Double covers of EPW-sextics ∗ Kieran G. O’Grady “Sapienza”Universit`a di Roma December 16 2011 1 1 Contents 0 2 0 Introduction 1 c e D 1 Symmetric resolutions and double covers 5 1.1 Product formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 7 1.2 Double covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1 1.3 Local models of double covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 ] 1.4 Desingularization of certain double covers . . . . . . . . . . . . . . . . . . . . . . . . 14 G A 2 Families of double EPW-sextics 17 . h t 3 The divisor ∆ 19 a 3.1 Parameter counts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 m 3.2 First order computations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 [ 2 4 Simultaneous resolution 22 v 2 5 Double EPW-sextics parametrized by ∆ 25 5 5.1 EPW-sextics and K3 surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 9 4 5.2 Xǫ for A∈(∆\Σ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 A . 7 0 6 Appendix: Three-dimensional sections of Gr(3,C5) 35 0 1 : 0 Introduction v i X EPW-sextics are defined as follows. Let V be a 6-dimensional complex vector space. Choose a r volume-form vol: 6V −∼→C and equip 3V with the symplectic form a V V (α,β) :=vol(α∧β). (0.0.1) V Let LG( 3V) be the symplectic Grassmannian parametrizing Lagrangian subspaces of 3V - of courseLVG( 3V)doesnotdependonthe choiceofvolume-form. LetF ⊂ 3V ⊗OP(V) bVe the sub vector-bundle with fiber V 3 V F :={α∈ V |v∧α=0} (0.0.2) v over [v] ∈ P(V). Notice that (,) is zero on^F and 2dim(F ) = 20 = dim 3V; thus F is a V v v Lagrangian sub vector-bundle of the trivial symplectic vector-bundle on P(V) with fiber 3V. V ∗SupportedbyPRIN2007 V 1 Next choose A∈LG( 3V). Let V 3 F −λ→A ( V/A)⊗OP(V) (0.0.3) ^ be the composition of the inclusion F ⊂ 3V ⊗ OP(V) followed by the quotient map. Since rkF =dim(V/A) the determinat of λ makes sense. Let A V Y :=V(detλ ). A A A straightforward computation gives that detF ∼= OP(V)(−6) and hence detλA ∈ H0(OP(V)(6)). It follows that if detλ 6= 0 then Y is a sextic hypersurface. As is easily checked detλ 6= 0 for A A A A ∈ LG( 3V) generic - notice that there exist “pathological”A’s such that λ = 0 e.g. A = F . A v0 An EPW-sextic (after Eisenbud, Popescu and Walter [6]) is a sextic hypersurface in P5 which is V projectively equivalent to Y for some A ∈ LG( 3V). Let Y be an EPW-sextic: one constructs A A a double cover of Y as follows. Since A is Lagrangian the symplectic form defines a canonical A V isomorphism 3V/A ∼=A∨; thus (0.0.3) defines a map of vector-bundles λA: F →A∨⊗OP(V). Let i: YA ֒→(cid:16)PV(V) be(cid:17)the inclusion map: since a local generator of detλA annihilates coker(λA) there is a unique sheaf ζ on Y such that we have an exact sequence A A 0−→F −λ→A A∨⊗OP(V) −→i∗ζA −→0. (0.0.4) Let ξ :=ζ (−3). (0.0.5) A A We will equip O ⊕ξ with a structure of commutative O -algebra. Choose B ∈ LG( 3V) YA A YA transversalto A; thus we have a direct-sum decomposition V 3 V =A⊕B. (0.0.6) Then (,) defines an isomorphism B ∼=^A∨. Decomposition (0.0.6) defines a projection map V 3V → A; thus we get a map µA,B: F → A⊗OP(V). We claim that there is a commutative diagram with exact rows V 0 → F −λ→A A∨⊗OP(V) −→ i∗ζA → 0 µA,B µtA,B βA (0.0.7)  λt   0 → A⊗OP(V) −→A F∨ −→ Ext1(i∗ζA,OP(V)) → 0 y y y InfactthesecondrowisobtainedbyapplyingtheHom(·,OP(V))-functorto(0.0.4)andtheequality µtA,B ◦λA = λtA ◦µA,B holds because F is a Lagrangian sub-bundle of 3V ⊗OP(V). Lastly βA is defined to be the unique map making the diagram commutative; it exists because the rows are V exact. Noticethatthemapβ isindependentofthechoiceofB assuggestedbythenotation. Next A by applying the Hom(i ζ , ·)-functor to the exact sequence ∗ A 0−→OP(V) −→OP(V)(6)−→OYA(6)−→0 (0.0.8) we get the exact sequence 0−→i∗Hom(ζA,OYA(6))−∂→Ext1(i∗ζA,OP(V))−n→Ext1(i∗ζA,OP(V)(6)) (0.0.9) where n is locally equal to multiplication by detλ . Since the second row of (0.0.7) is exact a A local generator of detλA annihilates Ext1(i∗ζA,OP(V)); thus n = 0 and hence we get a canonical isomorphism ∂−1: Ext1(i∗ζA,OP(V))−∼→i∗Hom(ζA,OYA(6)). (0.0.10) We define m by setting A ζA×ζA −me→A OYA(6) (0.0.11) e (σ ,σ ) 7→ (∂−1◦β (σ ))(σ ). 1 2 A 1 2 2 Let ξ be given by (0.0.5). Tensorizing both sides of (0.0.11) by O (−6) we get a multiplication A YA map ξ ×ξ −m→A O . (0.0.12) A A YA Thus we have defined a multiplication map on O ⊕ξ . YA A Proposition 0.1. Let A ∈ LG( 3V) and suppose that Y 6= P(V). Let notation be as above. A Then: V (1) β is an isomorphism. A (2) The multiplication map m is associative and commutative and hence it equips O ⊕ξ with A YA A a structure of commutative ring. Proposition 0.1 will be proved in Subsection 1.1. In fact we will give an explicit formula for (0.0.12) (known to experts, see [4]). Granting Proposition 0.1 we let X :=Spec(O ⊕ξ ), f : X →Y (0.0.13) A YA A A A A where f is the structure morphism. Then X is a double EPW-sexticandf is its structuremap. A A A Thecovering involution ofX isthe automorphismφ : X →X correspondingtothe involution A A A A of O ⊕ξ with (−1)-eigensheaf equal to ξ . Let YA A A Y (k)= {[v]∈P(V)|dim(A∩F )=k}, (0.0.14) A v Y [k]= {[v]∈P(V)|dim(A∩F )≥k}. (0.0.15) A v Thus Y (0) = (P(V)\Y ) and Y = Y [1]. Clearly Y (1) is open in Y , it is dense in Y if A is A A A A A A A generic. Claim 0.2. The map f−1(Y (1))→Y (1) defined by restriction of f is a topological covering of A A A A degree 2 and its non-trivial deck transformation is equal to the restriction of φ to f−1(Y (1)). A A A Proof. The restriction of ξ to Y (1) is an invertible sheaf. By Item (1) of Proposition 0.1 the A A restriction of m to Y (1) defines an isomorphism ξ ⊗ξ ∼=O ; the claim follows. A A A A YA Double EPW-sextics come with a natural polarization; we let O (n):=f∗O (n), H ∈|O (1)|. (0.0.16) XA A YA A XA Let Σ:= {A∈LG( 3V)|∃W ∈Gr(3,V) s. t. 3W ⊂A}, (0.0.17) ∆:= {A∈LG( 3V)|Y [3]6=∅}. (0.0.18) V A V ThenΣ and∆areclosedsubsetsofLG( 3V).VAstraightforwardcomputation,see [16],givesthat Σ is irreducible of codimension 1. A similar computation, see Proposition 3.2, gives that ∆ is V irreducible of codimension 1 and distinct from Σ. Let 3 3 LG( V)0 :=LG( V)\Σ\∆. (0.0.19) ^ ^ Thus LG( 3V)0 is open dense in LG( 3V). In [13] we provedthat if A∈LG( 3V)0 then X is A a hyperk¨ahler (HK) 4-fold which can be deformed to (K3)[2]. Moreoverwe showed that the family V V V of polarized HK 4-folds {(X ,H )} A A A∈LG(V3V)0 is locally complete. Three other explicit locally complete families of projective HK’s of dimension