TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 350, Number 4, April 1998, Pages 1359–1378 S 0002-9947(98)02136-9

COVERS OF ALGEBRAIC VARIETIES III. THE DISCRIMINANT OF A COVER OF DEGREE 4 AND THE TRIGONAL CONSTRUCTION

G. CASNATI

Abstract. For each Gorenstein cover %: X Y of degree 4 we define a scheme ∆(X) and a generically finite map ∆(%):→ ∆(X) Y of degree 3 called the discriminant of %. Using this construction we deal→ with smooth degree 4covers%:X n with n 5. Moreover we also generalize the trigonal PC construction of→ S. Recillas. ≥

1. Introduction and notations In [C–E] a general structure theorem for Gorenstein covers of degree d has been proved. Such a result has been used for characterizing covers of degree 3, 4 and 5 (see [C–E] and [Cs] respectively) of algebraic varieties. More precisely, in section 4 (resp. 3) of [C–E] it is proved that the total space X of each Gorenstein cover %: X Y of degree 4 (resp. 3) is obtained inside a suitable projective bundle → π : P Y of rank 2 (resp. 1) as the base locus of a pencil of relative conics (resp. → the zero locus of a relative cubic form in two variables), and % = π X . 2 | t For each pencil of conics b in Pk spanned by two quadratic forms a := x Ax and b := xtBx (A and B being symmetric 3 3 matrices) the discriminant ∆(b) is the cubic (possibly zero) polynomial ∆(b)=det(× sA + tB). The roots of ∆(b) correspond to the degenerate conics of b. In this paper we investigate the geometry of the Galois–theoretic relationship between the general equations of degree 4 and 3. We are thus able to generalize 1 in arbitrary dimension the trigonal construction due to S. Recillas for covers of Pk (see [Re]). We do this globalizing the description above (exploiting an idea due to T. Ekedahl: see [Ek]). In section 4 we define for each Gorenstein cover %: X Y of degree 4 its discriminant ∆(%): ∆(X) Y, getting the following result. → → Theorem 1.1. Let Y be an integral scheme defined over an algebraically closed field k of characteristic p =2. To each Gorenstein cover %: X Y of degree 4 one can associate its discriminant6 ∆(%): ∆(X) Y, which is generically→ finite of degree 3. The points y Y over which ∆(%) does→ not have finite fibres are exactly ∈

Received by the editors December 1, 1995. 1991 Mathematics Subject Classification. Primary 14E20, 14E22. Key words and phrases. Cover, Gorenstein, discriminant. This work was done in the framework of the AGE project, H.C.M. contract ERBCHRXCT 940557.

c 1998 American Mathematical Society

1359

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the points such that k(y)[u, v] % 1(y) = spec , − ∼ (u2,v2)   and in this case ∆(%) 1(y) 1 . If there are not such y’s, then ∆(%) is a − ∼= Pk(y) Gorenstein cover of degree 3. Moreover we also study the reducedness, connectedness and smoothness of the discriminant cover, giving many examples exhibiting various possible behaviours. In section 5 we determine the class of Sing(∆(X)) under the hypotheses of gener- icity of %: X Y . Section 6 is→ devoted to the generalization of the above mentioned trigonal con- struction to covers of arbitrary smooth schemes Y satisfying some mild technical conditions (see Theorem 6.3). In section 7 we deal with covers of n. PC Theorem 1.2. If X is integral and smooth, n 5,and%:X n is a cover of PC ≥ i n → i degree 4 such that % Y / n is uniform, then %∗ : H , H X, is an PC i PC C C isomorphism for i ∗0O nO 1 and a monomorphism (but not an→ isomorphism) for i = n. ≤ ≤ −

Taking into account Fujita’s description of covers %: X Y of degree 3 with both X and Y smooth (see [Fj]), we are also able to give a qualitative→ study of the locus of points of total ramification of a cover of degree 4. Theorem 1.3. If %: X n is a cover of degree 4 with X smooth and n 5,then PC Xis a quadrisecant of an→ ample line bundle if and only if for each y n≥there is PC an embedding X := % 1(y) , 1 . More precisely, either X is a quadrisecant∈ of y − AC an ample line bundle, or there→ is a point y n such that the fibre of % over y is PC isomorphic to ∈

1 C[u, v] % (y) = spec . − ∼ (u2,v2)   In this case y has at least multiplicity 4 in the branch locus of %. Now we recall some definitions, notations and results which will be used in what follows. A local ring R is Cohen–Macaulay if dim(R) = depth(R). A Cohen–Macaulay ring R is called Gorenstein if its injective dimension idR(R) is finite. An arbitrary ring R is called Cohen–Macaulay (resp. Gorenstein) if RM is Cohen–Macaulay (resp. Gorenstein) for every maximal ideal M R. In this paper all schemes are assumed to be⊆ noetherian. All schemes over a field k of characteristic p are assumed to be separated and of finite type over k,andwe always take p =2. AschemeX6 is Cohen–Macaulay (resp. Gorenstein) if for each point x X the ∈ local ring X,x of X in x is Cohen–Macaulay (resp. Gorenstein). If X is defined over k, thenO it is Gorenstein if and only if it is Cohen–Macaulay and its dualizing sheaf ωX k is invertible. Let Y | be a scheme. A flat morphism %: X Y is said to be Gorenstein of → 1 relative dimension r if for each y Y the scheme–theoretic fibre Xy := %− (y)is a Gorenstein scheme over k(y)ofdimension∈ r. In particular, the relative dualizing sheaf is defined and invertible.

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Let Y be an scheme. A morphism %: X Y is called a cover of degree d if % X → ∗O is a locally free Y –sheaf of rank d: % is a cover if and only if it is flat and finite. If Y is smooth andO X is Cohen–Macaulay, then every finite surjective morphism is acover. ˇ There exists an exact sequence of the form 0 Y % X 0, where ˇ →O → ∗O →E→ is a locally free Y –sheaf of rank d 1 called the Tschirnhausen module of %. EIf the characteristicOp of k does not divide− d, then the above sequence splits and ˇ % X = Y (generalize the proof of lemma 2.2 in [Mi]). If the cover %: X Y ∗O ∼ O ⊕ E → is Gorenstein, then % X ˇ = % ωX Y (see [Ha], exercise III 6.10), and hence ∗O ∼ ∗ | % ωX Y = Y . ∗ | ∼ O ⊕E
If is a locally free Y –sheaf of rank d +1,wedenotebyπ:P( ) Y the E O E → corresponding projective bundle, i.e. P( ):=Proj( )( denotes the symmet- ric algebra of , n its component ofE degree n), andSEπ isSE induced by the natural E S E monomorphism Y  (see [Ha]). 0 O SE If η H Y, is a section of a locally free Y –sheaf ,wedenotebyD0(η) its zero–locus.∈ F O F I would like to express
my thanks to F. Catanese for a number of helpful discus- sions and interesting suggestions.

