On the extendability of projective varieties: a survey
aa r X i v : . [ m a t h . AG ] F e b ON THE EXTENDABILITY OF PROJECTIVE VARIETIES: A SURVEY.(with an appendix by Thomas Dedieu)
ANGELO FELICE LOPEZ
Dedicated to Ciro Ciliberto on the occasion of his 70th birthday.What I learned from him, both from a mathematical and a human point of view, is invaluable.
Abstract.
We give a survey of the incredibly beautiful amount of geometry involved with theproblem of realizing a projective variety as hyperplane section of another variety. Introduction
At the beginning of the 90’s I attended a seminar talk by Ciro Ciliberto at the Universityof Milan. The main upshot, beautifully conveyed by the speaker, was to make a connectionbetween two apparently unrelated theorems that had appeared recently and to investigate theconsequences of this awareness.The story that followed in the subsequent years, and the research still going on today, will berecollected in this survey, through the magnifying lens of my views on the subject, very muchinfluenced by that talk.To state the theorems in question we start with some definitions and notation.
Definition 1.1.
Let X ⊂ P r be an irreducible nondegenerate variety of codimension at least1. Let k ≥ X is k -extendable if there exists a variety Y ⊂ P r + k different from a cone over X , with dim Y = dim X + k and having X as a section by an r -dimensional linear space. We say that X is precisely k -extendable if it is k -extendable but not( k + 1)-extendable. The variety Y is called a k -extension of X . We say that X is extendable ifit is 1-extendable.It is of course a basic question in projective geometry to understand when a variety is extend-able and if so, how much. Even when a variety is extendable, it can be extendable in differentways and with chains of extensions of different length (see the example in [Z]).On the other hand, unless one has a very good knowledge of the variety and its very ampleline bundles, it is usually a difficult but fascinating problem.A general remark to be made is that two general approaches have been predominant. Re-searchers have often taken an ”optimistic” or a ”pessimistic” approach. The first one being: fix X and try to prove that it is not extendable or to classify all of its extensions. The second one:start with Y , study its hyperplane sections and try to prove that such a Y cannot exist. Wewill give examples of both.To give a measure in this direction let us define Definition 1.2.
Let X ⊂ P r be a smooth irreducible nondegenerate variety of codimension atleast 1 with normal bundle N X . We set α ( X ) = h ( N X ( − − r − . The first result, due to F.L. Zak [Z] and S. L’vovsky [Lv] (see also [Bad2, BS]), is as follows. * Research partially supported by PRIN “Advances in Moduli Theory and Birational Classification” andGNSAGA-INdAM.
Mathematics Subject Classification : Primary 14N05, 14J40. Secondary 14J28, 14H51. heorem 1.3. (Zak-L’vovsky’s theorem) Let X ⊂ P r be a smooth irreducible nondegenerate variety of codimension at least andsuppose that X is not a quadric. If α ( X ) ≤ , then X is not extendable.Given an integer k ≥ , suppose that either (i) α ( X ) < r (L’vovsky’s version), or (ii) H ( N X ( − (Zak’s version).If α ( X ) ≤ k − , then X is not k -extendable. Now to shift to the second result, let us first give the following
Definition 1.4.
Let C be a smooth irreducible curve. The Wahl map of C is the mapΦ ω C : Λ H ( ω C ) → H ( ω C )defined by Φ ω C ( f dz ∧ gdz ) = ( f g ′ − gf ′ ) dz .In the same years, J. Wahl [W1] introduced this map (so named in the subsequent years) andproved the following seminal Theorem 1.5. (Wahl’s theorem)
Let C be a smooth irreducible curve. If C sits on a K3 surface, then Φ ω C is not surjective. A beautiful geometric proof of this theorem was also given by Beauville and M´erindol [BM].The connection between Zak-L’vovsky’s and Wahl’s theorem will be made in the next sections.In this survey we will focus on extendability of projective varieties. The question was investi-gated already by the Italian and British school, it went on with the school of adjunction theoryand received a big push after the theorems of Zak-L’vovsky and Wahl. It is a very active area ofresearch still going strong today, where Ciro Ciliberto stands as one of the main contributors.We apologize for not treating, due to lack of space, several important aspects very muchrelated to extendability, such as projective duality, deformation theory and graded pieces of the T sheaf. 2. Extendability in general
In order to understand Zak-L’vovsky’s theorem, we start with some simple but useful obser-vations. Even though we do not know a precise reference, they have been all well-known for along time.