greaterthan2areknown-see[2,3,9,10]. InalloftheexamplestheHKmanifoldsaredeformations ofthe HilbertsquareofaK3: theyaredistinguishedbythe valueofthe Beauville-Bogomolovform on the polarization class (it equals 2 in the case of double EPW-sextics and 6, 22 and 38 in the other cases). Inthe presentpaper wewill analyzeX forA∈∆, mainly under the hypothesis that A A6∈Σ. Let A∈(∆\Σ). We will prove the following results 3 (1) Y [3] is a finite set and it equals Y (3). If A is generic in (∆\Σ) then Y (3) is a singleton. A A A (2) One may associate to [v ] ∈ Y (3) a K3 surface S (v ) ⊂ P6 of genus 6, well-defined up to 0 A A 0 projectivities. Conversely the generic K3 of genus 6 is projectively equivalent to S (v ) for A 0 some A∈(∆\Σ) and [v ]∈Y (3). 0 A (3) ThesingularsetofX isequaltof−1Y (3). Thereisasinglep ∈X mappingto[v ]∈Y (3) A A A i A i A and the cone of X at p is isomorphic to the cone over the set of incident couples (x,r) ∈ A i P2×(P2)∨ (i.e.P(ΩP2)). Thuswehavetwostandardsmallresolutionsofaneighborhoodofpi in X , one with fiber P2 over p , the other with fiber (P2)∨. Making a choice ǫ of local small A i resolution at each p we get a resolution Xǫ → X with the following properties: There is i A A a birational map Xǫ 99KS (v )[2] such that the pull-back of a holomorphic symplectic form A A i on S (v )[2] is a symplectic form on Xǫ. If S (v ) contains no lines (true for generic A by A i A A i Item (2)) then there exists a choice of ǫ such that Xǫ is isomorphic to S (v )[2]. A A i (4) Given a sufficiently small open (classical topology) U ⊂ (LG( 3V)\Σ) containing A the familyofdoubleEPW-sexticsparametrizedbyU hasasimultaneousresolutionofsingularities V (no base change) with fiber Xǫ over A (for an arbitrary choice of ǫ). A A remark: if Y (3) has more than one point we do not expect all the small resolutions to be A projective (i.e. K¨ahler). Items (1)-(4) should be compared with known results on cubic 4-folds - recall that if Z ⊂P5 is a smooth cubic hypersurface the variety F(Z) parametrizing lines in Z is a HK 4-fold which can be deformed to (K3)[2] and moreover the primitive weight-4 Hodge structure of Z is isomorphic (after a Tate twist) to the primitive weight-2 Hodge structure of F(Z), see [2]. Let D ⊂ |OP5(3)| be the prime divisor parametrizing singular cubics. Let Z ∈ D be generic: the following results are well-known. (1’) singZ is a finite set. (2’) Given p ∈ singZ the set S (p) ⊂ F(Z) of lines containing p is a K3 surface of genus 4 and Z viceversa the generic such K3 is isomorphic to S (p) for some Z and p∈singZ. Z (3’) F(Z) is birational to S (p)[2]. Z (4’) After a local base-change of order 2 ramified along D the period map extends across Z. ThusItems(1’)-(2’)-(3’)areanalogoustoItems (1),(2)and(3)above,Item(4’)isanalogousto(4) but there is an important difference namely the need for a base-change of order 2. (Actually the paper [14] contains results showing that there is a statement valid for cubic hypersurfaces which is even closer to our result for double EPW-sextics, the rˆole of Σ being played by the divisor parametrizing cubics containing a plane.) We explain the relevance of Items (1)-(4). Items (3) and (4) prove the theorem of ours mentioned above i.e. that if A ∈ LG( 3V)0 then X is a A HK deformation of (K3)[2] (the family of polarized double EPW-sextics is locally complete by a V straightforward parameter count). The proof in this paper is independent of the proof in [13]. Beyondgivinganew proofofan“old”theoremthe aboveresults showthatawayfromΣ the period map is regular, it lifts (locally) to the relevant classifying space and the value at A∈(∆\Σ) may be identified with the period point of the Hilbert square S (v )[2]. We remark that in [15] we had A 0 provedthattheperiodmapisaswell-behavedaspossibleatthegenericA∈(∆\Σ),howeverwedid not have the exact statement about Xǫ and we had no statement about an arbitrary A∈(∆\Σ). A Thepaperisorganizedasfollows. InSection1wewillgiveformulaethatgivethelocalstructure of double EPW-sextics - the formulae are known to experts, see [4], we will go through the proofs because wecouldnotfind asuitable reference. We willalsoperformthe localcomputationsneeded to prove Item (4) above. In the very short Section 2 we will discuss (local) families of double EPW-sextics: we will determine the singular locus of the total space. In Section 3 we will go through some standard computations involving ∆. In Section 4 we will prove Items (1), (4) and the statements of Item (3) which do not involve the K3 surface S (v ). In Section 5 we will A 0 4 proveItem(2)andtheremainingstatementofItem(3). InSection 6wewillprovesomeauxiliary results on 3-dimensional linear sections of Gr(3,C5). Notation and conventions: Throughout the paper V is a 6-dimensional complex vector space. Let W be a finite-dimensional complex vector-space. The span of a subset S ⊂ W is denoted by hSi. Let S ⊂ qW. The support of S is the smallest subspace U ⊂W suchthat S ⊂im( qU −→ qW): we denote it by supp(S), if S = {α} is a singleton we let supp(α) = supp({α}) (thus if V V q =1 we have supp(α)=hαi). We define the support of a set of symmetric tensors analogously. If V α∈ qW orα∈SymdW the rank of αis the dimensionofsupp(α). AnelementofSym2W∨ may be viewed either as a symmetric map or as a quadratic form: we will denote the former by q,r,... V and the latter by q,r,... respectively. e e Let M = (M ) be a d×d matrix with entries in a commutative ring R. We let Mc = (Mij) be ij the matrix of cofactors of M, i.e. Mi,j is (−1)i+j times the determinant of the matrix obtained from M by deleting its j-th row and i-th column. We recall the following interpretation of Mc. Supposethatf: A→B isalinearmapbetweenfreeR-modulesofrankdandthatM isthematrix associated to f by the choice of bases {a ,...,a } and {b ,...,b } of A and B respectively. Then 1 d 1 d d−1f may be viewed as a map V d−1 d d−1 d−1 d f: A∨⊗ A∼= A−→ B ∼=B∨⊗ B. (0.0.20) ^ ^ ^ ^ ^ (Here A∨ := Hom(A,R) and similarly for B∨.) The matrix associated to d−1f by the choice of bases{a∨⊗(a ∧...∧a ),...,a∨⊗(a ∧...∧a )} and{b∨⊗(b ∧...∧b ),...,b∨⊗(b ∧...