2. Covers of degree 3 and 4 For the following results we refer to sections 3 and 4 of [C–E]. Let X and Y be schemes, Y integral, and let %: X Y be a Gorenstein cover of degree d =3 → or d = 4. There exists a natural factorization % = π i,whereπ:P Y is ◦ → the natural projection, P := P( ), ˇ being the Tschirnhausen module of % and i: X, is a closed embedding.E Moreover,E ω (1) ,andforeachy Y P X Y ∼= P X → 1 | O | 1 ∈ the fibre Xy := %− (y) is a complete intersection in Py := π− (y). If d =3then X 1 is the zero locus of a single cubic polynomial, whereas if d =4then y Py ∼=Pk(y) ⊆ 2 Xy Py =P is the base locus of a pencil of conics without fixed components. ⊆ ∼ k(y) If d = 3 there is a unique exact sequence

δ 0 π∗ det ( 3) 0. → E − −→OP →OX → If d = 4 there is a unique locally free Y –sheaf of rank 2 such that det ∼= det and an exact sequence O F F E

δ (2.1) 0 π∗ det ( 4) π∗ ( 2) 0. → E − → F − −→OP →OX → In particular, in both the above cases, X = D0(δ). Using the natural isomorphisms 0 3 1 0 1 (2.2) Φ3 : H Y, det − ∼ H P, (π∗ det − )(3) , S E⊗ E −→ E 0 ˇ 2 0 ˇ Φ4: H Y,
∼ H P,(π∗ )(2) ,
F⊗S E −→ F 1 1 one gets sections η := Φ3− (δ)orη:= Φ
4− (δ). Note that dim(
D0(δy)) = 0 for each point y Y . In this case we say briefly that η has the right codimension at y Y . ∈ ∈ Theorem 2.3. Let Y be an integral scheme. Any Gorenstein cover %: X Y of degree 3 such that ˇ = coker %# determines, up to scalars, → E ∼ 0 3 1 η H Y, det − ∈ S E⊗ E having the right codimension at every y Y .
∈

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0 3 1 Conversely, given η H Y, det − having the right codimension at ∈ S E⊗ E every y Y , let X := D0 Φ3(η) P. Then the restriction % of the canonical map ∈ ⊆
π : P Y to X is a Gorenstein cover of degree 3 such that ˇ = coker %#. →
E ∼ Proof. See [C–E], theorem 3.4. Theorem 2.4. Let Y be an integral scheme. As explained above, any Gorenstein ˇ # cover %: X Y of degree 4 such that ∼= coker % determines a locally free Y – → E 0 ˇ 2 O sheaf of rank 2 with det ∼= det and, up to scalars, η H Y, having the rightF codimension at everyF y EY . ∈ F⊗S E ∈
Conversely, given a locally free Y –sheaf of rank 2 with det ∼= det and η H0 Y, ˇ 2 having theO right codimensionF at every y FY , let EX := ∈ F⊗S E ∈ D Φ (η) . Then the restriction % of the canonical map π : Y to X is a 0 4 P
P ⊆ ˇ # 2→ 2 Gorenstein cover of degree 4 such that = coker % and = ker %ωXY .
E ∼ F ∼ S E→ ∗ | Proof. See [C–E], theorem 4.4.
It is helpful to know when sections having the right codimension at each point exist. Theorem 2.5. Let Y be a projective, smooth, connected scheme over a field k, E and be locally free Y –sheaves of ranks 3 and 2 respectively, det ∼= det ,and defineF := 2 ˇO.Thesets E F H S E⊗F H := η H0 Y, η has the right codimension at every y Y , rc { ∈ H | ∈ } Hs := η Hrc D0(Φ4(η)) P is smooth {
∈ | ⊆ } are open (but possibly empty). If k = C, dim(Y ) 3 and is globally generated, ≤ H 0 then H = . D0(Φ4(η)) is connected for any η H if and only if h Y, ˇ =0. s 6 ∅ ∈ rc E Proof. See [C–E], theorem 4.5. See also [C–E], theorem 3.6, for the degree
3 case.

3. The branch locus of a cover of degree 4 Let %: X Y be a Gorenstein cover of degree 4 with invariants , and defined → 2 E F π by a section η H0 Y, ˇ . From now on we will denote by P := P( ) Y , π ∈ F⊗S E E −→ P := P( ) Y the canonical projections.
AccordingF −→ to the results of section 2, for each y Y we can define the Segre ∈ number of Xy , denoted by S(Xy), as the Segre number of the pencil of conics of 2 cutting out the scheme X (see [Wa]). We have the Table 3.1 (see [Wa], Py ∼= Pk(y) y table 0). Definition 3.2. Let %: X Y be a Gorenstein cover of degree 4 and y Y . → ∈ We say that % is planar (resp. even) over y if S(Xy)=[1,1;;1](resp. S(Xy)= [(1, 1), 1], [(2, 1)]) and we define the two sets Rplanar(%):= x X %is planar at %(x) (resp. R (%):= x X %is even at %(x) . { ∈ | } even { ∈ | } ˇ Let := π∗ .Since ∼= det and det ∼= det , there is an isomorphism 0G ˇ E 2 F0 F⊗2 F 1 F E Ψ: H Y, H P, det − (1) . In the following we will identify F⊗S E → S G⊗ G ϑ := Ψ(η) with the corresponding symmetric map ϑ: ˇ( 1) det 1.Its

− adjoint adj(ϑ): ˇ(2) is symmetric too. By applyingG π− we→G⊗ get G G→G ∗ 2 2 2 η Hom Y , = Hom , (2) . ∈ O S E S F ∼ OP S G OP

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2 S (Xy ) Xy P ⊆ k(y) 2 2 2 2 i) [1, 1, 1] V+(u + w ,v +w ) 2 2 ii) [2, 1] V+(uv, v + w ) 2 2 2 iii) [(1, 1), 1] V+(u + v ,w ) 2 iv) [3] V+(v +2uw, vw) 2 2 v) [(2, 1)] V+(w +2uv, v ) 2 2 vi) [1, 1;;1] V+(u ,v )

Lemma 3.3. η has rank 1 (resp. 2)atyif and only if % is planar (resp. even) over y. Proof. We treat only case i) of Table 3.1, the other ones being similar. In this 3 3 case = k(y)[s, t]⊕ and ϑy corresponds to the morphism k(y)[s, t]( 1)⊕ H|Py ∼ − → k(y)[s, t] 3 given, up to a suitable linear transformation, by the matrix ⊕
e s 00 0t0. 00s+t

 2 3 2 2 Therefore it is easy to check that η : k(y)⊕ k(y)⊕ is given by the matrix y S →S 000100 100101, 100000 which has rank 3.  

Suppose that Y = spec(A), where A is local with maximal ideal M (y the ∼ 3 2 corresponding point of Y ), ∼= A⊕ with basis u, v, w , ∼= A⊕ , and let %: X Y be a Gorenstein cover ofE degree 4. Theorem{ 2.4 implies} F that X is given inside→ P( ) = P2 by two quadratic forms a, b A[u, v, w]. On the other hand, X is affine; E ∼ A ∈ thus we can assume that Xy V+(w)= . This means that a and b have no common factors modulo (w)+M; hence∩ we can assume∅ that a = u2 + other terms in u, v, w and b = v2 + other terms in u, v, w. Finally, with proper linear transformations one easily gets that X = spec(R), where A[u, v] (3.4) R := (u2 +2αv + β,v2 +2γu + δ) for suitable α, β, γ, δ A. If we set L (y):=∈zy for z R and Q: R R A is the bilinear form defined z ∈ × → by Q(x, y):=Tr(xy):=Tr(Lxy), then an equation of B% is b =det(Q)(see[A–K], proposition 6.6). By (3.4), 1,u,v,uv is a basis of R over A;then { } 400 12αγ 0 4β 12αγ 8αδ b= − 012αγ 4δ 8βγ (3.5) − 12αγ 8αδ 8βγ 48α2γ2 +4βδ

