Deformations of Hyperelliptic and Generalized Hyperelliptic Polarized Varieties

The purpose of this article is twofold. Firstly, we address and completely solve the following question: Let (X, L) be a smooth, hyperelliptic polarized variety and let φ:X⟶Y⊂PN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi : X \longrightarrow Y \subset \textbf{P}^N$$\end{document} be the morphism induced by |L|; when does φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document} deform to a birational map? Secondly, we introduce the notion of “generalized hyperelliptic varieties” and carry out a study of their deformations. Regarding the first topic, we settle the non trivial, open cases of (X, L) being Fano-K3 and of (X, L) having dimension m≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \ge 2$$\end{document}, sectional genus g and Lm=2g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^m=2g$$\end{document}. This was not addressed by Fujita in his study of hyperelliptic polarized varieties and requires the introduction of new methods and techniques to handle it. In the Fano-K3 case, all deformations of (X, L) are again hyperelliptic except if Y is a hyperquadric. By contrast, in the Lm=2g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^m=2g$$\end{document} case, with one exception, a general deformation of φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document} is a finite birational morphism. This is especially interesting and unexpected because, in the light of earlier results, φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document} rarely deforms to a birational morphism when Y is a rational variety, as is our case. The Fano-K3 case contrasts with canonical morphisms of hyperelliptic curves and with hyperelliptic K3 surfaces of genus g≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g \ge 3$$\end{document}. Regarding the second topic, we completely answer the question for generalized hyperelliptic polarized Fano and Calabi–Yau varieties. For generalized hyperelliptic varieties of general type we do this in even greater generality, since our result holds for Y toric. Standard methods in deformation theory do not work in the present setting. Thus, to settle these long standing open questions, we bring in new ideas and techniques building on those introduced by the authors concerning deformations of finite morphisms and the existence and smoothings of certain multiple structures. We also prove a new general result on unobstructedness of morphisms that factor through a double cover and apply it to the case of generalized hyperelliptic varieties.


Introduction
In this article we study the deformations of finite, degree 2 morphisms ϕ.Two things may happen: (I) either there are deformations of ϕ which are of degree 1; or (II) all deformations of ϕ are of degree 2.
The paradigmatic situation in which (I) happens is when ϕ is the canonical morphism of a hyperelliptic curve of genus g, g > 2. Thus, from the point of view of deformations, moduli and related issues, it is natural to look for a generalization of the notion of hyperellipticity to higher dimensions.There will naturally be more than one analogue of this notion and we study such analogues from the perspective of deformations and moduli in this article.We first deal with the notion of hyperellipticity that carries over from the case of algebraic curves, before we introduce another, more general notion (see Definition 0.4).A canonical curve comes with a natural polarization, namely, we have a polarized variety (C, K C ).Therefore, if we want to know in which situations (I) occurs, we look at the concept of polarized hyperelliptic varieties, which were first introduced by Fujita in [Fuj83].We reformulate [Fuj83, Definition 1.1] in a way more suitable for us: Definition 0.1.Let X be a smooth variety and let L be a polarization (i.e., an ample line bundle) on X.We say that the polarized variety (X, L) is hyperelliptic if L is base-point-free and the dimensional) Fano-K3 variety (X, L) (see Definition 2.8) has a general surface section which is a polarized K3 surface.If (X, L) is a Fano-K3 variety, then L m = 2g − 2. Conversely, if L m = 2g − 2, m ≥ 3 and H 1 (L) = 0, then (X, L) is a Fano-K3 variety (see [Fuj83,Propositions 5.18 (3), 5.19 and (6.7)]).It is well known that, unless g = 2, the general deformation of a hyperelliptic K3 surface is nonhyperelliptic (this is because, if g > 2, hyperelliptic K3 surfaces form a closed locus in the moduli space of polarized K3 surfaces).For m > 2, the study of the deformations of Fano-K3 varieties (X, L) is partially addressed by Fujita (see [Fuj83,Remark 7.7, Corollary 7.14]), who finds that there is a hyperelliptic deformation of (X, L).In Section 2 we settle this question completely and our result contrasts with the results for surfaces and curves which are general sections of (X, L) (those sections are, respectively, polarized hyperelliptic K3 surfaces and canonically polarized hyperelliptic curves).Indeed, if the image i(Y ) of the morphism induced by |L| is smooth, we show that any deformation of a Fano-K3 variety (X, L) is hyperelliptic except if Y is embedded as a hyperquadric (see Theorems 2.10 and 2.11).On the contrary, if g > 2 and (X, L) is either a canonically polarized hyperelliptic curve or a polarized hyperelliptic K3 surface, then the general deformation of (X, L) (with h 0 (L t ) constant throughout the deformation if X is a curve) is nonhyperelliptic.This is because, if g > 2, then being hyperelliptic is not a numerical condition, but a closed condition on both the moduli of curves and the moduli of polarized K3 surfaces.
The interesting case L m = 2g was not previously known.In Section 3 we settle the question completely when Y is smooth.We prove that, except for one case, (see Theorems 3.6, 3.8, and 3.10) (X, L) deforms to a nonhyperelliptic polarized variety, in clear contrast with Fano-K3 varieties.If Y is as in (2e) of Proposition 3.2 we overcome an added difficulty: in this case (see Proposition 3.3), it is not clear whether H 1 (N ϕ ) vanishes, and hence, whether ϕ is unobstructed or not.Still, we find a way to study the deformations of ϕ (see the proof of Theorem 3.8).
The outcome of all explained so far is that (I) seems to be more rare than (II), as only may occur if L m = 2g − 2 or L m = 2g.In addition, as dimension increases, (I) becomes even more rare (see Theorem 2.10) and also, families of hyperelliptic varieties become fewer than one would expect (see Propositions 2.9 and 3.2).
The above results compel us to investigate a deeper structural reason on why these phenomena happen.In this direction, we introduce another, broader generalization of hyperellipticity: Definition 0.4.Let (X, L) a smooth polarized variety with L base-point-free.Let the morphism from X to P N , induced by |L| be of degree 2 onto its image i(Y ) (i is the embedding of Y in P N ).Let the variety Y be smooth and isomorphic to any of this: (1) projective space; (2) a hyperquadric; (3) a projective bundle over P 1 .The variety Y is not necessarily embedded by i as a variety of minimal degree in P N .Then we say that (X, L) is a generalized hyperelliptic polarized variety.
In Section 2 and 4 we show that, in most cases, the deformations of generalized hyperelliptic Fano, Calabi-Yau and general type polarized varieties are morphisms of degree 2 (see Theorems 2.5, 4.5 and 4.14).In particular, the results on Calabi-Yau varieties are in sharp contrast to the results for their lower dimensional analogue, namely, K3 surfaces.This contrast can also be traced to the question of the existence of double structures.Indeed, in [GP97] and [BMR20] it is shown the existence of locally Gorenstein multiplicity two structures (named, in this case, carpets and, in general, ribbons; see e.g.[BE95] for definitions), nonsplit and with the same invariants as smooth K3 surfaces, supported on rational normal scrolls and, more generally, on general embeddings of Hirzeburch surfaces and other rational surfaces.Contrarily, we show in Corollary 4.4 that there are no nonsplit ribbons with Calabi-Yau invariants supported on varieties Y as in Definition 0.4.Deformations of finite morphisms of degree 2 are linked to the existence or non existence of ribbons with right invariants.Indeed, the relationship between the deformation of morphisms and these double structures appears in [Fon93], [GGP08a], [GGP08b], [GGP13a] or [GGP13b], to mention just a few references.This relationship is at the very heart of the proofs of results like [GGP10, Theorem 1.4] and [GGP13a, Theorem 1.5], that will be used later on (for a deeper insight on all this, see [Gon06]).Thus, in this article, we do not only address the problem of the existence of Calabi-Yau ribbons but we also look at the existence of ribbons with other invariants.In Corollaries 2.3 and 4.12 we prove (with one exception, see Example 2.12), that Fano and general type nonsplit ribbons, supported on varieties Y as in Definition 0.4, do not exist.

