∞ ∩ A = ∅ , on this locus a ∞ never vanishes, and setting a = a /a ∞ we obtain the equations (26). Under our hypotheses Theorem6.1 applies, hence X is smooth if u v = 0.Consider now the locus v u = 0 and observe that there F nevervanishes. Let us define u = u /u , v = v /v and g = 1 /F . Then theequations (31) become ( uv = ga ∞ u n − a g n + a ∞ v n = 0 . Since A ∞ ∩ A = ∅ , a ∞ never vanishes in this locus. So, defining a = a /a ∞ , we get the following equations for X : u n − ag n u n + g n = 0 , v = g/u . Notice that this is the equation of a D n -cover, with action σ ( u ) = ζ u , τ ( u ) = g/u . Since g = 0 everywhere, h σ i ∼ = Z /n Z acts freely and so X is smooth if and only if the intermediate double cover is smooth. Theintermediate double cover has equation z − ag n z + g n = 0 and itsbranch divisor is { g n ( g n a − 1) = 0 } = { a − F n = 0 } . By hypothesis(ii) this locus is smooth, so X is smooth where v u = 0.It remains to consider the case where v u = 0. Here, setting v =1 = u , the equations (31) reduce to ( u − v F = 0 a ∞ − a v n + a ∞ v n u n = 0 . Substituting u = v F in the second equation we obtain: a ∞ − a v n + a ∞ F n v n = 0 . Notice that, if v = 0, then we have already seen that X is smooth.On the other hand, when v = 0, we must have a ∞ = 0, and then thesmoothness of A ∞ implies that of X .The finiteness of π follows from the fact that π has finite fibre andits restriction to each open subset of Y where L is trivial is projective([Hart77], III, Ex. 11.1).To describe the branch divisor, recall that the restriction of π to theopen set u v = 0 is a simple dihedral cover and so its branch divisor is F n − a = 0, where a := a /a ∞ . Notice that, if u v = 0, then a ∞ = 0everywhere, on the other hand, if u v = 0, then u v = 0 by (31), so X ∩ { u v = 0 } ⊂ { u v = 0 } ∪ { u v = 0 } . The claim now followsfrom the previous explicit description of π on u v = 0.Finally, if A ∩ { F = 0 } 6 = ∅ , then X is irreducible since the opensubset X ∩ { u v = 0 } is irreducible by Theorem 6.1. (cid:3) The invariants of almost simple D n -covers π : X → Y can be com-puted in the same way as in the simple case, once a description of π ∗ O X in terms of L and A ∞ is provided. In the remaining part of this sectionwe show that there is an isomorphism as follows: π ∗ O X ∼ = O Y ⊕ O Y ( − nL − A ∞ ) ⊕ (32) (cid:0) ⊕ n − i =1 [ O Y ( − iL ) ⊕ O Y ( − ( n − i ) L )] (cid:1) ( − A ∞ ) . Notice that, on the open subset Y \ A ∞ where π is a simple cover, theprevious formula reduces to (27).To prove (32), we consider the following Cartesian diagram Q ˜ p −−−→ P p y y p P p −−−→ Y where Q := P × Y P is a P × P -bundle with projection p : Q → Y , p = p i ◦ ˜ p i , ∀ i = 1 , 2. Recall that the Picard group of Q is isomorphic IHEDRAL GALOIS COVERS 39 to Pic( Y ) × Z ⊕ via the usual isomorphism that sends ( L , m, n ) ∈ Pic( Y ) × Z ⊕ to p ∗ L ⊗ O Q ( m, n ) := p ∗ L ⊗ O P ( m ) ⊠ O P ( n ).Let us define D i := { Φ i = 0 } to be the divisor of Q given by theequation Φ i = 0 in (31), for i = 1 , 2, and notice thatΦ ∈ H ( Q , O Q (1 , ⊗ p ∗ O Y (2 L )) , Φ ∈ H ( Q , O Q ( n, n ) ⊗ p ∗ O Y ( nL + A ∞ )) . (33)Then we consider the usual short exact sequence(34) 0 → I X → O Q → O X → , where I X is the sheaf of ideals of X . Since X is the complete intersec-tion of two divisors in Q , the Koszul resolution of I X is as follows:(35) 0 → O Q ( − D − D ) → O Q ( − D ) ⊕ O Q ( − D ) → I X → . Applying p ∗ to (34) we obtain the following split short exact se-quence: 0 → O Y ∼ = p ∗ O Q → p ∗ O X → R p ∗ I X → , where we have used the fact that p ∗ I X = 0 = R p ∗ O Q .In order to compute R p ∗ I X , we apply p ∗ to (35) and we obtain thefollowing exact sequence:0 → R p ∗ I X → R p ∗ O Q ( − D − D ) → R p ∗ ( O Q ( − D ) ⊕ O Q ( − D )) , where we have used the equality R p ∗ ( O Q ( − D ) ⊕ O Q ( − D )) = 0 thatfollows from the K¨unneth formula. Furthermore, since R p ∗ O Q ( − D ) =0, we have that R p ∗ I X = ker (cid:2) R p ∗ O Q ( − D − D ) µ → R p ∗ O Q ( − D ) (cid:3) , where µ is induced by the morphism O Q ( − D − D ) → O Q ( − D )in the Koszul resolution of I X , given by ψ ∧ ψ ψ (Φ ) ψ , for ψ i ∈ O Q ( − D i ).To describe µ explicitly, we use the following isomorphisms: R p ∗ O Q ( − D − D ) ∼ = S n − ( O Y ⊕ O Y ( L )) ⊗ ⊗ O Y ( − nL − A ∞ ) ,R p ∗ O Q ( − D ) ∼ = S n − ( O Y ⊕ O Y ( L )) ⊗ ⊗ O Y ( − ( n − L − A ∞ ) , that follow applying the projection formula, K¨unneth formula, andthe standard isomorphisms (see e.g. [Hart77, Ex. 8.4, III]). Then,if we choose local sections x , y of O Y , and x , y of O Y ( L ), thatgenerate the corresponding sheaves, we obtain the following local basisfor S n − ( O Y ⊕ O Y ( L )) ⊗ : E ij := x i x n − − i ⊗ y j y n − − j , ≤ i, j ≤ n − . Similarly, G km := ( x k x n − − k ⊗ y m y n − − m ) ⊗ ( x ⊗ y ), for 0 ≤ k, m ≤ n − 2, is a local basis for S n − ( O Y ⊕ O Y ( L )) ⊗ ⊗ O Y (2 L ). The morphism µ in these basis is given as follows: µ ( E ij ) = G ij − f G i − ,j − , where F = f x ⊗ y , and G km := 0, for k, m 6∈ { , . . . , n − } .Hence from elementary linear algebra, we have that ker( µ ) (twisted by O Y ( nL + A ∞ )) is generated by E n − ,j and E i,n − , for 0 ≤ i, j ≤ n − µ ) = (cid:0) [ ⊕ n − j =0 O Y (( n − − j ) L )] ⊕ [ ⊕ n − i =0 O Y (( n − − i ) L )] (cid:1) ⊗ O Y ( − nL − A ∞ ) , and hence (32) follows.7. Deformations of simple dihedral covers Let X be a simple dihedral covering of Y : this means that X isthe subvariety of the vector bundle V = L ⊕ L which is (see 26) thecomplete intersection of two hypersurfaces, one in | p ∗ (2 L ) | , the otherin | p ∗ ( nL ) | ; here p : V → Y is the natural projection.Observe now that the cotangent sheaf of V is an extension of p ∗ Ω Y by p ∗ ( O Y ( − L ) ⊕ ),(36) 0 → p ∗ Ω Y → Ω V → p ∗ ( O Y ( − L ) ⊕ ) → , where p ∗ Ω Y → Ω V is the cotangent map of p .Then the conormal sheaf exact sequence of X reads out, if we denoteby L ′ := π ∗ ( L ), as0 → N ∗ X | V = O X ( − L ′ ) ⊕ O X ( − nL ′ ) → Ω V ⊗ O X → Ω X → , and the dual sequence is0 → Θ X → Θ V ⊗ O X → N X | V = O X (2 L ′ ) ⊕ O X ( nL ′ ) → , whose direct image under π ∗ yields the tangent exact sequence:0 → π ∗ Θ X → π ∗ (Θ V ⊗ O X ) → ( O Y (2 L ) ⊕ O Y ( nL )) ⊗ π ∗ ( O X ) → . Passing to the long exact cohomology sequence, we get the Kodaira-Spencer exact sequence(37) H (( O Y (2 L ) ⊕O Y ( nL )) ⊗ π ∗ ( O X )) → H (Θ X ) → H ( π ∗ (Θ V ⊗O X )) → . The meaning of the first linear map is given through the followingdefinition. Definition 7.1. The space of natural deformations of a simple di-hedral covering is the family of complete intersections of V : ( uv − F = 0 u n − a + v n + P n − ( b i u i + c i v i ) + d ( u n − v n ) = 0 , (38) where b i , c i ∈ H ( O Y ( n − i ) L ) , d ∈ H ( O Y ) . (39) Remark 7.2. The reader can see that in the second equation − a + v n + n − X ( b i u i + c i v i ) + d ( u n − v n ) IHEDRAL GALOIS COVERS 41 can be any section in H ( O Y ( nL ) ⊗ π ∗ ( O X )), in view of our basicformulae; instead, any section in H ( O Y (2 L ) ⊗ π ∗ ( O X )) is of the form − F + βu + αv + λu + µu , where α, β ∈ H ( O Y ( L )) , λ, µ ∈ C . But the new equation uv − F + βu + αv + λv + µu is the old form u ′ v ′ = F ′ (up to a multiplicative