4 4 2 2 3 2 2 3 2 2 = 64(27α γ 18α βγ δ 4β γ 4 α δ β δ ). − − − − −

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If X := spec(R)whereRis defined by 3.4, then ϑ is given by s 0 γt 0 tαs. γt αs βs + δt It follows that η is induced by  α2 00 β α0 −βαγ0 δ−01,  δ 0 γ γ2 00 −− which has rank less than 3 at y if and only if αγ, αδ,βγ, βδ are all zero at y. More precisely, up to a permutation of α, β, γ and δ the only distinct cases are α, β, γ, δ M;thenS(Xy)=[1,1;;1], α M, β,γ,δ M,andthenS(Xy)= [(2, 1)], and∈ α, β M and γ,δ M.ThenS6∈(X) = [(1, 1)∈, 1]. 6∈ ∈ y With the above considerations in mind let us describe the singularities of B%, using (3.5) and Lemma 3.3 above. Definition 3.6. Let Y be smooth and integral and let B Y be a divisor through apointy Y. ⊆ ∈ Let M A := Y,y be the maximal ideal. We say that y is a cuspidal double point (resp.⊆ a transverseO double point) if the local equation of the tangent cone 2 2 2 at B around y is ` (resp. `1`2)where` M M (resp. `1,`2 M M are transversal). ∈ \ ∈ \ A cuspidal double point is called ordinary if (`, b) M3 M4, non–ordinary otherwise. ⊆ \ Assume that both X and Y are smooth over a point y Y . ∈ If S(Xy)=[2,1], % is either ´etale or analytically a double cover. Therefore B% must be smooth. If S(Xy) = [3], % is either ´etale or analytically a totally ramified triple cover. Therefore (see [Mi], section 4) B% has at least a double point at y.Ifyis exactly a double point then it is cuspidal (and, in general, ordinary). If S(Xy) = [(1, 1), 1] then we can assume α, β M and γ,δ M in (3.4). If γ,δ M M2 are not proportional the tangent cone6∈ at y is 4β3γ∈2 + β2δ2,andso yis∈ a transverse\ double point. If either γ and δ are proportional or at least one of them belongs to M2, then it is easy to check by direct computation that y is a non–ordinary cuspidal double point. If S(Xy) = [(2, 1)], then we can assume α M and β,γ,δ M.Inthiscasey is at least a triple point. 6∈ ∈ Finally, if S(Xy)=[1,1;;1], then α, β, γ, δ M.Inthiscaseyis at least a fourfold point. ∈

4. The Discriminant of a cover of degree 4

Let , be locally free Y –sheaves of ranks 3 and 2 respectively, and such that det =E detF . As in sectionO 3, we consider the isomorphism Ψ, and we identify E ∼ F ˇ ϑ 1 ϑ := Ψ(η) with a symmetric map ( 1) det − . Taking the determinant of ϑ and applying the projection formula,G − −→G⊗ we finally obtainG a map of sets 0 2 0 3 1 ∆: H Y, ˇ H Y, det − F⊗S E → S F⊗ F (recall that ˇ det = ).

F⊗ F ∼ F

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Let Φ3 be as in (2.2). Definition 4.1. Let %: X Y be a Gorenstein cover of degree 4 corresponding to 0 2 →ˇ a section η H Y, . The scheme ∆(X):=D0 Φ3(∆(η)) P (resp. the ∈ S E⊗F ⊆ map ∆(%):=π∆(X)) is called the discriminant scheme of X (resp. the discriminant map of %). |

If X = spec(R), where R is defined by 3.4, then

A[s, t] 1 (4.2) ∆(X) = proj PA. ∼ α2t3 βt2s δts2 + γ2s3 ⊆  − −  1 1 Note that either ∆(Xy) P is a subscheme of length 3, or ∆(Xy) = P . ⊆ k(y) ∼ k(y) This second case is characterized by the vanishing of α, β, γ, δ at the point y.In particular, ∆(X ) 1 if and only if y ∼= Pk(y) k(y)[u, v] X = spec . y ∼ (u2,v2)   Remark 4.3. We identify ∆(η) with a map 3 ˇ det ˇ,and S F→ F 1 Rplanar(%)=%− D0(∆(η))

by the above description. Thus Rplanar(%) is closed, and
either codimX Rplanar(%) 3 1 4orR (%)= ,sincerk det − =4. ≤ planar ∅ S F⊗ F
Proposition 4.4. Let % : X Y be a Gorenstein
cover of degree 4 with Y integral and invariants and . Then→ the following assertions are equivalent. i) ∆(%): ∆(XE) YFis a cover (of degree 3). ii) ∆(%): ∆(X)→ Y is a Gorenstein cover. → iii) Rplanar(%)= . iv) For each y Y∅ the Zariski tangent space at x X has dimension 1. ∈ ∈ y ≤ Moreover, if ∆(%) is a cover its Tschirnhausen module is ˇ,andifY is smooth F at y the branch loci of ∆(%) and %, B∆(%) and B%, coincide as divisors around y. Proof. i) and ii) are trivially equivalent since ∆(X) P( ) is a divisor. ∆(%)isnot ⊆ F a cover if and only if there is y Y such that ∆(Xy) is not finite; hence i) and iii) are equivalent by the above description.∈ The Zariski tangent space T at x X x ∈ y is the intersection of the Zariski tagent spaces of all the conics of Py through Xy. Hence Tx has dimension 2 if and only if all the conics through Xy are singular at x, i.e. if and only if % is planar over y. The equality B∆(%) = B% must be checked locally; hence we assume Y = spec(A) and X ∼= spec(R), where R is defined by (3.4). Thus B% = div(b), where b is defined by (3.5). On the other hand, if ∆(%) is a cover we can also suppose that α does not vanish on Y .Inthiscasebcoincides, up to the invertible factor α2,withthe discriminant of the polynomial p(t):=α2t3 βt2 δt + γ2 representing ∆(X)in spec(A[t]). − − From now on we will assume both X and Y are smooth. With these hypotheses we give some results about the reducibility, connectedness and smoothness of ∆(X). First of all, note that there is a natural structure of regular conic bundle having ∆(X) as degeneration divisor (see [Sa], definitions 1.1 and 1.4), p: BlXP P, → defined in the following way. If x P X,lety:= π(x) and let Cx be the unique ∈ \ conic inside Py containing both x and Xy.ThenCxcorresponds to a point of Py.

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Proposition 4.5. Let % : X Y be a cover of degree 4 with X smooth and Y smooth and integral. Then: → i) ∆(X) is reduced; ii) if ∆(X) is reducible there is an invertible –sheaf and an epimorphism OY M  ; F iii)M∆(X) is singular at x over y Y if and only if y R (%) R (%). x ∈ ∈ even ∪ planar corresponds to the conics of rank 1 cutting out Xy Py. x is a Gorenstein double point with tangent cone of rank at least 2 (briefly, a⊆ pseudo–node). Proof. The statements i) and iii) follow from the above description, [Sa], proposition 1.8, corollary 1.9, and the proof of proposition 1.2 iii) in [Bea]. Assume that ∆(X)=Z Z0.BothZand Z0 are Gorenstein (they are divisors ∪ inside P), and since the degree of ∆(%) is 3 we can suppose that σ := ∆(%) Z : Z Y | → is generically finite of degree 1; hence Z P is a unisecant divisor. This implies the existence of an epimorphism ⊆where is invertible. F  M M The following example shows that in Proposition 4.5 we cannot hope to prove the irreducibility of ∆(X).