General results on deformations of morphisms
In this section we prove several general results on deformations of morphisms, that we will apply in the remaining of the article.First we make clear what we will mean when we talk of a deformation of a morphism: Definition 1.1.Let X be an algebraic, projective variety and let ϕ : X −→ P N be a morphism.By a deformation of ϕ we mean a flat family of morphisms Φ : X −→ P N Z over a smooth, irreducible algebraic variety Z (i.e., Φ is a Z-morphism for which X −→ Z is proper, flat and surjective) with a distinguished point 0 ∈ Z such that (1) X is irreducible and reduced; (2) X 0 = X and Φ 0 = ϕ.
Unless otherwise specified, when we say that certain property (*) is satisfied by a deformation Φ, we will mean that there exists an open neighborhood U of 0 such that, for all z ∈ U , property (*) is satisfied by the fibers Φ z : X z −→ P N of Φ.That certain property (*) is satisfied by a general deformation will mean that there exists an open set U of the base of an algebraic formally semiuniversal deformation Φ of ϕ such that, for all z ∈ U , the morphism Φ z satisfies (*).We will use analogous definitions for a deformation of a variety X or for a deformation of a polarized variety (X, L).

Notation and conventions:
Throughout this article, unless otherwise stated, we will use the following notation and conventions: (1) We will work over an algebraically closed field k of characteristic 0.
(3) ω X and ω Y denote respectively the canonical bundle of X and Y ; K X and K Y denote respectively the canonical divisor of X and Y .We use the same notation for all the other varieties that appear.(4) i will denote a projective embedding i : Y ֒→ P N .In this case, I will denote the ideal sheaf of i(Y ) in P N .We will often abridge i * O P N (1) as O Y (1).(5) π will denote a finite morphism π : X −→ Y of degree n = 2; in this case, E will denote the trace-zero module of π (E is a line bundle on Y ) and B will be the branch divisor of π (then We introduce a homomorphism defined in [Gon06, Proposition 3.7]: Proposition 1.3.Let N π and N ϕ be respectively the normal bundles of π and ϕ and let N i(Y ),P N be the normal bundle of i(Y ) in P N .There exists a homomorphism that appears when taking cohomology on the exact sequence the homomorphism Ψ has two components Taking cohomology on (1.3.1), the homomorphism Ψ fits in this long exact sequence of cohomology: We now prove a useful generalization of [GGP10, Theorem 2.6].The proof given here also settles an unclear point in the proof given in [GGP10].
and then an exact sequence 0 One can show that Def (Y,B) has a semiuniversal formal element by checking Schlessinger's conditions (see e.g.[Ser06, Theorem 2.

3.2]).
There is a map defined as follows.Let ( Ȳ , B) be an element in Def (Y,B) (A), where B = (r) 0 , with r ∈ H 0 ( Ē −2 ) lifting r ∈ H 0 (E −2 ) such that B = (r) 0 .Then, on the total space of Ē −1 , there is a tautological section t lifting the tautological section on the total space of E −1 .Then ( X, ϕ) ∈ Def ϕ (A) is given by x x q q q q q q q q q q q q q q Ȳ , and Then X is flat over A and ( X, ϕ) is a lifting of (X, ϕ).
Recall that Def ϕ (k[ǫ]) = H 0 (N ϕ ) and the long exact sequence of cohomology (1.3.2).Since the restriction of π becomes an isomorphism between the ramification divisor and the branch locus B, ) be a first order deformation of (Y ⊂ P N , B ∈ |E −2 |) and ( X, ϕ) ∈ H 0 (N ϕ ) be the first order deformation of (X, ϕ) associated to ( Ȳ , B) by dF .From the construction we made for F , we see that . Therefore im ϕ = Ȳ .Then, using [Gon06, Theorem 3.8 (2) and Propositions 3.11, 3.12], we see that there is a commutative We see, from diagram (1.4.3), that if Ψ 2 = 0, then dF is an isomorphism.Since Def (Y,B) and Def ϕ have a semiuniversal formal element, Def (Y,B) is smooth and dF is an isomorphism it follows that Def ϕ and F are smooth.This shows that ϕ is unobstructed.Now we prove that any deformation of ϕ is of degree 2. Let p : Y ֒→ P N U → (U, u 0 ) be an algebraic formally universal embedded deformation of Y .We can, by assumption, take U and the total family Y smooth.Since, from (1), the line bundle E can be lifted to any infinitesimal deformation of Y , it follows that, after an etale base change centered at u 0 ∈ U , there exists a lifting of E to a line bundle L on Y .Since, by the election of Y , for any infinitesimal lifting ( Ȳ , Ē ) of (Y, E ) there is a base change Spec A → U such that Y × Spec A = Ȳ then, by the uniqueness of Ē , we see that By the hypothesis h 1 (E −2 ) = 0, we can assume h 1 (L −2 |Yu ) = 0 for any u ∈ U and p * (L −2 ) is a free sheaf on U and the formation of p * commute with base extension (see [Gro61, Chapitre III, §7] or [MFK94, Chapter 0, §5]).Let P(p * (L −2 )) → U be the associated projective bundle.On Y × U P(p * (L −2 )), we consider the divisor Since on any fiber the line bundle associated to , where q : Y × U P(p * (L −2 )) − → Y is the projection.The divisor B is the zero locus of some section r ∈ H 0 (q * L −2 ).Let t ∈ H 0 (q ′ * q * L −2 ) denote the tautological section on the total space of q * L −1 , where q ′ : q * L −1 → Y × U P(p * (L −2 )) is the projection.Then we can construct a relative double covering x x q q q q q q q q q q q Y × U P(p * (L −2 )) , whose fiber over any point (u, [r]) In fact, we restrict the construction to the open set V ⊂ P(p * (L −2 )), where the divisors B |Yu×{[r]} are reduced and smooth, in order to obtain integral, smooth, double coverings X (u,[r]) → Y u .The open set V contains the point (u 0 , [r 0 ]) that corresponds to the pair (Y, B), so we can assume V maps surjectively onto U .Let Φ denote the composite map Then Φ is an algebraic deformation of X ϕ − → P N .We will compute the tangent space to V at (u 0 , [r 0 ]) by considering the fiber at (u 0 , [r 0 ]) of the sequence 0 → T V /U → T V → τ * T U → 0, associated to the projection V τ − → U .This way, since H 1 (O Y ) = 0, we obtain the sequence Comparing (1.4.4) to (1.4.1) we see that the Kodaira-Spencer map for the family (Y × U V → V, B, (u 0 , [r 0 ]) ∈ V ) is bijective, so this family is an algebraic formally semiuniversal deformation of (Y Since the differential dF is an isomorphism then the Kodaira-Spencer map for the family (X ) is an algebraic formally semiuniversal deformation for ϕ, which shows that any deformation of ϕ is a (finite) morphism of degree 2.
Proposition 1.5.Let X be a smooth, algebraic projective variety, let φ : X −→ P N be a morphism and let L be a polarization on X.If H 2 (O X ) = 0, then ϕ and (X, L) have an algebraic formally semiuniversal deformation.
Proof.Since X is a projective variety, it has a formal versal deformation (see e.g.[DI74] or [Zar95]).Since H 2 (O X ) = 0, then, by Grothendieck's existence theorem (see [Ser06, Theorem 2.5.13]), this formal versal deformation of X is effective.It follows from general deformation theory the existence of an algebraic formally versal (even semiuniversal) deformation of ϕ (see [Ser06,Theorem 3.4.8]).In this case, the formal semiuniversal deformation of φ is also effective, so it is algebraizable by Artin's algebraization theorem (see [Art69]).By [Ser06, Theorem 3.3.11.(i)], the functor Def (X,L) has a semiuniversal formal element.Then arguing as above, this element is algebraizable, so there exists an algebraic formally semiuniversal deformation for (X, L).