Proposition 2.1.
Let X ⊂ P r = P V be a smooth irreducible nondegenerate variety of dimension n ≥ and of codimension at least . Then (i) h ( T P r | X ( − ≥ r + 1 and equality holds if n ≥ ; (ii) N X ( − is globally generated; (iii) α ( X ) ≥ ; (iv) If h ( O X ) = 0 and either n ≥ or n = 2 and the multiplication map V ⊗ H ( ω X ) → H ( ω X (1)) is surjective, then α ( X ) = h ( T X ( − .Proof. From the twisted Euler sequence(2.1) 0 → O X ( − → V ∨ ⊗ O X → T P r | X ( − → T P r | X ( −
1) is globally generated, h ( T P r | X ( − ≥ r + 1 and that equality holds if n ≥ → T X ( − → T P r | X ( − → N X ( − → H ( T X ( − X is aplane conic, but then α ( X ) = 0), we get that α ( X ) = h ( N X ( − − r − ≥ h ( T P r | X ( − − r − ≥ hat is (iii). Now assume that h ( O X ) = 0. If n ≥
3, then H ( O X ( − H ( T P r | X ( − n = 2 the same is achieved byobserving that the map H ( O X ( − → V ∨ ⊗ H ( O X ) is injective. Then (2.2) gives (iv). (cid:3) Let us take a look at the behaviour of α ( X ) under hyperplane sections. Proposition 2.2.
Let X ⊂ P r be a smooth irreducible nondegenerate variety of dimension n ≥ and of codimension at least . Let Y ⊂ P r +1 be a smooth extension of X . If H ( N X ( − ,then H ( N Y ( − and α ( Y ) ≤ α ( X ) − .Proof. Since ( N Y/ P r +1 ) | X ∼ = N X/ P r , for every i ≥ → N Y ( − i − → N Y ( − i ) → N X ( − i ) → . Since H ( N X ( − h ( N Y ( − i )) = h ( N Y ( − i − i ≥ h ( N Y ( − i = 1, we see that α ( Y ) = h ( N Y ( − − r − ≤ h ( N X ( − − r − α ( X ) − . (cid:3) We will now give an idea of how to prove a much-simplified version of Zak-L’vovsky’s theorem(Zak’s version). We will consider only the case of a chain of smooth extensions. For singularversions see [BF, Bad2]. In section 5 we give an example of Zak of a variety that is extendablebut not smoothly extendable.
Proof of Theorem 1.3(simplified version).Proof.