Example 4.6. Let Y be an integral smooth scheme, projective over C, Pic(Y ) 2 2 2 0 L∈ 1 globally generated, := , := .Letu, v H Y, − = F L ⊕L E L⊕L⊕L ∈ E⊗L ∼ H0 , (1) π 1 , w H0 Y, 2 H0 , (1) π 2 (resp. s, t P P ∗ − − ∼= P P ∗ − 0 O ⊗2 L 0 ∈ E⊗L2 O ⊗ L
∈ H Y, − =H P, (1) π∗ − ) be independent sections. For a general F⊗L ∼
OP ⊗ L

choice of β,δ H0 Y, 2 , γ H0 Y, the subscheme X P( ) defined by ∈
L ∈ L
⊆ E η := (u2 +
βw2,v2 +2 γuw
+ δw2) H0 Y, ˇ 2 ∈ F⊗S E 0 1 is smooth and connected, since h Y, − = 0. On the other hand
∆(X) P( ) has the equation L ⊆ F
p(s, t):=t(βs2 + δst γ2t2), − and hence ∆(X) is reducible.

Corollary 4.7. Let %, X and Y be as above. If B% is reduced and ∆(X) is integral, then it is also normal. In particular, if dim(Y )=2,then∆(X) has at most isolated normal double points as singularities.

Proof. Since Sing(∆(X)) lies exactly over Sing(B%)(seesection2)ifB% is reduced, then Sing(∆(X)) has codimension at least 2. On the other hand, ∆(X)isCohen– Macaulay. Hence the statement follows from Proposition 4.5 and [Sa], Proposition 1. From now on we will assume that ∆(%) is a cover, i.e., that R (%)= . planar ∅ q Definition 4.8. Let Y be an integral scheme and Q Y a conic bundle. We define −→ 1 Sing(q):= x Q qis not smooth at x = x Q x Sing q− (q(x)) . { ∈ | } { ∈ | ∈ } Sing(q) is closed in Q,andq(Sing(q)) is exactly the discriminant curve of
the conic bundle Q.NotethatSing(q)=Sing(Q). 6 Now assume that ∆(X) is not connected. Then ∆(X)=Z Z0 is a reduced ∪ scheme with Z Z0 = . We can suppose that σ := ∆(%) Z : Z Y is quasi–finite of degree 1, so∩ that it∅ is a cover of degree 1 (see [Ha], exercises| → III 10.9 and III

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1 11.2), i.e., an isomorphism. In particular, σ− is a section of π . It follows that q there exists a conic bundle P Q Y containing X such that Qy := Q Py is a ⊇ −→ n ∩ conic of rank 2 for each y Y , and Sing(q) X = (recall that if b Pk is a pencil of conics, then C b corresponds∈ to a simple∩ root∅ of the discriminant⊆ if and only ∈ if Sing(C) Bs(b)= ). In particular, q Sing(q) :Sing(q) Yis an isomorphism (it is quasi–finite∩ of degree∅ 1); thus there is| an invertible →–sheaf such that the OY L section Y ∼ Sing(q) P corresponds to the exact sequence 0 0 0. −→ ⊆ →E →E→L→ Let f : P Sing(q) P( 0) be the induced dominant map (which is, fibrewise, the \ → E projection from the point Sing(Qy) onto the line P( 0)y) and define X0 := f(X) E (which is, fibrewise, Qy P( 0)y Py). Let α := f X : X X0. The projection ∩ E ⊆ | → P( 0) Y induces a double cover β : X0 Y such that % = β α. The ramification E → → ◦ points of β correspond to conics Qy of rank 1; hence β is necessarily ´etale. Then we assume that ( ) % does not factor as β α with β a non–trivial ´etale double cover of Y . ♥ ◦ Condition ( ) is obviously satisfied if Y is simply connected. ♥ If ( ) is satisfied one gets that X0 is trivial, and this allows us to find a de- ♥ composition Q = P1 P2 into two projective subbundles Pi P of rank 1. But, if ∪ ⊆ this is the case, X = X1 X2, with Xi := X Pi = , since, generically, the fibre X := % 1(y) consists of∪ four points in general∩ position6 ∅ in π 1(y) 2. y − − ∼= Pk Proposition 4.9. Let % : X Y be a cover of degree 4 with X and Y smooth and integral and R (%)= .If→Ysatisfies ( ) then ∆(X) is connected. planar ∅ ♥ Corollary 4.10. If %, X and Y are as above, then h0 Y, ˇ =0. F Proof. The number of connected components of ∆(X)is
h0 ∆(X), = h0 Y, + h0 Y, ˇ . O∆(X) OY F

Corollary 4.11. If %, X and Y are as above and B% is reduced (resp. has at most ordinary cuspidal double points as singularities), then ∆(X) is integral (resp. smooth). Proof. We already know that ∆(X) is reduced. If ∆(X) P( ) were reducible then we would have dim Sing(∆(X)) dim(X) 1(since∆(⊆ FX) is connected). On the other hand, ≥ −
1 Sing(∆(X)) ∆(%)− y Y S(X ) = [(1, 1), 1], [(2, 1)] ⊆ { ∈ | y } 1 1 ∆(%)− (Sing(B )) = ∆(%)− (Sing(B )). ⊆ ∆(%) % Hence dim Sing(∆(X)) dim(Sing(B )) dim(X) 2. ≤ % ≤ − If B% has at most ordinary cuspidal double points as singularities, then the smoothness of ∆(X) follows
from section 3. Both Proposition 4.9 and its Corollaries 4.10 and 4.11 are sharp, as the following easy examples show. Example 4.12. Consider Example 4.6. As we saw, ∆(X) is in any case reducible. On the other hand, using the expression of b given by (3.5), one finds that the branch locus B% has global equation b =4β3γ2+β2δ2 =(4βγ2 + δ2)β2, which is not reduced.

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Example 4.13. Consider an elliptic curve Y over C, and let β : X0 Y be a → double ´etale cover with X0 smooth and connected. Then there is an invertible 2 1 sheaf such that = Y and β X = Y − . Now let be an invertible L L ∼ O ∗O 0 ∼ O ⊕L M and globally generated Y –sheaf and let α: X X0 be any smooth and connected double cover with α O= β 1. The→ morphism % := β α: X Y is a X ∼ X0 ∗ − Gorenstein cover of degree∗O 4O whose⊕ TschirnahausenM module is dual◦ to :=→ E L⊕M⊕ . Moreover, ωX k = ωX X ωX Y = %∗ . It is not difficult to check L⊗M | ∼ | 0 ⊗ 0| ∼ L⊗M that = 2 2 = 2. Hence h0 Y, ˇ = h0 Y, + h0 Y, 2 =1, ∼ ∼ Y
Y − so thatF ∆(LX)⊕M is not connected.O ⊕M Note that, sinceF X is smooth,O ∆(%): ∆(MX) Y is actually a cover of degree 3.