We give now a generalization of [GGP10, Lemma 2.4].We recall before the definition of Fano variety and variety of general type: Definition 1.6.Let X be a smooth variety of dimension m, m ≥ 2. We say X is a Fano variety if −K X is ample.We say that X is of general type if K X is big.
Lemma 1.7.Let X be a smooth variety of dimension m, m ≥ 2, which is either a Fano variety or a regular variety of general type.Let l ∈ N. (1) X and L are ample.Then semicontinuity implies h 0 (L z ) is constant.If X is of general type we use that the plurigenera are deformation invariants.Shrinking Z if necessary we may assume L z is base-point-free for all z ∈ Z. Then Φ z is induced by the complete linear series of |L z |.

Deformations of hyperelliptic Fano-K3 varieties and generalized hyperelliptic polarized Fano varieties
Notation 2.1.Unless otherwise stated, in the remaining of this article the variety Y has dimension m, m ≥ 2, and is isomorphic to a variety of one of these three types: (i) P m .(ii) A smooth hyperquadric of dimension m ≥ 3; in this case, let h be the restriction of the hyperplane section of P m+1 to Y .(iii) A (smooth) projective bundle on P 1 .
In order to study the deformations of hyperelliptic Fano-K3 varieties we need first to carry out certain cohomology computations.We do them in the broader setting of generalized hyperelliptic polarized Fano varieties.
Proposition 2.2.Let X, Y and E be as in notations 1.2 and 2.1, let T Y be the tangent bundle of Y and assume follows from suitably twisting and taking cohomology on the Euler sequence of tangent bundle of P m and by the vanishing of the intermediate cohomology of line bundles on P m .If m = 2, we also need to check that obtained by tensoring the normal sequence of Y in P m+1 by E .We want to see Consider the exact sequence which is obtained by restricting to Y the Euler sequence of the tangent bundle of P m+1 and tensoring with E .Taking cohomology on (2.2.3) we get Then we need Both vanishings follow from suitably twisting and taking cohomology on the exact sequence because of the vanishing of the intermediate cohomology of line bundles on P m+1 (recall m+ 1 ≥ 4).Now we study Case 3: Y is a projective bundle on P 1 .We use the following notation: (2.2.6) (i) Let Y = P(E 0 ) with E 0 normalized, and let (iii) Let p be the structure morphism from Y to P 1 and let F be a fiber of p.
Let T Y /P 1 be the relative tangent bundle to p and consider the exact sequence and the relative Euler sequence First we argue for α > 1.We see that because of the vanishing of the intermediate cohomology of P m−1 and, for the topmost cohomology, because 0 using the same arguments.Then exact sequences (2.2.7) and (2.2.8) and the vanishing of Then exact sequences (2.2.7) and (2.2.8) and the vanishing of Then exact sequences (2.2.7) and (2.2.8) and the vanishing of We now study the existence or non existence of nonsplit Fano ribbons on varieties Y as in Notation 2.1 (for the definitions of ribbon and nonsplit ribbon, see [BE95,§1]; by a Fano ribbon we mean that h 0 (O Y ) = 1 and that the dual of the dualizing sheaf ω Y of Y is ample): , so E can be thought as the trace-zero module of a cover π : X −→ Y as in Proposition 2.2, except for the fact that | E | might not contain a smooth divisor.Since we do not use this property of E in the proof of Proposition 2.2 and since and, taking cohomology on (2.2.1), we see that H 1 (T Y ⊗ E ) is the cokernel of the map of multiplication of global sections, which is, trivially, an isomorphism.Thus Ext Corollary 2.4.Let X, Y , ϕ, E and I be as in notations 1.2 and 2.1.Assume X is Fano and ϕ is induced by a complete linear series (i.e., H 0 (E (1)) = 0).Then Hom( Proof.Taking cohomology on the conormal sequence of i(Y ) in P N we get (2.4.1)Hom(Ω We are going to see For this we need H 0 (E (1)) = 0 and H 1 (E ) = 0.The former holds because ϕ factors through π and ϕ is induced by a complete linear series.The latter holds by Kodaira vanishing theorem, because Theorem 2.5.Let X, Y and ϕ be as in notations 1.2 and 2.1.Let X be a Fano variety and let the morphism ϕ from X to P N be induced by a complete linear series (i.e., H 0 (E (1)) = 0). (1 Then ϕ is unobstructed and any deformation of ϕ is a finite morphism of degree 2 onto its image, which is a deformation of i(Y Thus hypotheses (1), (3) and (4) of Theorem 1.4 are satisfied.If Y is P m or a hyperquadric, then H 1 (E −2 ) = 0 because the vanishing of cohomology in projective space.If Y is a projective bundle over P 1 , then condition (2) of the statement is equivalent to β ≥ αe m−1 , with α, β and e m−1 as in (2.2.6).Then condition (2) implies H 1 (E −2 ) = 0. Thus hypothesis (2) of Theorem 1.4 is also satisfied.Finally, hypothesis (5) of Theorem 1.4 follows from Corollary 2.4.
The cases dealt with in the next proposition have already been covered by Theorem 2.5 except maybe if assumption (2) of Theorem 2.5 does not hold; that is why we include it here.
Proposition 2.6.Let X, Y and ϕ be as in notations 1.2 and 2.1.Let X be a Fano variety and let the morphism ϕ from X to P N be induced by a complete linear series |L|, where L = ω −l X for some l ∈ N. Let Y be a projective bundle over P 1 .Then any deformation of ϕ is a finite morphism of degree 2 onto its image, which is a deformation of i(Y ) in P N .
Proof.Let Z be a smooth algebraic variety with a distinguished point 0 and let (X , Φ) be a flat family over Z, with (X 0 , Φ 0 ) = (X, ϕ).Proposition 2.2 and [Weh86, Corollary 1.11] (see also [Hor76,Theorem 8.1]) imply that X is equipped, over an analytic neighborhood U of 0 in Z, with a deformation (X , Π) of (X, π), where Π is finite, surjective and of degree 2 (we make an abuse of notation and keep calling the restriction of the family X to U as X ).We call Y to the image of Π.Since This and [Hor76, Theorem 8.1] imply that the deformation Y of Y can be realized as a deformation (Y , i) of (Y, i).Since i is an embedding, we can assume, after shrinking U , that i is a relative embedding of Y in P N U .Then i • Π = Φ ′ is a deformation of ϕ.By Lemma 1.7, after shrinking U if necessary, for all z ∈ U we have Φ z and Φ ′ z are induced by |ω −l Xz |.Thus, for all z ∈ U , Φ z and Φ ′ z are equal up to automorphisms of P N .Thus, shrinking U again if necessary, Φ z is of degree 2 for all z ∈ U .
Remark 2.7.If X, Y and ϕ are as in Theorem 2.5 or Proposition 2.6 and Y is a projective bundle over P 1 , but neither condition (2) of Theorem 2.5 nor the condition on L in Proposition 2.6 hold, still something can be said about the deformations of ϕ.Indeed, it follows from Proposition 2.2, [Weh86, Corollary 1.11] and [Hor76, Theorem 8.1] that, for any deformation X of X, there exists a deformation of (X , Φ) of (X, ϕ) which is finite and of degree 2 onto its image.
We give the definition of Fano-K3 polarized variety, which is equivalent to [Fuj83, Definition 1.5] (note that the condition of the ring R(L) = ⊕ ∞ n=0 H 0 (L ⊗n ) being Cohen-Macaulay, which is required by [Fuj83, Definition 1.5], is deduced from Definition 2.8, Kodaira vanishing theorem and Serre duality).