Recall that α ( X ) ≥ Claim 2.3. If α ( X ) = 0 , then H ( N X ( − .Proof. If X has codimension 1 and degree d , then d ≥ N X ( − ∼ = O X ( d − α ( X ) ≥ h ( O X (2)) − r − >
0a contradiction. Therefore codim X ≥
2. This implies that, at every point x ∈ X , the O X,x -module N X ( − x has rank at least 2.Assume that H ( N X ( − = 0 and let σ ∈ H ( N X ( − { τ , . . . , τ r } be abasis of Im { H ( O P r (1)) → H ( O X (1)) } . Then σ ⊗ τ , . . . , σ ⊗ τ r ∈ H ( N X ( − N X ( −
1) is globally generated by Propo-sition 2.1(ii), hence the sections σ ⊗ τ j , ≤ j ≤ r generate N X ( − x . But the σ x ⊗ ( τ j ) x generatean O X,x -module of rank 1, a contradiction. (cid:3)
Therefore, if α ( X ) = 0, the theorem follows by Claim 2.3 and Propositions 2.2 and 2.1(iii).Next let k ≥
2. Suppose that H ( N X ( − X is k -extendable. We show, byinduction on k , that α ( X ) ≥ k . This will of course complete the proof of the theorem.Let Y ⊂ P r +1 be a smooth extension of X . By Proposition 2.2 we have that H ( N Y ( − α ( X ) ≥ α ( Y ) + 1. If k = 2 we have that Y is extendable, hence α ( Y ) ≥ α ( X ) ≥
2. If k ≥
3, since Y is ( k − α ( Y ) ≥ k −
1, hence α ( X ) ≥ k . (cid:3) Remark . It is easy to see that the conditions (i) and (ii) in Zak-L’vovsky’s theorem areessential. For example one can take a hypersurface X ⊂ P n +1 of degree d ≥
3. Then α ( X ) >n + 1 and H ( N X ( − = 0. On the other hand X is infinitely extendable. See Remark 5.25for an example on canonical curves.One immediate consequence of Zak-L’vovsky’s theorem is the one mentioned below. As amatter of fact this appeared before Theorem 1.3, at least for smooth or normal extensions, andit actually has stronger consequences [Fu2, So, Bad1, BS], but we will not be concerned withthem here. roposition 2.5. Let X ⊂ P r be a smooth irreducible nondegenerate variety of dimension n ≥ , of codimension at least and suppose that X is not a quadric. If H ( T X ( − , then X is not extendable.Proof. By (2.2) and [MS, Thm. 8], [W2, Thm. 1] we see that h ( N X ( − h ( T P r | X ( − α ( X ) = 0 by Proposition 2.1(i), hence X is not extendable by Theorem 1.3. (cid:3) How to estimate α ( X ) : a fortunate coincidence Let X ⊂ P r be a smooth irreducible nondegenerate variety of dimension n ≥ α ( X )?Unless one has a very good knowledge of the normal bundle N X (or of T X in the case ofProposition 2.1(iv)), it turns out that it is quite difficult to compute α ( X ).However, in dimension 1, a fortunate coincidence, explained below, comes to help.All the results that follow are of course not new (see for example [CM2, W5]), but we includethem for the benefit of the reader. Lemma 3.1.
Let C ⊂ P r be a smooth irreducible nondegenerate linearly normal curve of positivegenus. Let µ O C (1) ,ω C : H ( O C (1)) ⊗ H ( ω C ) → H ( ω C (1)) be the multiplication map on sections.Consider the map, called Gaussian map, Φ O C (1) ,ω C : H (Ω P r | C ⊗ ω C (1)) → H ( ω C (1)) . Then (i) H (Ω P r | C ⊗ ω C (1)) ∼ = Ker µ O C (1) ,ω C ; (ii) Given the identification in (i), we have that Φ O C (1) ,ω C ( s ⊗ t ) = sdt − tds ; (iii) α ( C ) = corkΦ O C (1) ,ω C .Proof. The twisted dual Euler sequence0 → Ω P r | C ⊗ ω C (1) → H ( O C (1)) ⊗ ω C → ω C (1) → → H (Ω P r | C ⊗ ω C (1)) → H ( O C (1)) ⊗ H ( ω C ) µ O C (1) ,ωC −→ H ( ω C (1)) →→ H (Ω P r | C ⊗ ω C (1)) → H ( O C (1)) ⊗ H ( ω C ) → . This gives (i). Moreover, as µ O C (1) ,ω C is surjective by [Ci], [AS, Thm. 1.6], we have from (3.1)that(3.2) h (Ω P r | C ⊗ ω C (1)) = r + 1 . Now the twisted dual normal bundle sequence0 → N ∨ C ⊗ ω C (1) → Ω P r | C ⊗ ω C (1) → ω C (1) → H (Ω P r | C ⊗ ω C (1)) Φ O C (1) ,ωC −→ H ( ω C (1)) → H ( N ∨ C ⊗ ω C (1)) → H (Ω P r | C ⊗ ω C (1)) → . Thus we get (ii). Also, from (3.3), (3.2) and Serre duality, we get h ( N C ( − r + 1 + corkΦ O C (1) ,ω C that is (iii). (cid:3) Remark . Both the Gaussian map Φ O C (1) ,ω C and the Wahl map Φ ω C have an ancestor in themap µ : Ker µ W,ω C ( − L ) → H ( ω C ), where W ⊆ H ( L ). This was introduced and studied in1981 by Arbarello and Cornalba in [AC, §
4] (see also [ACG, Bib. notes to Chapt. XXI], [AS, § y the above lemma, now the connection between Zak-L’vovsky’s theorem and Wahl’s theo-rem is clear from the following Proposition 3.3.