→

5. Singularities of the discriminant In section 4 we gave a first rough description of Sing(∆(X)). In this section we want to study the nature of the singularities of the discriminant of a cover %: X Y of degree 4 under the hypothesis of genericity. We will assume that X is smooth→ and Y is smooth and integral. To that end we inspect the construction of the discriminant as the degeneration divisor of the conic bundle p: BlXP P used in the previous section. For each → x P X let y := π(x), and let Cx be the unique conic contained in Py and ∈ \ containing both x and Xy.ThenCx Py. We have a commutative diagram ∈ p E, BlXP P → −→ (5.1) ϕ π X,  π Y  P  y → y −→ y where ϕ is the natural projection. Since the fibres of p are conics, it follows that p is a Gorenstein morphism (of relative dimension 1). Hence, so is π p,and the relative dualizing sheaf ω is invertible. Moreover, by semicontinuity,◦ BlXP Y 1 | 0 := p ω− is a locally free –sheaf of rank 2 and the natural epimorphism BlX Y P H ∗ P| O p∗ 0  ωBlX Y induces an embedding BlXP , P := P( 0) factoring p through H P| → H the natural projection p: P P. In particular, p: BlXP P is an embedded conic in the sense of [Sa], 1.5. → e → Since p is induced by thee relative conics through X, it follows that p∗ (1) = e OP ∼ ϕ∗ (2) ( E). In particular, OP ⊗OBlX P −

ωBl Y = ϕ∗ω Y BlX (E) = p∗ ( 1) ϕ∗ ( 1) ϕ∗π∗ det . XP| ∼ P| ⊗O P ∼ OP − ⊗ OP − ⊗ E Then = (1) p ϕ (1) π det 1 .Sinceπ,πand p are flat, then 0 ∼ P ∗ P ∗ − p ϕ =Hπ π Oas functors⊗ ∗ (seeO [Ha],⊗ propositionE III 9.3); hence = := (1) ∗ ∗ 0 P ∗ ∼ ∗ 1
H ∼ H O ⊗ π∗ det − . BlXP has an –resolution of the form E⊗ E OP 1
0 ( 2) ep∗ − Bl 0, →OP − ⊗ M →OP →O X P → 2 where = ( 1) π∗edet (see [Sa], 1.5e and 1.6). Note that = M ∼ OP − ⊗ F e S H⊗M∼ π 2 det 1 (1); in particular, π 2 = ˇ 2 ,andthere ∗ − P ∼ is aS naturalE⊗ isomorphismE ⊗O ∗ S H⊗M F⊗S E

0 2 0 2 sym Φ: H Y, ˇ ∼ H P, =Hom ( ˇ, ). F⊗S E −→ S H⊗M ∼ OP H H⊗M

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The symmetric map ϑ =Φ(η): ˇ yields an exact sequence H→H⊗M (5.2) 0 ˇ ϑ 0, → H −→H⊗M→P→ where is supported on ∆(X). Notice that ∆(X)=D2(ϑ) (by construction) and P Sing(∆(X)) = D1(ϑ) (Proposition 4.5 iii)). With this in mind we can use the results of [Ba] to describe the singularities of ∆(X). Proposition 5.3. Let , be locally free sheaves of ranks 3 and 2 respectively over an integral and smoothE F scheme Y of dimension 2. If 2 ˇis globally generated, then the discriminant ∆(X) of each general coverS%:E⊗X FY of degree 4 with invariants and is smooth in codimension 1 and has at most→ nodes as singularities. E F 2 Proof. The natural map π∗ (1) induces an epimorphism π∗ ˇ F  OP S E⊗F  2 .If 2 ˇis globally generated, then the same is true for 2 .
SThusH⊗M the propositionS E⊗F follows from proposition 1 of [Ba]. S H⊗M Proposition 5.4. Let % : X Y be a cover of degree 4 with X smooth, Y smooth, integral and projective, and invariants→ , .If∆(X) is smooth in codimension 1, then Sing(∆(X)) represents the class ofE F

(5.4.1) 4π∗ (c2( ) c2( )) ξ 4π∗c3( ), E − F · − E ξ Pic(P) being the tautological class. In particular, if dim(Y )=2and ∆(X) has at∈ most nodes as singularities, then

(5.4.2) #Sing(∆(X)) = 4 (c2( ) c2( )) . E − F Proof. To prove the statement we consider formula 9 of [Ba]. In this case r =3, 1 ci =ci( ), λ = c1( ), where ∼= π( 1) π∗ det and = π∗ − .A direct substitutionH inM Barth’s formulaM O then− proves⊗ thatE Sing(∆(HX)) representsE⊗M 3 2 4 ξ π∗c1( ) ξ + π∗c2( ) ξ π∗c3( ) . − E · E · − E 3 2 Since ξ π∗c1( ) ξ + π∗c2( ) ξ =0andc1( )=c1( ), we
get formula (5.4.1). Formula− (5.4.2)F follows· by projectingF · via π andE the factF that the singularities of ∆(X) are nodes. ∗ Example 5.5. Let X be an Enriques surface not containing nodal curves. There 2 3 always exists a cover %: X of degree 4 whose invariants are = 2 (2)⊕ Pk ∼ Pk → 3 E O and the kernel of an epimorphism 2 (4)⊕ 2 (6) (see section 6 of [C–E] Pk  Pk F O O 2 and [Ve]). In particular, c2( )=12=c2( ). Since in this case ˇis globally generated, for each such generalE % the discriminantF ∆ is smooth.S E⊗F Indeed, A. Verra proved in [Ve] that the branch locus of % is a reduced curve of degree 12 having ordinary cuspidal points as singularities; thus Reven(%)= by Corollary 4.11. ∅ 6. The trigonal construction In this section we want to investigate the connection between the discriminant of a cover of degree 4 and a generalization of the well known trigonal construction due to S. Recillas (see [Re]). We maintain the notations of sections 4 and 5. The first step is to study the sheaf fitting into the sequence (5.2). P

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Definition 6.1. Let ∆ be a scheme defined over a field of characteristic p =2, 6 L an invertible ∆–sheaf and a coherent reflexive ∆–sheaf. We say that is an –quadratic sheafO if there existsP a symmetric isomorphismO P L σ : ∼ ˇ = om , . P −→ P⊗L∼H O∆ P L Lemma 6.2. If ∆(X) is normal, then the ∆(X)–sheaf
fitting in the sequence 2 O P 5.2 above is an ω∆(X) Y –quadratic sheaf. is not invertible| exactly on Sing(∆(X)). P Proof. Taking the tensor product of sequence 5.2 with itself, we obtain (6.2.1) ˇ ˇ δ1 0, H⊗H⊗M ⊕ H⊗M⊗H −→ H⊗M ⊗ H⊗M →P⊗P→ where δ1(α a, b
β)= ϑ(α) a+
b ϑ(β) and
which is exact
(use the spectral ⊗ ⊗ ⊗ ⊗ sequence of a double complex). On the other hand, let S2 := 1,i .Themap { } T0:= 1 + i acts on sending a b to a b + b a. In particular, H⊗M ⊗ H⊗M ⊗ ⊗ ⊗ T0 extends to an endomorphism T of (6.2.1) whose image is a direct summand, which is isomorphic to