Definition 2.8.We say that a polarized variety (X, L) of dimension m, m ≥ 3, is Fano-K3 if it is a Fano polarized variety of index m − 2, i.e., if ω −1 X = L ⊗m−2 .Proposition 2.9.Let X, Y , E , i and ϕ be as in Notation 1.2, 2.1 and (2.2.6), and assume (X, L) is a hyperelliptic Fano-K3 variety of dimension m ≥ 3 and that ϕ is induced by |L|.Then i(Y ) is a variety of minimal degree and Y , E −2 and B are as follows: (1) If Y = P m , then E −2 = O P m (6) (in this case, g = 2).
( As a consequence of Theorem 2.5 we obtain this: Theorem 2.10.Let X and Y be as in notations 1.2 (2).Let (X, L) a hyperelliptic Fano-K3 variety of dimension m ≥ 3.If Y is P m or a rational normal scroll, then any deformation of (X, L) is hyperelliptic.
Proof.If Y = P m , the result follows from Remark 0.3 (2).Let now Y be a projective bundle over P 1 and let ϕ be the morphism induced by |L|.Let Z be a smooth, algebraic variety with a distinguished point 0 ∈ Z.Let (X , L ) be a flat family over Z such that the fiber (X 0 , L 0 ) over 0 is isomorphic to (X, L).Recall that H 1 (L) = H 1 (π * L) and that, by the projection formula, the latter equals ) also vanishes because of the ampleness of −K X , hence H 1 (L) = 0.Then, shrinking Z if necessary, by semicontinuity H 1 (L z ) = 0 for all z ∈ Z.Then, shrinking Z if necessary, the push-out (X → Z) * (L ) is a free sheaf on Z and the formation of (X → Z) * commute with base extension (see [Gro61, Chapitre III, §7] or [MFK94, Chapter 0, §5]).Thus h 0 (L z ) is the same for all z ∈ Z, so we obtain from L a Z-morphism Φ : X −→ P N Z such that Φ 0 = ϕ, i.e., Φ is a deformation of ϕ.Now we apply Theorem 2.5 to Φ.By Proposition 2.9 (3), the line bundle E −2 satisfies condition (2) of Theorem 2.5.Then Theorem 2.5 implies that, for all z ∈ Z, the morphism Φ z has degree 2 onto its image, which is a deformation of i(Y ) in P N .Since any deformation of i(Y ) is a variety of minimal degree, the morphism Φ z is induced by the complete linear series H 0 (L z ) and L m z = L m , we conclude that (X z , L z ) is a hyperelliptic polarized variety.
Theorem 2.11.Let X and Y be as in notations 1.2 (2).Let (X, L) a hyperelliptic Fano-K3 variety of dimension m ≥ 3 and let ϕ be the morphism induced by the complete linear series |L|.If Y is a hyperquadric, then we have: (1) The morphism ϕ and the polarized variety (X, L) are unobstructed; (2) A general deformation of ϕ is an embedding.Likewise, a general deformation of (X, L) is nonhyperelliptic but its complete linear series induces an embedding.The images of those embeddings are quartic hypersurfaces of P m+1 . Proof.
we consider the sequence (2.11.1) Note that, by the Kodaira vanishing theorem, ) and H 1 (N π ) vanish.From (1.3.2) and from the vanishing of H 1 (O Y ) and H 1 (O Y (2h)), it follows that H 1 (N ϕ ) = 0, so ϕ is unobstructed.Now let Σ L be the sheaf of first order differential operators in L and let be the Atiyah extension of L (see [HM98,p. 96] or [Ser06, (3.30)]).By [Ser06, Theorem 3.3.11(ii)], if H 2 (Σ L ) = 0, then (X, L) is unobstructed.To prove the vanishing of H 2 (Σ L ), by exact sequence (2.11.2) and the vanishing of H 2 (O X ), we only need to show H 2 (T X ) = 0.For that we take cohomology on the exact sequence By the vanishing of H 1 (N ϕ ), the projection formula and the Leray's spectral sequence, it is enough to show the vanishings of H 2 (T For that we use the restriction to i(Y ) of the Euler sequence of the tangent bundle of P N , so the wanted vanishings will follow from the vanishings of All those vanishings follow from the vanishing of higher cohomology of line bundles of P m+1 (recall that m ≥ 3).Therefore H 2 (T X ) = 0 and so H 2 (Σ L ) = 0 and (X, L) is unobstructed.This completes the proof of (1).
To prove (2) we are going to use [GGP13a, Theorem 1.5].By Corollary 2.4, Hom(I /I 2 , E ) has dimension 1.In fact, since I /I 2 = E , any nonzero element of Hom(I /I 2 , E ) is an isomorphism.Recall the homomorphism Ψ 2 , introduced in Proposition 1.3, and cohomology sequence (1.3.2).Since H 1 (N π ) vanishes, the homomorphism Ψ 2 is surjective.Then, given a nonzero µ ∈ Hom(I /I 2 , E ), there exists ν in H 0 (N ϕ ) such that Ψ 2 (ν) = µ.Since there is an algebraic formally semiuniversal deformation of ϕ with base Z and ϕ is unobstructed, we see that all the hypotheses of [GGP13a, Theorem 1.5] are satisfied, so [GGP13a, Theorem 1.5] and its proof imply that there exists a smooth, algebraic curve T in Z, passing through 0 and tangent to the tangent vector v of Z corresponding to ν (recall that H 0 (N ϕ ) is isomorphic to the tangent space of Z at 0) and a deformation Φ T of ϕ over T such that Φ 0 = ϕ and Φ t is an embedding for all t ∈ T, t = 0. Taking the pullback by Φ T of O P N T (1) gives us a deformation (X T , L T ) of (X, L).Since for all t ∈ T {0}, Φ t is an embedding, then (X t , L t ) is non hyperelliptic.Let Z ′ be the base of an algebraic formally semiuniversal deformation (X , L ) of (X, L).Then, after shrinking T if necessary, (X T , L T ) is obtained from (X , L ) by etale base change, so there is a point z ′ ∈ Z ′ such that (X z ′ , L z ′ ) is nonhyperelliptic but L z ′ is very ample.Since very ampleness is an open condition, this finishes the proof of (2).
Example 2.12.Let m ≥ 3. Given a hyperquadric i(Y ) in P m+1 , the dualizing sheaf of the (unique) ribbon structure Y on i(Y ) embedded in P m+1 is ω Y = O Y (−m + 2).In addition, since Y is a divisor in P m+1 , it is arithmetically Cohen-Macaulay.Thus, we can say that Y is a Fano-K3 ribbon.The ribbon Y can be smoothed inside P m+1 , i.e., there exists an embedded deformation of Y whose general fiber is a smooth (Fano-K3) variety in P m+1 .This follows from the general theory developed in [GGP13a, §1] but, in this case, can be achieved in an ad-hoc, more straight forward fashion, by deforming Y in the linear system of degree 4 divisors on P m+1 .Remark 2.13.In [Isk77, (7.2)] Iskovskih classified hyperelliptic (X, L) Fano-K3 threefolds.They are of five types and their sectional genera are g = 2, 3, 4, 5 and 7 (compare with Proposition 2.9).Iskovskih also proved (see [Deb13, Theorem 3]) that, if X is a prime Fano threefold of genus g and g ≥ 4, then −K X is very ample.Proposition 2.9 and Theorem 2.10 say that hyperelliptic Fano-K3 threefolds with g ≥ 4 deform only to hyperelliptic Fano-K3 threefolds, so for each genus 4, 5 and 7 there are at least two kinds of Fano-K3 threefolds, hyperelliptic and anticanonically embedded, that do not deform to each other.