∧b )} 1 1 d d 1 d 1 1 d V d 1 d is equal to Mc. Let W be a finite-dimensional complex vector-space. We will adhere to pre-Grothendieck conven- tions: P(W) is the set of 1-dimensional vector subspaces of W. Given a non-zero w ∈ W we will denote the span of w by [w] rather than hwi; this agrees with standard notation. Suppose that T ⊂P(W). ThenhTi⊂P(W)is the projective span of T i.e.the intersectionofalllinearsubspaces of P(W) containing T. Schemes are defined over C, the topology is the Zariski topology unless we state the contrary. Let W be finite-dimensional complex vector-space: OP(W)(1) is the line-bundle on P(W) with fiber L∨ on the point L ∈ P(W). Let F ∈ SymdW∨: we let V(F) ⊂ P(W) be the subscheme defined by vanishing of F. If E → X is a vector-bundle we denote by P(E) the projective fiber-bundle with fiber P(E(x)) over x and we define OP(W)(1) accordingly. If Y is a subscheme of X we let Bl X −→X be the blow-up of Y. Y 1 Symmetric resolutions and double covers Wewillexaminelocaldoublecovers. Theproofof Proposition 0.1isgivenattheendof Subsec- tion 1.1. Subsection 1.4 contains the local construction needed to construct the simultaneous desingularization described in Item (3) of Section 0. 1.1 Product formula Let R be an integral Noetherian ring. Let N be a finitely generated R-module of rank zero and projective dimension 1 i.e. Exti(N,R)6=0 if and only if i=1. (1.1.1) LetFitt(N)bethe0-thFittingidealofN;itislocallyprincipalby(1.1.1). Givenahomomorphism β: N →Ext1(N,Fitt(N)) (1.1.2) 5 one defines a product m : N ×N → R/Fitt(N) as follows. Apply the functor Hom(N, ·) to the β exact sequence 0−→Fitt(N)−→R−→R/Fitt(N)−→0. (1.1.3) Arguing as in the proof of Isomorphism (0.0.10) we get a coboundary isomorphism ∂: Hom(N,R/Fitt(N))−∼→Ext1(N,Fitt(N)). (1.1.4) We let N ×N −m→β R/Fitt(N) (1.1.5) (n,n′) 7→ (∂−1β(n))(n′). The goal of this subsection is to give an explicit formula for m . A local formula will be sufficient. β Thus we may assume that N has a free resolution λ π 0−→U −→U −→N −→0. (1.1.6) 1 0 where rk U =rk U =d>0. (1.1.7) 1 0 Let{a ,...,a }and{b ,...,b }bebasesofU andU repectively. LetM bethematrixassociated 1 d 1 d 0 1 λ to λ by our choice of bases. Then Fitt(N)=(detM ). We choose the ismorphism λ ∼ R −→ Fitt(N) (1.1.8) a 7→ adetM λ From now on we view β and m as β β: N →Ext1(N,R), m : N ×N −→R/(detM ). (1.1.9) β λ Applying the Hom(·,R)-functor to (1.1.6) we get the exact sequence 0−→U∨ −λ→t U∨ −ρ→Ext1(N,R)−→0. (1.1.10) 0 1 Thus we get that β◦π lifts to a homomorphisms µt: U →U∨ - the map is written as a transpose 0 1 in order to conform to (0.0.7). It follows that there exists α: U →U such that 1 0 λ π 0 → U −→ U −→ N → 0 1 0 α µt β (1.1.11) 0 → U0∨ −λ→t U1∨ −ρ→ Ext1(N,R) → 0 y y y is a commutative diagram. Let {a∨,...,a∨} and {b∨,...,b∨} be the bases of U∨ and U∨ which 1 d 1 d 0 1 are dual to the chosen bases of U and U . Let M be the matrix associated to µt by our choice 0 1 µt of bases. Proposition 1.1. Keeping notation as above we have m (π(a ),π(a ))≡(Mc·M ) mod (detM ) (1.1.12) β i j λ µt ji λ where Mc is the matrix of cofactors of M . λ λ Proof. Equation (1.1.10) gives an isomorphism ν: Ext1(N,R)→∼ U∨/λt(U∨). (1.1.13) 1 0 Let det(U ):= dU∨⊗ dU . We will define an isomorphism • 1 0 V Vθ: U∨/λt(U∨)−∼→Hom(N,det(U )/(detλ)). (1.1.14) 1 0 • 6 First let U∨ = d−1U ⊗ dU∨ −θ→b d−1U ⊗ dU∨ = Hom(U ,det(U )) 1 1 1 0 1 0 • (1.1.15) ζ⊗ξ 7→ d−1(λ)(ζ)⊗ξ V V V V We claim that V im(θ)={φ∈Hom(U ,det(U ))|φ◦λ(U )⊂(detλ)}. (1.1.16) 0 • 1 In fact by Cramer’s formula b Mc·Mt =Mt ·Mc =detM ·1 (1.1.17) λ λ λ λ λ and Equation (1.1.16) follows. Thus θ induces a surjective homomorphism θ: U1∨b−→Hom(N,det(U•)/(detλ)). (1.1.18) One checks easily that