(6.2.2) ˇ d1 2 2 0 H⊗H⊗M−→S H⊗M →S P→ via the canonical projection. It follows that 6.2.2
is exact. The adjoint of ϑ maps to ˇ(∆(X)); hence it induces a morphism φ1 : 2 H⊗M H (∆(X)). Denoting by φ2 : ˇ the usual contraction, we thenS H⊗M get a commutative→M diagram with exact rowsH⊗H⊗M
ˇ d1 2 2 0 H⊗H⊗M −→S H⊗M →SP→ φ2 φ1
 (∆(X)) (∆(X)) ∆(X) 0, M→M →M | → y 2 y 2 giving rise to a map φ0 : (∆(X)) ∆(X) = ω∆( ) , hence to a morphism S P→M | ∼ X Y ˇ 2 | σ : ω∆(X) Y . Since Sing(∆(X)) = D1(ϑ), we see that φ1 is surjective P→P⊗ | outside Sing(∆(X)) and the same is true for φ0. It follows that σ is an isomorphism outside Sing(∆(X)), since is invertible there (see [Ba], Lemma 5). Taking the dual of sequenceP (5.2) twisted by ,onegets M (6.2.3) 0 ˇ xt1 , 0. →H→H⊗M→E OP P M → The short exact sequence 0 (∆(X))
(∆(X)) ∆(X) 0 yields →M→M →M | → 1 ∼ ˇ 2 an isomorphism ∂ : xt , ω∆(X) Y . In particular, locally on Y , σ E OP P M −→ P⊗ lifts to a chain map between sequences (5.2) and| (6.2.3), and any two such maps
are homotopic. For degree reasons any homotopy must be zero, so we obtain a commutative diagram 0 ˇ ϑ 0 → H −→H⊗M→ P → (6.2.4) s2 s1 σ ˇ ˇ ϑ  ˇ 2 0   ω∆(X) Y 0 . → Hy −→H⊗M→y P⊗ y | → Sequences (5.2) and (6.2.3) are minimal, so s1 and s2 are isomorphisms outside Sing(∆(X)). The normality of ∆(X) then implies that they are both isomorphisms everywhere, so the same must be true for σ.

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is reflexive. Indeed, is torsion–free, so the natural map µ: ˇˇ i s P P t 1 P→P injective. On the other hand, and ˇˇare isomorphic via (σ )− σ.Thus and ˇˇ have the same Hilbert function,P P and it follows that coker(µ)=0.◦ P P Since is reflexive and σ is induced by φ0 it follows that σ is symmetric, i.e. 2 P P is an ω∆(X) Y –quadratic sheaf. Finally, | is invertible outside Sing(∆(X)). On the other hand, if x Sing(∆(X))P then y := π(x) is even or planar (Proposition 4.5 iii)). In the first∈ 2 2 2 2 case X = C D,whereC=V+(u +v )andD=V+(w ); thus t s is an equation y ∩ for ∆(X)y Py and ϑy is defined by the matrix ⊆ t 00 0t0. 00s Note that D ∆(X) corresponds to s = 1. We conclude that ∈ y dim k(D) =2, k(D) P⊗ and so cannot be invertible at D.Thecases
S(Xy) = [(2, 1)], [1, 1;;1]canbe treatedP analogously. Definition 6.3. Let Y be integral, defined over a field of characteristic p =2,and 6 let and be locally free Y –sheaves of rank 3 and 2 respectively. E 4 F O % Let Y, , denote the set of Gorenstein covers X Y of degree 4 with invariants T E F −→ , and such that X is smooth and codim R (%) 2, R (%)= . E F X even ≥ planar ∅ Let 3 denote the set of pairs ( , ∆ τ Y ) such that τ is a Gorenstein Y, ,
cover ofR degreeE F 3 with invariant , ∆P is normal−→ with at most pseudo–nodes as singularities, and is an ω2 –quadraticF sheaf which is invertible exactly outside P ∆ Y Sing(∆) and such that τ =| . ∗P ∼ E % 4 Up to now we have shown how to associate to each X Y Y, , an element −→ ∈T E F % 3 Trig(X Y) Y, , . −→ ∈R E F Definition 6.4. The map 4 3 Trig: Y, , Y, , . T E F →R E F defined above is called the “trigonal construction”. Theorem 6.5. The map Trig is bijective. Proof. We only need to show how to recover a Gorenstein cover of degree 4 from a fixed pair ( , ∆ τ Y ). First of all,P assume−→ that Y = spec(k) is a point, i.e. 1 .ThenH0 Y, P ∼= Pk ∼= 3 P k⊕ , is 0–regular since dim(∆) = 0 and it is supported on ∆. In particular the P 0 0 0
natural maps H Y, H Y, 1 (n) H Y, (n) are surjective for each Pk P ⊗ O → 3 P n 0. Thus there exists an epimorphism ϕ: ⊕1  whose kernel is free, say

Pk
≥ 3 O P ker(ϕ) = 1 ( αi), as follows from the Auslander–Buchsbaum formula and ∼ i=1 Pk the decompositionO theorem− of Grothendieck. It follows that an exact sequence of the form L 3 3 ϕ (6.5.1) 0 1 ( α ) 0 P i ⊕1 → O k − →OPk −→P→ i=1 M

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is defined. Since the map H0 ϕ) is an isomorphism, taking the cohomologies of (6.5.1) one obtains αi =1fori=1,2,3. Now consider an arbitrary scheme Y and let and be as in section 5. One obtains the exact sequence M H (6.5.2) 0 θ 0, →K−→H⊗M→P→ whose restriction to coincides with (6.5.1). Since h1 , =0and Py Py Py 0 1 K| h Py, =3,thusRπ (1) = 0 and := π (1) is locally free of rank K|Py ∗K N ∗K
3. Moreover, the natural map π (1) is an isomorphism since this is true
∗ fibrewise. N→K 2 2 The symmetric map σ induces ω∆Y = (∆) ∆, hence a diagram S P→ | ∼M | 0 Λ2 d2 d1 2 2 0 → K −→ K⊗H⊗M −→S H⊗M →SP→
φ0 0 (∆) (∆)∆ 0. →M→M →M| → y Locally on the base the map φ0 canbeliftedtoachainmap,andanytwosuchmaps are homotopic. On the other hand, since ∼= π∗ ( 1), for degree reasons each homotopy must be zero, so we obtain a wellK definedN chain− map and a commutative diagram

0 Λ2 d2 d1 2 2 0 → K −→ K⊗H⊗M −→S H⊗M →SP→ φ3 φ2 φ1
φ0 0  (∆) (∆)∆ 0.  →M→M  →M| → y y y y The map φ2 induces a morphism s1 : ˇ, defined locally by s1(α),β := H→K h i φ2(α β). Let s2 be the transpose of s1: we claim that s1 θ = θˇ s2. Indeed, we have⊗ locally ◦ ◦

0=φ3(α β)=φ2 d2(α β)=φ2(α θ(β) β θ(α)). ∧ ◦ ∧ ⊗ − ⊗ Thus

s1 θ(α),β =φ2(β θ(α)) = φ2(α θ(β)) = s2(α),θ(β) = θˇ s2(α),β . h ◦ i ⊗ ⊗ h i h ◦ i It follows that there exists a chain map 0 θ 0 →K−→H⊗M→ P → s2 s1 σ ˇ ˇ θ ˇ ˇ 2 0   ω∆Y 0 → Hy −→K⊗My → P⊗y| → extending the isomorphism σ. By construction each row of the above diagram is minimal; thus the maps s1 and s2 are isomorphisms and we get the exact sequence 0 ˇ ϑ 0, → H −→H⊗M→P→ 1 0 2 where ϑ := θ s− . Notice that ϑ = ϑˇ. Hence we obtain η H Y, ˇ via ◦ 2 ∈ F⊗S E the isomorphism Φ 1 defined in section 5. −
We claim that η has the right codimension at each point y Y . Indeed, η defines ∈ a subscheme X := D0(η) P which is fibrewise over y Y the base locus of a ⊆ ∈ 1 pencil of conics by whose discriminant is ∆y.Sinceτis a cover we have ∆y = P . 6 k(y)