3. Deformations of hyperelliptic polarized varieties with L m = 2g.Notation 3.1.If Y is a projective bundle embedded by i as a rational normal scroll, then (i) E is the very ample vector bundle such that Y = P(E) and i is induced by . ., a n ) and N = a + n − 1); (iii) H will be the divisor corresponding to O P(E) (1); (iv) p will be the structure morphism from Y to P 1 and F will be the fiber of p. Proposition 3.2.Let X, Y , E , i and ϕ be as in Notation 1.2, 2.1, (2.2.6) and 3.1.Assume (X, L) is a hyperelliptic variety of dimension m ≥ 2 with L m = 2g and ϕ is induced by |L|.Then i(Y ) is a variety of minimal degree and Y , E −2 and B are as follows: (1) Proof.The result follows from [Fuj83, Proposition 5.18 (2), (6.7)].
Proof.We first see that ϕ is induced by the complete linear series |L|.This is equivalent to (3.3.1)H 0 (E (1)) = 0, so we check this in each case of Proposition 3.2 (2a), (2c), (2d) or (2e).Indeed, if Y and B are as in Proposition 3.2 (2a), (2c), (2d) or (2e), then E (1) equals respectively, and none of these line bundles have global sections.Now we prove Hom(I /I 2 , E ) and Ext 1 (Ω Y , E are isomorphic.this we will prove that the connecting homomorphism γ of (2.4.1) is an isomorphism in this case and then we will compute the dimension of Ext 1 (Ω Y , E ), i.e., h 1 (T Y ⊗ E ).To prove γ is an isomorphism, we will show For that, we consider the restriction to i(Y ) of the Euler sequence of P N and get ) and H 0 (O Y (−F )) and both cohomology groups vanish.This completes the proof of γ being an isomorphism.Now we prove (1).We compute the dimension Ext 1 (Ω Y , E ), which is isomorphic to H 1 (T Y ⊗ E ).In order to compute the dimension H 1 (T Y ⊗ E ) we consider the exact sequence (2.2.7).If Y and B are as in Proposition 3.2 (2a), then we consider the dual of the relative Euler sequence , which is the same as Then, if follows from all the above that the dimension of Hom(I /I 2 , E ) is g if Y and B are as in Proposition 3.2 (2a).Now we compute h 1 (T Y ⊗ E ) when Y and B are as in Proposition 3.2 (2c), (2d) or (2e).Since in this case Y has dimension 2, with notation (2.2.6), exact sequence (2.2.7) becomes where e = a 1 − a 2 , i.e., e = 0 in case (2c), e = 1 in case (2d), e = 2 in case (2e).Tensoring with E , (3.3.5)becomes ) and hence, the dimension of Hom(I /I 2 , E ), is 2 if Y and B are as in Proposition 3.2 (2c), (2d) or (2e).Now we prove (2).By the projection formula and by the Leray spectral sequence, If Y and B are as in Proposition 3.2 (2c), (2d) or (2e), then Using the projection formula and the Leray spectral sequence, we see that h 1 (O Y (B)) is the sum of the dimensions of the first cohomology group of certain line bundles on P 1 .If Y and B are as in Proposition 3.2 (2a), then the smallest among the degrees of those line bundles is 2a n + 2, which is greater than or equal to 4; if Y and B are as in Proposition 3.2 (2c), then the smallest among the degrees of those line bundles is 2; and if Y and B are as in Proposition 3.2 (2d), then the smallest among the degrees of those line bundles is 0. In all these three cases the first cohomology groups of the line bundles vanishes, so H 1 (O Y (B)) vanishes and so do H 1 (O B (B)) and H 1 (N π ).If Y and B are as in Proposition 3.2 (2e), then Now we prove (4).We use cohomology sequence (1.3.2).First we will prove the vanishing of Then, for H 1 (N Y,P N ⊗ E ) to vanish, it suffices to have the vanishings of H 1 (T The first cohomology group is isomorphic to H 2 (O Y (−H + F )) and this group vanishes.For the vanishings of the second cohomology group it suffices to show the vanishings of H 2 (O Y (−(a i + 1)F )) for all i = 1, . . ., n; all of these vanishings occur.Finally , which also vanishes.Now we argue for Y and B as in Proposition 3.2 (2c), (2d) or (2e).To prove the vanishing of H 2 (T Y ⊗ E ) in these cases we look at the exact sequence (3.3.6).Note that H 2 (O Y (−F )) = 0. On the other hand, by Serre duality Because H 1 (π * N i(Y ),P N ) vanishes, H 1 (N π ) surjects onto H 1 (N ϕ ).If Y and B are as in Proposition 3.2 (2a), (2c) or (2d), then H 1 (N ϕ ) = 0 because H 1 (N π ) = 0.If Y and B are as in Proposition 3.2 (2e), then h 1 (N ϕ ) is 0 or 1 because h 1 (N π ) = 1.
so ϕ is induced by the complete linear series |L|.In view of (2.4.1), in order to prove the vanishing of Hom(I /I 2 , E ) it will suffice to prove the vanishing of H 0 (T Arguing as in the proof of Proposition 3.3 it suffices to prove H 0 (E (1)) and H 1 (E ) vanish.We have already seen H 0 (E (1)) = 0.For H 1 (E ) = 0, since B ∼ 4H − 4F , we have and We use exact sequences (2.2.7) and (3.3.4).Then it suffices to prove the vanishings of The vanishing of the second one follows from H 1 (O Y (−H + F )) = 0.The vanishing of the third one follows because Remark 3.5.
(2) It follows from Proposition 3.4 and [BE95, Corollary 1.4] that there do not exist nonsplit ribbons supported on Y and with conormal bundle E , where Y and E are as in Proposition 3.2 (2b).
Theorem 3.6.Let X and Y be as in notations 1.2 (2).Let (X, L) be a hyperelliptic variety such that L m = 2g and let ϕ be the morphism induced by the complete linear series |L|.If Y and B are as in (2a), (2c) or (2d) of Proposition 3.2, then we have: (1) The morphism ϕ and the polarized variety (X, L) are unobstructed.
(2) A general deformation of ϕ is a finite morphism of degree 1 onto its image.Likewise, a general deformation of (X, L) is nonhyperelliptic but its complete linear series induces a finite morphism of degree 1 onto its image.
Proof.By Proposition 3.3 (2), H 2 (O X ) = 0. Thus, by Proposition 1.5, there exist algebraic formally semiuniversal deformations of ϕ and (X, L).It follows from Proposition 3.3 (4) that ϕ is unobstructed.To prove that (X, L) is also unobstructed, we see that H 2 (Σ L ) = 0.For this we use exact sequence (2.11.2).Since H 2 (O X ) = 0, we only need to see that H 2 (T X ) vanishes.Since H 1 (N ϕ ) vanishes, taking cohomology on exact sequence (2.11.3), we see that, by the projection formula and the Leray's spectral sequence, it is enough to prove the vanishing of H 2 (T P N | Y ) and Taking cohomology on the restriction to Y of the Euler sequence of the tangent bundle of P N , since ) and H 3 (E ) vanish, we obtain the desired vanishings.Then H 2 (Σ L ) = 0 and by [Ser06, Theorem 3.3.11(ii)], the polarized variety (X, L) is unobstructed.This completes the proof of (1).