λt(U0∨)e= kerθ - use Cramer again. We define θ to be the homomorphism induced by θ; we have proved that it is an isomorphism. We claim that e e θ◦ν =∂−1, ∂ as in (1.1.4). (1.1.19) In fact let K be the fraction field of R and 0 → R →ι I0 → I1 → ... be an injective resolution of R with I0 =det(U )⊗K and ι(1)= detλ⊗1. Then Ext•(N,R) is the cohomology of the double • complex Hom(U ,I•) andof coursealsoof the single complexes Hom(U ,R)andHom(N,I•). One • • checks easily that the isomorphism ∂ of (1.1.4) is equal to the isomorphism H1(Hom(N,I•)) →∼ H1(Hom(U ,I•)) i.e. • ∂: Hom(N,det(U )/(detλ))=Hom(N,I0/ι(R))−∼→H1(Hom(U ,I•)). (1.1.20) • • Let f ∈ Hom(N,det(U )/(detλ)); a representative of ∂(f) in the double complex Hom(U ,I•) is • • given by g0,1 := f ◦π ∈ Hom(U ,I1). Let g0,0 ∈ Hom(U ,det(U )) be a lift of g0,1 and g1,0 ∈ 0 0 • Hom(U ,det(U )) be defined by g1,0 := g0,0 ◦λ. One checks that im(g1,0) ⊂ (detλ) and hence 1 • there exists g ∈ Hom(U ,R) such that g1,0 = ι◦g. By construction g represents a class [g] ∈ 1 H1(Hom(U ,R))=U∨/λt(U∨)and[g]=ν◦∂(f). Anexplicitcomputationshowsthat[g]=θ−1(f). • 1 0 This proves (1.1.19). Now we prove Equation (1.1.12). By (1.1.19) we have m (π(a ),π(a ))=(∂−1βπ(a ))(π(a ))=(θνβπ(a ))(π(a )). (1.1.21) β i j i j i j Unwinding the definition of θ one gets that the right-hand side of the above equation equals the right-hand side of (1.1.12). Letm begivenby(1.1.9): wedefineaproducton(R/(detM )⊕N)asfollows. Let(r,n),(r′,n′)∈ β λ (R/(detM )⊕N): we set λ (r,n)·(r′,n′):=(rr′+m (n,n′),rn′+r′n). (1.1.22) β In general the above product is not associative nor commutative. The following is an example in which the product is both associative and commutative. Let U be a free finitely generated R-module, and assume that we have an exact sequence 0−→U∨ −γ→U −π→N −→0 (1.1.23) with γ symmetric. Then we get a commutative diagram (1.1.11) by setting α=IdU, µt =IdU∨, (1.1.24) and letting β be the map induced by the above choices. We denote by m the correponding map γ N ×N →R/(detM ). γ Proposition 1.2. Suppose that we have Exact Sequence (1.1.23) with γ symmetric. The product defined on (R/(detM )⊕N) by m is associative and commutative. γ γ 7 Proof. Let d := rk U > 0. Let {a ,...,a } be a basis of U and {a∨,...,a∨} be the dual basis of 1 d 1 d U∨. Let M = M i.e. the matrix associated to γ by our choice of bases. Since γ is a symmetric γ map M is a symmetric matrix. By (1.1.12) we have m (π(a ),π(a ))≡Mc mod (detM). (1.1.25) γ i j ji Since M is symmetric so is Mc and hence we get that m is symmetric. It remains to prove that γ m is associative. For 1 ≤ i < k ≤ d and 1 ≤ h 6= j ≤ d let Mi,k be the (d−2)×(d−2)-matrix γ h,j obtained by deleting from M rows i,k and columns h,j. Let X =(Xh )∈Rd be defined by ijk ijk (−1)i+k+j+hdetMi,k if h<j, j,h Xh :=0 if h=j. (1.1.26) ijk (−1)i+k+j+h−1detMi,k if j <h. j,h A tedious but straightforwardcomputation gives that d Mca −Mc a =γ( Xh a∨). (1.1.27) ij k jk i ijk h h=1 X The above equation proves associativity of m . γ Hypothesis1.3. In CommutativeDiagram (1.1.11)wehave thatµt isan isomorphism andα=µ. Proposition 1.4. Assume that Hypothesis 1.3 holds. Set γ :=λ◦µ−1, U :=U . (1.1.28) 1 Then (1.1.23) is an exact sequence, γt =γ and m (π(a ),π(a ))≡(Mc) mod (detM ). (1.1.29) β i j γ ij γ where {a ,...,a } is the basis of U∨ = U introduced right after Equation (1.1.7). In particular 1 d 0 (R/(detM )⊕N) equipped with the product given by (1.1.22) is a commutative (associative) ring. λ Proof. The first statement is obvious. Equation (1.1.29) follows from Proposition 1.2. The last statement follows from (1.1.29). Definition 1.5. Suppose that Hypothesis 1.3 holds: the symmetrization of (1.1.11) is Exact Sequence (1.1.23) with γ given by (1.1.28). Proof of Proposition 0.1 Let [v ] ∈ P(V). Choose B ∈ LG( 3V) transversal to F (and to 0 v0 A of course). Then µ is an isomorphism in an open neighborhood U of [v ]. It follows that A,B V 0 β is an isomorphism in a neighborhood of [v ]. This proves Item (1). Let’s prove Item (2). A 0 Let B ∈ LG( 3V) and U be as above; we may assume that U is affine. Let N := H0(i ζ | ) ∗ A U and β := H0(β | ). Thus β: N → Ext1(N,C[U]). By Commutativity of Diagram (0.0.7) and VA U by Proposition 1.4 we get that the multiplication map m is associative and commutative. On β the other hand m is the multiplication induced by m on N; since [v ] is an arbitrary point of β A 0 P(V) it follows that m is associative and commutative. A 1.2 Double covers Suppose that we are given β as in (1.1.2) and assume that Hypothesis 1.3 holds. We let X :=Spec(R/(detM )⊕N), Y :=Spec(R/(detM )). (1.2.1) β λ β λ Let f : X →Y be the structure map. The covering involution of f is the automorphism φ of β β β β β X which corresponds to the isomorphism of the R/(detM )-algebra (R/(detM )⊕N) acting as β λ λ 8 multiplicationby(−1)onN. LetY [c]betheclosedsubschemeofSpecRdefinedbyimposingthat β corkM ≥c i.e. λ Y [c]= V(detMI,J). (1.2.2) β λ |I|=|J\|=d+1−c (I,J,K,H are multi-indices, MI,J,MK,H the corresponding minors of M .) Thus Y [1]=Y . Let λ λ λ β β Y (c):=Y [c]\Y [c+1] (1.2.3) β β β Let γ be the simmetrization of β - see Definition 1.5. Then f : X → Y is identified with β β β f : X →Y . Thus we may replace throughout X by X . We will denote Y [c] by Y [c] etc. We γ γ γ β γ β γ let φ be the covering involution of X . We realize X as a subscheme of Spec(R[ξ ,...,ξ ]) as γ γ γ 1 d follows. Since the ring (R/(detM )⊕N) is associative and commutative there is a well-defined γ surjective morphism of R-algebras R[ξ ,...,ξ ]−→R/(detM )⊕N (1.2.4) 1 d γ mapping ξ to a . Thus we have an inclusion i i X ֒→Spec(R[ξ ,...,ξ ]). (1.2.5) γ 1 d Claim1.6. ReferringtoInclusion (1.2.5) theideal ofX is generatedbytheentries ofthematrices γ M ·ξ, ξ·ξt−Mc. (1.2.6) γ γ (We view ξ as a column matrix.) Furthermore the map f is induced by the natural projection γ Spec(R[ξ ,...,ξ ])→SpecR and the covering involution φ is induced by the automorphism of the 1 d γ R-algebra R[ξ ,...,ξ ] which multiplies each ξ by (−1). 1 d i Proof. By Equation(1.1.25) the ideal of X is generatedby detM and the entries of the matrices γ γ M ·ξ, ξ·ξt−Mc. (1.2.7) γ γ By Cramer’s formula detM belongs to the ideal generated by the entries of the first and second γ matrices above. This proves that the ideal of X is as claimed. The statements regarding f and γ γ φ are obvious. γ Suppose that R is a finitely generated C-algebra and that we have (1.1.23) with γ symmetric. Let p ∈ SpecR be a closed point: we are interested in the localization