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Hence by has no fixed components, and % := π X : X Y is a Gorenstein cover of degree 4 without planar points. | → It remains to prove the smoothness of X.Since is not invertible exactly on Sing(∆), it follows that %(Sing(X)) %(R (%))P = τ(Sing(∆)). Let y ⊆ even ∈ %(Reven(%)) be such that S(Xy)iseither[(2,1)] or [(1, 1), 1]. Then Xy = C D 2 2 ∩ where C = V+(u )andD=V+(v +2u) (so Supp(Xy)=(0,0)) in the first case or 2 2 C = V+(u )andD=V+(v 1) (so Supp(X )= (0, 1) ) in the second. Locally − y { ± } on Y the subscheme X P is given by spec(R), where R is as in (3.4), and either α(y)=β(y)=δ(y)=0and⊆ γ(y) = 1 in the first case, or α(y)=β(y)=γ(y)=0 and δ(y)= 1 in the second (see (2.2) and the description above). Then ∆ is given by (4.2)− locally on Y . Moreover, C Sing(∆) corresponds to s =0.Let ∈ y1,...,y be a system of local regular parameters around y, and assume that X { n} is singular over y.IfS(Xy) = [(2, 1)], one can easily check that the jacobian matrix of X at u = v =0is 00 ∂β ∂yi , 20 ∂δ ∂yi !i=1,...,n ∂β whence ∂y =0.IfS(Xy) = [(1, 1), 1], the jacobian matrix at u =0,v= 1is i y ±  | 00 2∂α + ∂β ±∂yi ∂yi , 0 2 ∂δ ± ∂yi !i=1,...,n ∂α ∂β whence ∂y = 2i ∂y = 0. In both the cases the tangent cone at C ∆ i y ± i y ∈ would have rank| at most 1.| 2 Remark 6.6. It is not difficoult to verify that giving a ω∆ Y –quadratic sheaf on | P τ 0 ∆ is equivalent to giving a finite morphism ∆0 ∆ of degree 2 branched exactly −→ 1 at Sing(∆). Indeed, we define the ∆–algebra := ∆ ω− with the O A O ⊕ P⊗ ∆Y multiplication induced by σ and consider the canonical projection ∆ |:= Spec( ) 0
onto ∆. A We give the following classical example. 1 1 Example 6.7. Let X := Pk k Pk. The linear system X (1, 2) induces an em- 5 × O bedding i: X, Pk. Projecting i(X) onto a non–intersecting plane α, we finally → 2 3 get a cover %: X P of degree 4. The invariants of % are = 2 (1)⊕ and → k E ∼ OPk = 2 (1) 2(2). It follows from Proposition 5.3 and formula (5.4.2) that ∼ Pk Pk Ffor a generalO ⊕O projection the discriminant ∆ has exactly 4 nodes as singularities. If 3 ϕ 3 3 3 P = BlrP P is the blow-up of P along a line r,thenϕ(∆) P is a cubic ∼ k −→ k k ⊆ k and ϕ ∆ is an isomorphism. We conclude that ϕ(∆) is a Cayley cubic. On the | τ 0 other hand, we also have a double cover ∆0 ∆ branched along Sing(∆). One 4 −→ 6 has τ τ 0 ∆ = 2 2( 1)⊕ 2( 2). ∆0 is a Del Pezzo sextic in ,and 0 ∼ Pk Pk Pk Pk ∗ ∗O O ⊕O − ⊕O 6 − ττ0 is the projection from a space β Pk of dimension 3. It is a classical result that all the⊆ Cayley cubic surfaces arise in this way.

7. Covers of n. PC

We defined Rplanar(%) in section 4 for any cover %: X Y of degree 4 with both X and Y smooth and integral. In this section we want to→ restrict ourselves to the

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case Y = n, and we will deal with the behaviour of % with respect to R (%) ∼ PC planar when n 5. In this case there always exist points y Y of total ramification (see [G–L]). ≥ ∈ Proposition 7.1. Let X be integral and smooth, n 5, and let %: X n be a PC cover of degree 4.Then ≥ → i) there is k 2 such that ωX n = %∗ n (k); PC ∼ PC ≥ n | O ii) for each line r P there are k2 k1

(7.1.1) 0 π∗ det ( 4) π∗ ( 2) 0, → 0 E − → 0 F − →OP0 → where π := π . Twisting (7.1.1) by (2) and taking its direct image via π ,we 0 P0 P0 0 get | O 2 n (7.1.2) 0 0 P (k) 0. →F→S E →O C → 2 In particular, 3c1( 0)=c1( 0)=c1( )+k = c1( )+k = c1( 0)+2k, i.e. E S E F E E c1( 0)=kand c1( )=c1( )=2k. E E n F 1 For each line r P the scheme %− (r) is connected (theorem 7.1 of [Jo]). ⊆ C Since we wish to deal with r and r we can assume that n = 1. One has E| F| 1 (k ) 1(k ), k k 1(Xis connected) and k + k = k;then 0 = P 1 P 2 1 2 1 2 E ∼ O C ⊕O C ≥ ≥ k>k 1 (this completes the proof of i)). Let 1 (h ) 1(h ), h h . 1 = P 1 P 2 1 2 ≥ F ∼ O C ⊕O C ≥ Sequence (7.1.2) becomes a (7.1.3) 0 1(h ) 1(h ) 1(2k ) 1(2k ) 1(k) 1(k) 0. P 1 P 2 P 1 P 2 P P →O C ⊕O C →O C ⊕O C ⊕O C −→O C → The matrix of a is

a2k k,a2k k,a0 , 1− 2− 0 1 where a H , 1 (i) .Ifa = 0, then (7.1.3)
splits, and ii) is proved. i P P 0 ∈ C O C 6 If a = 0 there is a monomorphism 1 (k) 1 (h ) 1(h ). In particular 0 P  P 1 P 2
O C O C ⊕O C h1 k = k1 + k2; hence h1 2k2.Ifh1=2k2 then h2 =2k h1 =2k 2k2 =2k1. ≥ ≥ 0 1 0 − − Assume h1 > 2k2, and let w H , ( k) = H , (1) π 1 ( k) , PC ∼ P P ∗ P 0 1 0 ∈ E − O 0 ⊗1 O C − v H P , ( k1) = H P, (1) π∗ 1 ( k1) and u H P , ( k2) = C ∼ P PC
C ∼
0∈ E − O ⊗ O − ∈ E − H , (1) π 1 ( k ) be independent sections. Then = V (w) ,and P P ∗ P 2 P0 + P O ⊗ O
C −
⊆
X0:= P0 X P is defined by a system of equations of the form ∩ ⊆
2 2 w = α2k h u + αk +k h uv + α2k h v 2− 1 1 2− 1 1− 1 2 2 = α2k h u + αk +k h uv + α2k h v =0 2− 2 1 2− 2 1− 2

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0 1 where α H , 1 (i) .Sinceh >2k,thenα =0.If2k =h ,there i PC P 1 2 2k2 h1 2 2 1 ∈ O C − 6 is y P such that α2k2 h2 (y)=0and(y,[1, 0, 0]) X0 = , which is absurd. We ∈ C −
∈ 6 ∅ conclude that h1 =2k1,h2 =2k2. iii) is an easy computation from sequence (7.1.2).