Let C ⊂ P g − be a canonically embedded smooth curve of genus g ≥ . Then α ( C ) = corkΦ ω C . Moreover if S ⊂ P g is a smooth surface having C as hyperplane section, then S is a K3 surface.Proof. First of all, Lemma 3.1(iii) says that α ( C ) = corkΦ ω C ,ω C . Now Φ ω C ,ω C vanishes on symmetric tensors by Lemma 3.1(ii), hence α ( C ) = corkΦ ω C . For the second part, recall that C is projectively normal by M. Noether’s theorem. Hence thecommutative diagram H ( O P g ( l )) (cid:15) (cid:15) / / / / H ( O P g − ( l )) (cid:15) (cid:15) (cid:15) (cid:15) H ( O S ( l )) / / H ( O C ( l ))implies that the map H ( O S ( l )) → H ( O C ( l )) is surjective for every l ≥
0. From the exactsequence 0 → O S ( l − → O S ( l ) → O C ( l ) → h ( O S ( l − ≤ h ( O S ( l )) for every l ≥
0, and therefore h ( O S ( l )) = 0 for every l ≥
0. In particular q ( S ) = 0. Also h ( O S (1)) = 0 and the exact sequence0 → O S → O S (1) → ω C → h ( ω S ) = h ( O S ) ≥ h ( ω C ) = 1, hence K S ≥
0. On the other hand O C (1) ∼ = ω C ∼ = ω S (1) ⊗ O C hence K S · C = 0 and this implies that K S = 0. Thus S is a K3 surface. (cid:3) We remark that singular extensions of canonical curves can exist and are studied in [Ep1, Ep2].We can now pair Proposition 2.2 and Lemma 3.1 to get the following way to estimate α ( X ). Proposition 3.4.
Let X ⊂ P r be a smooth irreducible nondegenerate variety of dimension n ≥ . Let C be a smooth curve section of X and assume that it is linearly normal and H ( N C ( − . Then α ( X ) ≤ corkΦ O C (1) ,ω C − n + 1 . Proof.
Apply Lemma 3.1 and an iteration of Proposition 2.2. (cid:3)
The emerging philosophy is that if one can control the corank of the Gaussian maps of thecurve section of X , then on can give information on the extendability of X .As far as we know, aside from passing to the curve section, there is only one other generalresult that allows to study extendability of surfaces, without passing to the hyperplane section,but considering suitable linear systems on the surface. This will be discussed in section 8.However in [CDGK] another interesting method for calculating h ( T S ( − S is anEnriques surface, is employed. This method potentially generalizes to other surfaces.Moreover in [R1, R2] a very nice method of studying extendability of varieties covered bylines is introduced. Some applications of this method are mentioned in the next section. . Old days
Already in the beginning of the last century several mathematicians of the Italian and Britishschool, such as Castelnuovo, del Pezzo, Scorza, Terracini, Edge, Semple, Roth, just to mentiona few, studied the extendability problem.For example Scorza [Sc1, Sc2, Sc3] proved that a Veronese variety of dimension n ≥ n = 2) or a Segre variety, except a quadric surface, is not extendable. Terracini[Te] also proved non extendability of Veronese varieties. Di Fiore and Freni [DF] proved that aGrassmannian in its Pl¨ucker embedding, except G (1 , ⊂ P , is not extendable.We can quickly recover these results by applying Proposition 2.5. Proposition 4.1.