To prove (2) we are going to use [GGP10, Theorem 1.4].By Proposition 3.3, Hom(I /I 2 , E ) = 0.If µ ∈ Hom(I /I 2 , E ), µ = 0, then µ is a homomorphism of rank 1 because E is a line bundle.By Proposition 3.3, H 1 (N π ) = 0, so if follows from the long exact sequence of cohomology (1.3.2) that the map Ψ 2 of Proposition 1.3 is surjective.Then, given a nonzero µ ∈ Hom(I /I 2 , E ), there exists ν in H 0 (N ϕ ) such that Ψ 2 (ν) = µ.Since there is an algebraic formally semiuniversal deformation of ϕ with base Z and ϕ is unobstructed, we see that all the hypotheses of [GGP10, Theorem 1.4] are satisfied, so [GGP10, Theorem 1.4] and its proof imply that there exists a smooth, algebraic curve T in Z, passing through 0 and tangent to the tangent vector v of Z corresponding to ν and a deformation Φ T of ϕ over T such that Φ 0 = ϕ and Φ t is a finite morphism of degree 1 onto its image, for all t ∈ T, t = 0. Taking the pullback by Φ T of O P N T (1) gives us a deformation (X T , L T ) of (X, L), such that for all t ∈ T {0}, (X t , L t ) is nonhyperelliptic.Let Z ′ be the base of an algebraic formally semiuniversal deformation (X , L ) of (X, L).Then, after shrinking T if necessary, (X T , L T ) is obtained from (X , L ) by etale base change, so there is a point induces a finite morphism of degree 1 onto its image.Since this is an open condition, this finishes the proof of (2).
Remark 3.7.The proof of Theorem 3.6 yields a more precise statement for the deformations of ϕ.Indeed, note that the element ν in the proof is a general element of H 0 (N ϕ ).In fact, ν belongs to the complement U of a linear subespace of H 0 (N ϕ ), namely, the kernel of Ψ 2 , which has codimension h 0 (N i(Y ),P N ⊗ E ) in H 0 (N ϕ ) (for the value of h 0 (N i(Y ),P N ⊗ E ), see Proposition 3.3 (1)).Then, for any ν ∈ U , there exists a smooth, algebraic curve T in Z, passing through 0 and tangent to ν and a deformation Φ T of ϕ over T such that Φ 0 = ϕ and, for all t ∈ T, t = 0, the morphism Φ t , we have is finite of degree 1 onto its image.
Consider the commutative diagram (3.7.2) where the square arises from [Gon06, Proposition 3.7 (1)]), the homomorphism ζ is the one in (3.7.1), the homomorphism ζ ′ arises when taking cohomology in exact sequence (2.11.2) and ζ, ζ ′ and ζ ′′ are the homomorphisms that forget, in the obvious way, part of the information of the first order infinitesimal deformations of ϕ and (X, L).
Then, since there exist elements in H 0 (N ϕ ) kerΨ 2 , we have that ζ(kerΨ 2 ) is a proper vector subspace of Then, for any ̟ ∈ U ′ , there exists a smooth, algebraic curve T ′ in Z ′ , passing through 0 and tangent to ̟ and a deformation (X T ′ , L T ′ ) of (X, L) over T ′ such that (X 0 , L 0 ) = (X, L) and, for all t ′ ∈ T ′ , t ′ = 0, we have that |L t ′ | induces a finite of degree 1 onto its image.
An analogous remark can be made in relation to Theorem 2.11.Theorem 3.8.Let X and Y be as in notations 1.2 (2).Let (X, L) be a hyperelliptic variety such that L m = 2g and let ϕ be the morphism induced by the complete linear series |L|.If Y and B are as in (2e) of Proposition 3.2, then a general deformation of ϕ is a finite morphism of degree 1 onto its image and a general deformation of (X, L) is nonhyperelliptic but its complete linear series induces a finite morphism of degree 1 onto its image.
Proof.By Proposition 3.3 (2), H 2 (O X ) = 0, so, by Proposition 1.5, there exist algebraic formally semiuniversal deformations of ϕ and (X, L).We look now at long exact sequence of cohomology (1.3.2).Recall that in this case h 1 (N π ) = 1 and that we saw in the proof of Proposition 3.3 that η is surjective, so that h 1 (N ϕ ) is 0 or 1.Let Z be the base of an algebraic formally semiuniversal deformation of ϕ.We have (see [Ser06, Corollary 2.2.11]) We argue first for h 1 (N ϕ ) = 1.Then η is an isomorphism, so Ψ 2 is surjective.If dimZ = h 0 (N ϕ ), then ϕ is unobstructed, so we can argue as in the proof of Theorem 3.6 to find deformations Φ T and (X T , L T ) over a smooth algebraic curve T in Z. Now, If h 1 (N ϕ ) = 1 and dimZ = h 0 (N ϕ ) − 1, then the tangent cone of Z at 0 has codimension 1 in H 0 (N ϕ ).By Proposition 3.3, the dimension of Hom(I /I 2 , E ) = 2, so kerΨ 2 has codimension 2 in H 0 (N ϕ ).Thus the tangent cone of Z is not contained in kerΨ 2 and there exists an element ν in the tangent cone of Z such that Ψ 2 (ν) = µ = 0. Since E is a line bundle, µ is a homomorphism of rank 1.Since ν is in the tangent cone of Z, there exists an algebraic curve T , 0 ∈ T such that ν is tangent to T .Desingularizing T necessary we obtain a flat family of morphisms satisfying the hypotheses of [GGP10, Proposition 1.3], so there exists a deformation Φ T of ϕ over a smooth algebraic curve T such that the fiber Φ t over any t ∈ T {0} is a morphism to P N , which is finite and of degree 1 onto its image.Now, taking the pullback by Φ T of O P N T (1) gives us a deformation (X T , L T ) of (X, L).Since for all t ∈ T {0} is of degree 1, (X t , L t ) is non hyperelliptic.Let Z ′ the base of an algebraic formally semiuniversal deformation of (X, L).Since Φ T and (X T , L T ) are obtained, by etale base change, from the algebraic formally semiuniversal deformations over, respectively, Z and Z ′ , we may conclude that there are z ∈ Z and z ′ ∈ Z ′ such that Φ z and the morphism induced by |L z ′ | are finite and of degree 1 onto its image.We argue now for h 1 (N ϕ ) = 0.In this case ϕ is unobstructed.The kernel of the homomorphism ǫ of (1.3.2) has codimension 1 in Hom(I /I 2 , O Y ) ⊕ Hom(I /I 2 , E ).By Proposition 3.3, the linear subspace W = Hom(I /I 2 , O Y ) × {0} has codimension 2 in Hom(I /I 2 , O Y ) ⊕ Hom(I /I 2 , E ).Then the kernel of ǫ is not contained in W . Thus there exist ν ∈ H 0 (N ϕ ) such that Ψ 2 (ν) = 0. Then we can argue as in the proof of Theorem 3.6 to find deformations Φ T and (X T , L T ) over a smooth algebraic curve T in Z.In all cases, there are z ∈ Z and z ′ ∈ Z ′ such that Φ z and the morphism induced by |L z ′ | are finite and of degree 1 onto its image.Thus we may conclude the proof of (2) using that being a finite morphism of degree 1 is an open condition.Question 3.9.As seen in the proof of Theorem 3.8, since we do not know if h 1 (N ϕ ) = 0 (see Proposition 3.3), it is not clear whether ϕ is unobstructed or not.It would be interesting to settle the question one way or the other.
Theorem 3.10.Let X and Y be as in notations 1.2 (2).Let (X, L) a hyperelliptic variety such that L m = 2g and let ϕ be the morphism induced by |L|.If Y and B are as in Proposition 3.2 (1) or (2b), then ϕ is unobstructed and any deformation of (X, L) is hyperelliptic.
Proof.If Y = P m , although the claim about deformations follows from Remark 0.3 (3), the complete result, including the unobstructedness of ϕ, follows trivially from Theorem 1.4.