Corollary 7.2. Let %, X , n be as above. If either or 0 is uniform, then F E n n n n n n = P (k1) P (k2) P (k1 +k2), = P (2k1) P (2k2) and ωX P = E ∼ O C ⊕O C ⊕O C F ∼O C ⊕O C | C ∼ n %∗ P (k1 + k2). O C Moreover, the natural map % : Hi n, Hi X, is an isomorphism for i∗ PC C C 0 i n 1 and a monomorphism (but not an→ isomorphism) for i = n. ≤ ≤ −

Proof. If n 5 and either or 0 is uniform, then the other one is also uniform and hence they≥ both split (seeF [O–S–S],E theorem I 3.2.3). n Now fix k1 and k2.Covers%:X P with X smooth and invariants and → C E correspond to sections η of a suitable open set H0 n,ˇ 2 (see PC TheoremsF 2.4 and 2.5). We claim that = . Indeed,U⊆ let α: Y F⊗Sn and βE: X PC 0 U6 ∅ 0→ n
→ Y be two covers of degree 2 defined by general sections in H , n (k ) and PC P 2 0 O C n H Y,α∗ P (k1) respectively, so that one can assume both Y and X0 smooth. C
It is easyO to check that % := α β : X n corresponds to a section η . 0 0 PC 0 Obviously is connected,
and there◦ is a smooth→ family ∈U U χ := D0(Φ4(η)),η P X ⊆ ×C U −→U η [∈U
1 whose special fibres are X and X0. In particular bi(χ− (η)) is constant on ; hence n n U it suffices to prove that bi(P )=bi(X0), 0 i n 1, and bn(P )

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Proof. Proposition 4.4 and Remark 4.3 imply that ∆(%): ∆(X) Y is a Gorenstein cover of degree 3 and the map ∆(η): 3 ˇ det ˇ is surjective.→ It follows that we can imitate word for word the proof ofS [Fj],F→ theoremF 2.2 and corollary 2.3, taking into account that c1( )=c1( ). In particular, either ∆(%)is´etale (and the same E F 2 is true for % by Proposition 4.4), or ∼= for some ample Pic(Y )(see [Fj], proof of theorem 2.1; see also [La]).F L⊕L L∈ We now prove a more precise form of Theorem 1.3 which was stated in the introduction.

Theorem 7.6. Let % : X Pn be a cover of degree 4,withXsmooth and n 5. → C ≥ If Rplanar(%) = ,thencodimX Rplanar(%) 4 and the branch locus B% of % has at least a fourfold6 ∅ point. ≤ R (%)= if and only if there is h>
0such that X is a quadrisecant planar ∅ n n n of the line bundle associated to P (h). In particular, = P (2h) P (4h), O C F ∼ O C ⊕O C n n n n n = P (h) P (2h) P (3h),andωXP =%∗ P (3h). E ∼ O C ⊕O C ⊕O C | C ∼ O C Proof. The first assertion follows from Remark 4.3 and section 3. Obviously, if X is a quadrisecant of a line bundle, then R (%)= .Con- planar ∅ n versely, let Rplanar(%)= . Then the structure of the sheaves , and ωX P ∅ E F | C follows from Corollary 7.2 if we prove that = 2 for some Pic( n). ∼ PC Assume that ∆(X) is irreduciblel then we canF applyL⊕L Lemma 7.5. L∈ It remains only to study the case when ∆(X) is reducible. In this case by Propo- sition 4.5 ii) there is an exact sequence 0 n(a) n(b) 0. Since PC PC 1 n →O →F→O → H P , n (a b) =0,then = n(a) n(b) with, for example, a b.Thus, C PC ∼ PC PC O − F O ⊕O 3 2 2 ≤ 3 ∆(X) is represented in P by the single equation α2a bs +αas t+αbst +α2b at = 0
n 0 − 0 n − 0, where s H P , ( a) = H P, π∗ n ( a) , t H P , ( b) = C ∼ P PC C ∼ 0 ∈ F − O ⊗ O − ∈0 n F − H , π n ( b) are independent sections and α H , n (i) .Since P P ∗ P i PC P O ⊗ O C −
n
∈ O C
n 5, if b =2athere is y P such that αi(y) = 0, which contradicts the ≥ 6
∈ C
assumption Rplanar(%)= . Let u H0 n, ( h)∅, v H0 n, ( 2h) be independent sections. Up to a PC PC suitable linear∈ transformationE − ∈ on the variablesE − u and v, the above discussion implies

that X P has equations ⊆ 2 α(u, v):=α0u +α3hv+α4h, 2 β(u, v):=β0u+β000v + βhv + β2h, 0 n n where αi,βi H P , P (i) . Obviously β000 =0.Ifβ0 =0then ∈ C O C 6 n = y P α3h(y)=α
4h(y)=βh(y)=β2h(y)=0 %(Rplanar(%)); ∅6 { ∈ C| }⊆ hence we can assume β0 =0.Usingβ, we can write u explicitly as a quadratic polynomial in the variable6 v. Substituting in α,wegetXas a quadrisecant of the

n line bundle associated to P (h). O C Remark 7.7. In [G–L] the loci R (%):= x X the local degree at x is e (x) ` { ∈ | % ≥ ` +1 are defined. In particular, Rplanar(%) R3(%). In theorem 1 of [G–L] it is proved} that codim (R (%)) `. In our setting,⊆ R (%) = if ` 3. X ` ≤ ` 6 ∅ ≤ Example 7.8. The bound on n is sharp. Let us consider a globally generated n 2 3 invertible sheaf on P with n 4, and define = , = ⊕ .Since L C ≤ F ∼ L⊕L E ∼ L := ˇ 2 is globally generated, for any general η H0 Y, one has a smooth H F⊗S E ∈ H

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subscheme X := D0(Φ4(η)) P. Moreover, any such X has global equations of the form ⊆ αuv + βuw + γv2 + δvw + w2 = u2 vw =0; − hence any such η has the right codimension at each point y n, i.e. % := π : X PC X n ∈ | → P is a cover of degree 4 with Rplanar(%)= . C ∅ We cannot hope to show that each cover %: X n of degree 4 is a quadrisecant PC of some line bundle as for covers of degree 3 with→n 0 (see [Fj] or [La]).  Example 7.9. Let Y , , , and u, v, w be as in Example 4.6. := ˇ 2 L E F H F⊗S E is globally generated; hence X := D0(Φ4(η)) P is smooth for any general section η H0 Y, . Up to a suitable linear transformation⊆ of the coordinates u, v and ∈ H w, X P has global equations ⊆
u2 + αvw + βw2 = v2 + γuw + δw2 =0,

0 0 2 where α, γ H Y, , β,δ H Y, . In particular, X := X V+(w)= ; ∈ L ∈ L w ∩ ∅ hence % := π X : X Y is a cover of degree 4 with X smooth for each value of dim(Y ). On the| other →
hand, it cannot be
a quadrisecant of any line bundle, at least when Y = Pn with n 4. ∼ C ≥

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Dipartimento di Matematica Pura ed Applicata, Universita` degli Studi di Padova, via Belzoni 7, I–35131 Padova (Italy) E-mail address: [email protected]

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