constant) if we choose new variables u ′ := u + α + λv, v ′ := v + β + µu. Of course, one can deform not only the equations, but also simul-taneously the base Y , the vector bundle V , and the equations; thishowever leads in general to a deformation with non smooth base.We have at any rate an easy result which says that all small defor-mations are obtained by natural deformations. Theorem 7.3. Assume that π : X → Y is a simple dihedral cover-ing. Then all small deformations of X are natural deformations of π : X → Y , provided H ( π ∗ (Θ V ⊗ O X )) = 0 (that happens, for exampleif H ((Θ Y ⊕ O Y ( L ) ⊕ ) ⊗ π ∗ ( O X )) = 0 ). In particular, the Kuranishifamily of X is smooth, and the Kuranishi space Def( X ) is locally ana-lytically isomorphic to Def ′ := coker (cid:0) H ( π ∗ (Θ V ⊗ O X )) → H (( O Y (2 L ) ⊕ O Y ( nL )) ⊗ π ∗ ( O X )) (cid:1) . Proof. By our assumption, and the Kodaira-Spencer exact sequence, wehave an isomorphism of vector spaces Def ′ ∼ = H (Θ X ). Moreover, bythe previous remark, the family of natural deformations has Kodaira-Spencer map which is surjective onto Def; therefore, by the implicitfunctions theorem, Def( X ) is the germ of the analytic space H (Θ X )at the origin, hence our claim. (cid:3) Examples and applications Simple dihedral covers of projective spaces. We first con-sider the case where Y = P , the complex projective plane, n = 3 and O Y ( L ) = O P (1). Then a ∈ H ( P , O P (3)) and F ∈ H ( P , O P (2))are a cubic and a quadric curve respectively, such that the sextic curve { F − a = 0 } is smooth in the locus F = 0, and { a = 0 } intersects { F = 0 } transversely. By Theorem 6.1, we have a smooth D -cover π : X → P branched over B = { F − a = 0 } . Notice that B ∈ |− K P | and ω P = O Y ( − L ). Hence ω X = π ∗ ( ω P (3 L )) ∼ = O X . Furthermore, q ( X ) = 0, hence X is a K3 surface.Let W = X/ h τ i and let f : W → P be the induced triple cover (seeSection 5.3). Then W can be realised as a cubic surface in P in such a way that f is the projection from a point in P \ W . Indeed, considerthe equations (26) which define X . If we define w := u + v , then w = u + v + 3 uv ( u + v )= 2 a + 3 F w . The branch divisor of f is B = { F − a } . So, under the hypotheses ofTheorem 6.1, B is a sextic with 6 cusps lying on a conic.The fundamental group of the complement P \ B of a sextic curve B as above has been studied in [Zar29], where in particular it is proventhat π ( P \ B ) is generated by two elements of order 2 and 3 respec-tively. From this it follows that there exists a surjective group homo-morphism π ( P \ B ) → D , and hence it follows from the generalisedRiemann existence theorem of Grauert and Remmert, that there ex-ists a D -cover π : X → P . From Theorem 6.1 we have an explicitconstruction of such a cover.Let now n > D n -cover π : X → P associ-ated to O Y ( L ) = O P (1), a ∈ H ( P , O P ( n )) and F ∈ H ( P , O P (2)).Under the hypotheses of Theorem 6.1, X is a smooth surface with ω X = π ∗ O P ( n − ,K X = 2 n ( n − ,χ ( O X ) = 13 n − n + 136 n , in particular X is a surface of general type, which is minimal since K X · C > C ⊂ X .Finally, let n = 2, Y = P and O Y ( L ) = O P (1). Under the hy-potheses of Theorem 6.1, X is a smooth surface with the followinginvariants: ω X = π ∗ O P ( − χ ( O X ) = 1 q = 0 K X = 4 p n = dim H ( X, ω ⊗ nX ) = 0 . Hence X is a rational, non-minimal surface. Indeed, X is isomorphicto a del Pezzo surface of degree 4 in P , the complete intersection ofthe quadrics uv = F ( x : x : x ), u + v = 2 a ( x : x : x ), where( x : x : x : u : v ) are now homogeneous coordinates