Let now Y and B be as in Proposition 3.2 (2b).Let Z be a smooth, algebraic variety with a distinguished point 0 ∈ Z.Let (X , L ) be a flat family over Z such that the fiber (X 0 , L 0 ) over 0 is isomorphic to (X, L).Since H 1 (O Y (1)) = 0 and H 1 (E (1)) = 0, because H 1 (E (1)) = H 1 (O Y (−H + 2F )), we have H 1 (L) = H 1 (π * L) = 0, so, arguing as in the proof of Theorem 2.11 and shrinking Z if necessary, we obtain from Φ * L a morphism Φ : X −→ P N Z such that Φ 0 = ϕ, i.e., Φ is a deformation of ϕ.We apply Theorem 1.4 to Φ. Proposition 3.4 tells Hom(I /I 2 , E ) = 0, so Ψ 2 = 0.As already mentioned, H 1 (O Y ) = 0, and H 2 (O Y ) = 0 also.In addition, and it is well known that Y is unobstructed in projective space.Then all the hypotheses of Theorem 1.4 are satisfied so ϕ is unobstructed and, for all z ∈ Z, the morphism Φ z has degree 2 onto its image, which is a deformation of i(Y ) in P N .Since any deformation of i(Y ) is variety of minimal degree, the morphism Φ z is induced by the complete linear series H 0 (L z ) and L m z = L m , we conclude that (X z , L z ) is a hyperelliptic polarized variety.

Deformations of generalized hyperelliptic polarized varieties.
In this section we continue the study of deformations of certain generalized hyperelliptic polarized varieties, looking this time at Calabi-Yau and general type varieties.If (X, L) is either a hyperelliptic polarized Calabi-Yau variety of dimension m, m ≥ 3, or a hyperelliptic variety of general type, canonically polarized, of dimension m, m ≥ 2, then, by adjunction, L m < 2g − 2 (g is the sectional genus of (X, L)) and h 0 (L t ) is constant for any deformation of (X, L) (because H 1 (L) = 0, by the Kodaira vanishing theorem, if X is Calabi-Yau and because the invariance by deformation of the geometric genus otherwise); thus by Remark 0.3 (1), all deformations of (X, L) are hyperelliptic.Then in this context it is interesting to see if there are further reasons for this phenomenon, namely, all deformations of a hyperelliptic polarized variety are hyperelliptic, to happen.That is the case, since all the deformations of generalized hyperelliptic Calabi-Yau and general type varieties are generalized hyperelliptic, as we will see in Theorems 4.5 and 4.14.First, we recall the definition of Calabi-Yau variety: Definition 4.1.Let X be a smooth variety of dimension m, m ≥ 3. We say that X is a Calabi-Yau variety if (1) ω X = O X ; and (2) Lemma 4.2.Let X, Y , π and E be as in notations 1.2 and 2.1 with By the projection formula and the Leray's spectral sequence, Proposition 4.3.Let X, Y , π and E be as in notations 1.2 and 2.1 and assume by the projection formula and the Leray spectral sequence (recall m ≥ 3).In addition, Theorem 4.5.Let X, Y , π and ϕ be as in notations 1.2 and 2.1.Let X be a Calabi-Yau variety of dimension m, m ≥ 3 and let the morphism ϕ be induced by a complete linear series (i.e., H 0 (E (1)) = 0).Then any deformation of ϕ is a finite morphism of degree 2 onto its image, which is a deformation of i(Y ) in P N .
Proof.Let Z be a smooth algebraic variety with a distinguished point 0 and let (X , Φ) be a flat family over Z, with (X 0 , Φ 0 ) = (X, ϕ).Arguing as in the proof of Proposition 2.6 we show the existence of a deformation (X , Π) of (X, π), where Π is finite, surjective and of degree 2, a deformation (Y , i) of (Y, i) and a deformation Φ ′ of ϕ such Φ ′ = i • π.We now compare Φ and Φ ′ .Let X be a general member of |L|.By adjunction X is of general type, L| X = K X and, since H 1 (O X ) = 0, the complete linear series |L| restricts to the complete linear series |L| X |.We apply Lemma 1.7 to (X, L| X ).Then Φ and Φ ′ are the same when restricted to X, so Φ is generically of degree 2 onto its image.Since π is finite, shrinking Z if necessary, so is Φ onto its image.Therefore, Φ z is finite of degree 2 onto its image, for all z ∈ Z.
Remark 4.6.Assume X, Y and ϕ are as in Theorem 4.5 and that, if Y is a projective bundle over . ., e m−1 in Notation 2.2.6).Then we can argue as in the proof of Theorem 2.5 and, using Theorem 1.4, conclude that ϕ is unobstructed.
Although we already observed at the beginning of the section that next result follows from Remark 0.3 (1), it is interesting to see how it is deduced from the broader setting of Theorem 4.5: Corollary 4.7.Let X be a Calabi-Yau variety (of dimension m, m ≥ 3) and let L be a polarization on X.If (X, L) is hyperelliptic and the image of X by the morphism induced by |L| is smooth, then any deformation of (X, L) is hyperelliptic.
Example 4.8.If (X, L) is a polarized Calabi-Yau threefold with L 3 = 8 and h 0 (L) = 7, then, by adjunction and Clifford's theorem, (X, L) is hyperelliptic.Thus, if the image i(Y ) of the morphism induced by |L| is smooth, then i(Y ) is a rational normal scroll S(1, 1, 2) of P 6 .This suggests that polarized Calabi-Yau threefolds (X, L) with L 3 = 8 and h 0 (L) = 7 are parametererized by an irreducible scheme whose general point corresponds to a hyperelliptic polarized Calabi-Yau threefold.
Corollary 4.7 shows that a hyperelliptic polarized Calabi-Yau threefold (X, L) only deforms to hyperelliptic polarized Calabi-Yau threefolds.If the image of the morphism induced by |L| is a smooth rational normal scroll, then X is fibered by K3 surfaces (see [GP98, Proposition 1.6]); thus, in this case, any deformation of X carries also the K3 fibration.This motivates the following question.
Question 4.9.Let X be a Calabi-Yau threefold.If X carries a K3 fibration, does any deformation of X carry the K3 fibration?It is tempting but, probably, too optimistic, to expect that such a Calabi-Yau threefold X carries a K3 fibration if and only if there exists a polarization L on X so that (X, L) is hyperelliptic and the image of the morphism induced by |L| is a smooth rational normal scroll.If this expectation were true then one would easily show that the answer to our question is affirmative.Now we study the deformations of generalized hyperelliptic polarized varieties of general type: Proposition 4.10.Let X, Y , π and E be as in notations 1.2 and 2.1 and assume X is a minimal variety of general type (i.e., with K X ample) of dimension m, m ≥ 2. Then H 1 (T Y ⊗ E ) = 0.If Y is a (smooth) hyperquadric and m ≥ 3, the vanishing of H 1 (E ) follows from sequence (2.2.5) and the vanishing of intermediate cohomology in P N .If Y is a projective bundle on P 1 , then h 1 (E ) = h m−1 (E ′ ), with E ′ ample.Then the vanishing follows by the Leray's spectral sequence and the projection formula.
We say that a ribbon of dimension greater than or equal to 2 is a minimal ribbon of general type if its dualizing sheaf is ample.From Proposition 4.10 we deduce the non existence of nonsplit minimal ribbon of general type supported on Y as in Notation 2.1: Corollary 4.13.Let X, Y , ϕ, E and I be as in notations 1.2 and 2.1.Assume X is a minimal variety of general type (i.e., with K X ample) of dimension m, m ≥ 2 and ϕ is induced by a complete linear series (i.e., H 0 (E (1)) = 0).Then Hom(I /I 2 , E ) = 0.
Proof.The result follows from exact sequence (2.4.1), the vanishing of H 0 (E (1)) (this is because ϕ factors through π and ϕ is induced by a complete linear series), (4.11.1) and Proposition 4.10.
Theorem 4.14.Let X, Y and ϕ be as in notations 1.2 and 2.1.Assume X is a minimal variety of general type of dimension m, m ≥ 2 and ϕ is induced by a complete linear series.If Y is a projective bundle over P 1 , assume furthermore B is base-point-free.Then ϕ is unobstructed and any deformation of ϕ is a finite morphism of degree 2 onto its image, which is a deformation of i(Y ) in P N .