of X at points in f−1(p). γ γ Let J ⊂ U∨(p) be a subspace complementary to kerγ(p). Let J ⊂ U∨ be a free submodule whose fiber over p is equal to J. Let K⊂U∨ be the submodule orthogonalto J i.e. K:={u∈U∨ |γ(a)(u)=0 ∀a∈J}. (1.2.8) The localization of K at p is free. Let K := K(p) be the fiber of K at p; clearly K = kerγ(p). Localizing at p we have U∨ =K ⊕J . (1.2.9) p p p Corresponding to (1.2.9) we may write γp =γK⊕⊥γJ (1.2.10) where γK: Kp → K∨p and γJ: Jp → J∨p are symmetric maps. Notice that we have an equality of germs (Y ,p)=(Y ,p). (1.2.11) γ γK Let k := dimK and d := rk U. Choose bases of K and J ; by (1.2.9) we get a basis of U∨. The p p p dual bases of K∨, J∨ and U∨ are compatible with respect to the decomposition dual to (1.2.9). p p p Corresponding to the chosen bases we have embeddings X ֒→Y ×Ck, X ֒→Y ×Cd. (1.2.12) γK γK γ γ The decomposition dual to (1.2.9) gives an embedding j: (Y ×Ck)֒→(Y ×Cd). (1.2.13) γK γ 9 Claim 1.7. Keep notation as above. The composition X ֒→(Y ×Ck)−→j (Y ×Cd) (1.2.14) γK γK γ defines an isomorphism of germs in the analytic topology (X ,f−1(p))−∼→(X ,f−1(p)) (1.2.15) γK γK γ γ which commutes with the maps f and f . γK γ Proof. This follows from Decomposition (1.2.10) and Equation (1.1.25). We pass to the analytic topology in order to be able to extract the square root of a regular non-zero function. Proposition 1.8. Assume that R is a finitely generated C-algebra. Suppose that Hypothesis 1.3 holds. Then the following hold: (1) f−1Y (1)→Y (1) is a topological covering of degree 2. β β β (2) Let p ∈ (Y \Y (1)) be a closed point. The fiber f−1(p) consists of a single point q. If β is β β β the isomorphism associated to a symmetric map γ fitting into Exact Sequence (1.1.23) then ξ (q)=0 for i=1,...,d where ξ are the coordinates on X associated to Embedding (1.2.5). i i γ Proof. (1): By Proposition 1.4 we may assume that β is the map associated to a symmetric γ fitting into Exact Sequence (1.1.23). Localizing at p ∈ Y (1) and applying Claim 1.7 we get β Item (1). (2): Since corkM (p) ≥ 2 we have Mc(p) = 0 and thus Item (2) follows from Claim γ γ 1.6. 1.3 Local models of double covers In the present subsection we assume that R is a finitely generated C-algebra. Let W be a finite- dimensional complex vector-space. We will suppose that we have an exact sequence 0−→R⊗W∨ −γ→R⊗W −→N −→0, γ =γt. (1.3.1) Thus we have a double cover f : X → Y . Let p ∈ Y be a closed point. We will examine X γ γ γ γ γ in a neighborhood of f−1(p) when the corank of γ(p) is small. We may view γ as a regular map γ SpecR→Sym2W; thus it makes sense to consider the differential dγ(p): T SpecR→Sym2W. (1.3.2) p Let K(p):=kerγ(p)⊂W∨; we will consider the linear map T SpecR δ−γ→(p) Sym2K(p)∨ p (1.3.3) τ 7→ dγ(p)(τ)| K(p) Let d := dimW; choosing a basis of W we realize X as a subscheme of SpecR×Cd with ideal γ given by Claim 1.6. Since corkγ(p) ≥ 2 Proposition 1.8 gives that f−1(p) consists of a single γ point q - in fact the ξ -coordinates of q are all zero. Throughout this subsection we let i f−1(p)={q}. (1.3.4) γ Claim 1.9. Keep notation as above. Suppose that d=dimW =2 and that γ(p)=0. Then I(X ) γ is generated by the entries of ξ·ξt−Mc. γ Proof. Claim 1.6 together with a straightforwardcomputation. 10