in P .In a similar way one can construct examples of dihedral covers of P with O Y ( L ) = O P ( d ), d > D n -cover π : X → P associated to O Y ( L ) = O ( m ), m ≥ 1, are natural deformations, if ( m, n ) = (1 , m ≥ IHEDRAL GALOIS COVERS 43 n ≥ 2. This follows directly from Thm. 7.3, formula (27) and thecomputation of the cohomology of Ω q P d ( k ) ([Bott57, p. 256]). Remark 8.1. Notice that, in general, a simple D n -cover of P d , π : X → P d , associated to O Y ( L ) = O ( m ), F ∈ H ( P d , O (2 m )) and a ∈ H ( P d , O ( nm )),is isomorphic to a complete intersection X ′ in the weighted projectivespace P d +2 (1 , . . . , , m, m ), where X ′ := { ( x : . . . : x d : u : v ) ∈ P d +2 (1 , . . . , , m, m ) | uv = F, u n + v n = 2 a } . To see this, observe that X ⊂ L ⊕ L is the quotient of ˜ X ⊂ ( C d +1 \{ } ) × C via the linear diagonal action of C ∗ with weights (1 , . . . , , m, m ),where ˜ X := { ( x , . . . , x d , u, v ) | uv = F, u n + v n = 2 a } . The claim now follows since ( C d +1 \ { } ) × C ⊂ C d +3 \ { } , and theaction of C ∗ on ( C d +1 \ { } ) × C is the restriction of the linear diagonalaction on C d +3 \ { } with weights (1 , . . . , , m, m ).If d ≥ 3, then Theorem 7.3 implies that all the small deformations of X are natural deformations of the simple D n -cover π : X → P d . Indeed,in this case, we have that H (cid:0) (Θ P d ⊕ O P d ( m ) ⊕ ) ⊗ π ∗ O X (cid:1) = 0 , as it follows from the fact that H ( P d , O P d ( k )) = 0, ∀ k , and that H ( P d , Θ P d ( k )) ∼ = H d − ( P d , Ω P d ( − k − d − ∨ = 0, ∀ k ([Bott57]).8.2. An application to fundamental groups. According to thegeneralized Riemann existence theorem of Grauert and Remmert [GR58],coverings π : X → Y of a normal variety Y , of degree n , with branchlocus contained in a divisor B ⊂ Y , and with X normal, correspondto conjugacy classes of group homomorphisms µ : π ( Y \ B ) → S n . Inthis situation, X is irreducible, if and only if Im( µ ) is a transitive sub-group of S n ; π : X → Y is Galois with group G := Im( µ ), if and onlyif G coincides with the group of automorphisms of π . In particular, if π : X → Y is a G -cover with G non-abelian and X irreducible, then π ( Y \ B ) is necessarily non-abelian.Fundamental groups of complements of divisors in projective va-rieties have been extensively studied by many authors. As a directconsequence of Theorem 6.1, we have the following result. Proposition 8.2. Let Y be a smooth variety and L ⊂ Y be a divisor.Assume that there exist a ∈ H ( Y, O Y ( nL )) and F ∈ H ( Y, O Y (2 L )) ,such that the conditions (i), (ii) of Thm. 6.1 are satisfied and { a =0 } ∩ { F = 0 } 6 = ∅ . Then π ( Y \ B ) admits an epimorphism onto D n ,in particular it is non-abelian, where B = { a − F n = 0 } . Notice that similar results have been obtained using different meth-ods ([CKO03, Lemma 3], [ArCo10]). Briefly, one considers the pen-cil { λ a = 0) + µn ( F = 0) } ( λ : µ ) ∈ P ⊂ | nL | and the induced mor-phism Y \ B → P \ { (1 : 1) } . This gives a group homomorphism π ( Y \ B ) → π orb ( P \ { (1 : 1) } ), where π orb is the orbifold fundamentalgroup of P \ { (1 : 1) } with two orbifold points, (1 : 0) of order 2 and(0 : 1) of order n . Now, using the long exact homotopy sequence, oneconcludes that π ( Y \ B ) → π orb ( P \ { (1 : 1) } ) is surjective and so π ( Y \ B ) is not abelian. References [ArCo10] Artal-Bartolo, E.; Cogolludo-Agust´ın, J.; On the connection betweenfundamental groups and pencils with multiple fibers. J. Singul. 2(2010), 1–18.[BaCa08] Bauer, I.; Catanese, F. A volume maximizing canonical surface in3-space. Comment. Math. Helv. 83 (2008), no. 2, 387–406.