Proof.As seen in the proof of Theorem 2.5, hypotheses (1), (3) and (4) of Theorem 1.4 are satisfied.If Y is P m or a hyperquadric, then H 1 (E −2 ) = 0 because of the vanishing of cohomology in projective space.If Y is a projective bundle over P 1 , then B being base-point-free implies β ≥ αe m−1 , with α, β and e m−1 as in (2.2.6), so H 1 (E −2 ) = 0. Thus hypothesis (2) of Theorem 1.4 is also satisfied.Finally, hypothesis (5) of Theorem 1.4 follows from Corollary 4.13.Y is ample and free, and so is E −2 .The cases dealt with in the next proposition have already been covered by Theorem 4.14 except maybe if B is not base-point-free in the statement of Theorem 4.14; that is why we include it here.The proof is the same as the proof of Proposition 2.6.Proposition 4.16.Let X, Y and ϕ be as in notations 1.2 and 2.1.Let X be a minimal variety of general type and let the morphism ϕ from X to P N be induced by a complete linear series |L|, where L = lK X for some l ∈ N. Let Y be a projective bundle over P 1 .Then any deformation of ϕ is a finite morphism of degree 2 onto its image, which is a deformation of i(Y ) in P N .Remark 4.17.If X, Y and ϕ are as in Theorem 4.14 or Proposition 4.16 and Y is a projective bundle over P 1 , but neither is B base-point-free in the statement of Theorem 4.14 nor the condition on L in Proposition 4.16 hold, still something can be said about the deformations of ϕ.Indeed, it follows from the same reasons argued in Remark 2.7 that, for any deformation X of X, there exists a deformation of (X , Φ) of (X, ϕ) which is finite and of degree 2 onto its image.
Ā) is surjective.Therefore the argument above shows that the map Def (Y,B) ( A) → Def (Y,B) ( Ā) is surjective, which shows that the functor Def (Y,B) and the forgetful map Def (Y,B) → H P N Y are smooth.Therefore we have an exact sequence on tangent spaces (1.4.1) then the result follows from (2.4.1) and Proposition 2.2.If Y is a hyperquadric and E = O Y (−2h), then H 0 (E (1)) = 0 implies L = π * O Y (h) and i in the embedding of Y as a hyperquadric in P m+1 .Then I /I 2 ≃ O Y (−2h), so Hom(I /I 2 , E ) is isomorphic to H 0 (O Y ) in this case.
and Ext 2 (O Y , E ) all vanish.On the one hand, Hom(O N +1 i(Y ) (−1), E ) vanishes because of (3.3.1) and Ext 1 (O N +1 i(Y ) (−1), E ) vanishes because none of the line bundles of (3.3.2) has higher cohomology.On the other hand, if Y and B are as in Proposition 3.2 (2a), then Ext 1 (O Y , E ) and Ext 2 (O Y , E ) are isomorphic to H 1 (O Y (−H − F )) and H 2 (O Y (−H − F )) and both also vanish.If Y is as in the other three cases, Ext 1 (O Y , E ) and Ext 2 (O Y , E ) are, respectively, Serre dual of H 1 (O Y (−F ) Proposition 3.4.Let X, Y , E , i, π and ϕ be as in Notation 1.2, 2.1 and 3.1.Let L = ϕ * O Y (1).If Y and B are as in Proposition 3.2 (2b), then ϕ is induced the complete linear series |L| and Hom(I /I 2 so E ′ is ample.Any ample line bundle on Y is very ample, soE ′ is very ample.Then E = ω Y ⊗ E ′−1 and in the proofs of [GGP16, Propositions 1.4, 1.5, 1.6] (see also [GGP13b, Proposition 1.7]) we showed H 1 (T Y ⊗ ω Y ⊗ E ′−1 ) = 0.Remark 4.11.Under the hypotheses of Proposition 4.10, the variety X is regular.Indeed, π * (O X ) = O Y ⊕ E .We have H 1 (O Y ) = 0. We see that (4.11.1)H 1 (E ) = 0 also.If Y = P m , then H 1 (E ) vanishes because of the vanishing of intermediate cohomology in P m .

Corollary 4. 12 .
Let Y be as in Notation 2.1.There are no nonsplit minimal ribbon of general type on Y .Proof.Let Y be a minimal ribbon of general type supported on Y and let E be the conormal bundle of Y in Y .Since ω Y is ample, by [GGP08b, Lemma 1.4], so is ω Y ⊗ E −1 and, in fact, very ample.Then Ext 1 (Ω Y , E ) = H 1 (T Y ⊗ E ) = 0 by [GGP16, Propositions 1.4, 1.5, 1.6] so the result follows from [BE95, Corollary 1.4].

Remark 4. 15 .
The divisor B being base-point-free in the statement of Theorem 4.14 is not a very restrictive condition.Indeed,E −2 = ω −2 Y ⊗ E ′ , where E ′ is ample.If Y is "balanced", e.g. if e 1 = • • • = e m−1 = 0, then ω −2 be as in Notation 2.1.There are no nonsplit Fano ribbons Y on Y except if Y is a hyperquadric in P m+1 and Y is the unique ribbon in P m+1 supported on Y .Proof.Let Y be a Fano ribbon supported on Y and let E be the conormal bundle of Proposition 3.3.Let X, Y , E , i, π and ϕ be as in Notation 1.2, 2.1 and 3.1.LetL = ϕ * O Y (1).Assume furthermore that Y and B are as in Proposition 3.2 (2a), (2c), (2d) or (2e).Then ϕ is induced by the complete linear series |L|, the groups Hom(I /I 2 , E ) and Ext 1 (Ω Y , E ) are isomorphic, and the following holds:(1) If Y and B are as in Proposition 3.2 (2a), then the dimension of Hom(I /I 2 , E ) is g; otherwise, the dimension of Hom(I has been already proved (see (3.3.3)).For the vanishing of H 2 (T Y ⊗ E ) we use (2.2.7), (3.3.4) and (3.3.6).We argue first for Y and B as in Proposition 3.2 (2a).It suffices to prove the vanishings of H 2 0 by the vanishing of the intermediate cohomology of line bundles in projective space.In view of the Euler sequence for the tangent bundle of P m , this implies the vanishing ofH 1 (T Y ⊗ E ).If Y is a hyperquadric, then it follows from Lemma 4.2 that E = O Y (−mh).In view of exact sequence for the normal bundle of Y in P m+1 , we need to check the vanishings of H 0 (O Y ((−m+2)h)) and H 1 (T P m+1 | Y ⊗ E ).The first happens because m ≥ 3. The second follows from the Euler sequence for the tangent bundle of P m+1 restricted to Y , since H 1 (E (1)) = 0 by the Kodaira vanishing theorem andh 2 (E ) = h m−2 (O Y ) = 0.If Y is a projective bundle over P 1 , we use Notation 2.2.6 and exact sequences (2.2.7) and (2.2.8).Then it is enough to prove the vanishings of H 1 (p * T P 1 also by the projection formula and the Leray spectral sequence.The vanishings ofH 1 (E ⊗ O Y (H 0 + e 1 F )), . .., H 1 (E ⊗ O Y (H 0 + e m+1 F )) are argued analogously.Finally, h 2 (E ) = h m−2 (O Y ) = 0.In analogy with Definition 4.1, by a Calabi-Yau ribbon we mean a ribbon of dimension bigger than 2, with trivial dualizing sheaf and such that the intermediate cohomology of its structure sheaf vanishes.We now deduce from Proposition 4.3 the non existence of nonsplit Calabi-Yau ribbons: Proof.Let Y be a ribbon supported on Y and let E be the conormal bundle of Y in Y .The same argument of the proof of [GGP08b, Proposition 1.5] implies that Y is Calabi-Yau if and only if E = ω Y .Then the result follows from Proposition 4.3 and [BE95, Corollary 1.4].