[Bott57] Bott, R. Homogeneous vector bundles. Ann. of Math. (2) 66 (1957),203–248.[Bur66] Burniat, P., Sur les surfaces de genre P > . Ann. Mat. Pura Appl.(4) 71, 1–24 (1966).[Cat84] Catanese, F. On the moduli spaces of surfaces of general type . J.Differential Geom. 19 (1984), no. 2, 483–515.[Cat88] Catanese, F. Moduli of algebraic surfaces. Theory of moduli (Mon-tecatini Terme, 1985), 1–83, Lecture Notes in Math., 1337, Springer,Berlin, (1988).[Cat89] Catanese, F., Everywhere non reduced moduli spaces. Invent. math.98, 293–310 (1989).[Cat98] Catanese, F., Singular bidouble covers and the construction of inter-esting algebraic surfaces. Algebraic geometry: Hirzebruch 70 (War-saw, 1998), 97–120, Contemp. Math., 241, Amer. Math. Soc., Provi-dence, RI, 1999.[CKO03] Catanese, F.; Keum, J.; Oguiso, K.; Some remarks on the universalcover of an open K3 surface. Math. Ann. 325 (2003), no. 2, 279–286.[Cat10] Catanese, F., Irreducibility of the space of cyclic covers of alge-braic curves of fixed numerical type and the irreducible componentsof Sing ( M g ) . Advances in geometric analysis, 281–306, Adv. Lect.Math. (ALM), 21, Int. Press, Somerville, MA, 2012.[CHKS06] Catanese, F.; Ho¸sten, S.; Khetan, A.; Sturmfels, B., The maximumlikelihood degree . Amer. J. Math. 128 (2006), no. 3, 671–697.[CLP] Catanese, F., L¨onne, M., Perroni, F. (In preparation.)[Com30] Comessatti, A. Sulle superficie multiple cicliche. Rendiconti Semi-nario Padova 1, 1–45 (1930)[East11] Easton, R. W., S -covers of schemes. Canad. J. Math. 63 (2011), no.5, 1058–1082.[Enr49] Enriques, F., Le Superficie Algebriche. Nicola Zanichelli, Bologna(1949).[GH] Griffiths, P.; Harris, J.; Principles of algebraic geometry. Wiley Clas-sics Library. John Wiley & Sons, Inc., New York, 1994.[GR58] Grauert, H.; Remmert, R.; Komplexe R¨aume. Math. Ann. 136(1958), 245–318.[Hart77] Hartshorne, R., Algebraic geometry. Graduate Texts in Mathematics,No. 52. Springer-Verlag, New York-Heidelberg, 1977.[Mat89] Matsumura, H. Commutative ring theory. Second edition. CambridgeStudies in Advanced Mathematics, 8. Cambridge University Press,Cambridge, 1989. IHEDRAL GALOIS COVERS 45 [Mir85] Miranda, R. Triple covers in algebraic geometry. Amer. J. Math. 107(1985), no. 5, 1123–1158.[Mum84] Mumford, D., Tata lectures on theta. II. Reprint of the 1984 original.Birkh¨auser Boston, Inc., Boston, MA, 2007.[Namba87] Namba, M. Branched coverings and algebraic functions. Pitman Re-search Notes in Mathematics Series, 161. Longman Scientific & Tech-nical, Harlow; John Wiley & Sons, Inc., New York, 1987.[Par91] Pardini, R. Abelian covers of algebraic varieties. J. Reine Angew.Math. 417 (1991), 191–213.[ReidC3f] Reid, M., Canonical 3-folds. Journ´ees de G´eometrie Alg´ebriqued’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, pp. 273–310, Sijthoff & Noordhoff, Alphen aan den Rijn–Germantown, Md.,1980.[Sai80] Saito, K., Theory of logarithmic differential forms and logarithmicvector fields . J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), no.2, 265–291.[Serre77] Serre, J.-P., Linear representations of finite groups. Graduate Textsin Mathematics, Vol. 42. Springer-Verlag, New York-Heidelberg,1977.[Tok94] Tokunaga, H.-O, On dihedral Galois coverings. Canad. J. Math. 46(1994), no. 6, 1299–1317.[Zar29] Zariski, O., On the Problem of Existence of Algebraic Functions ofTwo Variables Possessing a Given Branch Curve. Amer. J. Math. 51(1929), no. 2, 305–328.[Zar58] Zariski, O., On the purity of the branch locus of algebraic functions .Proc. Nat. Acad. Sci. U.S.A. 44 1958 791–796.