The following varieties X ⊂ P r of dimension n ≥ and codimension at least , are not extendable: (i) any abelian variety or any finite unramified covering of such; (ii) a Veronese variety; (iii) a Segre variety, except a quadric surface; (iv) a Grassmannian in its Pl¨ucker embedding, except G (1 , ⊂ P .Proof. For the Veronese surface X ⊂ P observe that H (Ω P | X ) = C by (2.1). Now the map C = H (Ω P ) → H (Ω P | X ) → H (Ω X ) = C is an isomorphism, hence so is H (Ω P | X ) → H (Ω X ). Dualizing we get the isomorphism H ( T X ( − → H ( T P | X ( − . Now using (2.2) and Proposition 2.1(i) we get that α ( X ) = 0, hence Theorem 1.3 applies.In all other cases it is easily seen (for coverings use [Fu2, Prop. 2.7]) that H ( T X ( − (cid:3) More generally and recently Russo, using a description of the Hilbert scheme of lines passingthrough a general point, proves in [R1, R2] that irreducible hermitian symmetric spaces in theirhomogeneous embedding and adjoint homogeneous manifolds, with some obvious exceptions,are not extendable.A celebrated result that also deserves to be mentioned in the recollection of the old days, isthe so-called Babylonian tower theorem [BV, Bar, Tj, Fl, Co]:
Theorem 4.2.
Let X ⊂ P r be a l.c.i. closed subscheme of pure codimension c ≥ . If X extends to a l.c.i. closed subscheme Y ⊂ P r + m of pure codimension c for all m > , then X isa complete intersection. On the other hand, if one drops the l.c.i. condition for all the extensions, then it is easy to getexamples of smooth varieties that are infinitely extendable and are not complete intersections,such as arithmetically Cohen-Macaulay space curves (a more general result is in [Bal]).We end this section by just mentioning many very nice results obtained by Beltrametti,Sommese, Van de Ven, Fujita, Badescu, Serrano, Lanteri, Palleschi, and in general by thepeople working in adjunction theory. It would be too long, for the purposes of this survey, torecall them here. We merely recall a short list of some relevant papers here [SV, Fu1, Ser, Fu2,So, LPS1, LPS2, LPS3], referring the reader to [BS] and references therein.5.
Extendability of canonical curves, K3 surfaces and Fano threefolds
The most beautiful, in my view, part of the story is the incredible amount of nice mathematicsthat revolved around the study of extendability of canonical curves, still going strong today.This is very much intertwined with the study of extendability of K3 surfaces, as the second partof Proposition 3.3 shows. hroughout this section we will let C ⊂ P g − be a canonically embedded smooth irreduciblecurve of genus g ≥ canonical curve ).While for g = 3 , C is a complete intersection in P g − , namely a plane quarticor a complete intersection of a quadric and a cubic in P , the first case to be considered, interms of extendability, is g = 5. In that genus the first phenomenon occurs: the curve mightbe trigonal or not. It is a famous theorem of Enriques and Petri that then C is intersectionof three quadrics (hence extendable) if and only if it is not trigonal. Thus the Brill-Noethertheory of C comes into play, as far as extendability is concerned. As we will see, this is one ofthe main themes.We now concentrate on what happens for a general curve, the meaning of general to be clearalong the way.In a series of beautiful papers, the cases of genus 6 ≤ g ≤
11 were settled by Mori and Mukai.
Theorem 5.1. [M1, M2, M3, M4, MM]
Let C ⊂ P g − be a canonical curve of genus g . Then: (i) If g = 6 , then C extends to a quadric section of G (2 , if and only if C has finitelymany g ’s; (ii) If g = 7 , then C extends to a section of OG (5 , if and only if C has no g ’s; (iii) If g = 8 , then C extends to a section of G (2 , if and only if C has no g ’s; (iv) If g = 9 , then C extends to a section of SpG (3 , if and only if C has no g ’s; (v) If g = 10 , then a general C is not smoothly extendable; (vi) If g = 11 , then a general C is smoothly extendable. Taking into account the second part of Proposition 3.3, the above theorem clearly suggeststhat one should look more closely at the moduli spaces of curves and K3 surfaces.Let M g be the moduli space of curves of genus g . Let K g be the moduli stack of K g , that is pairs ( S, L ) with S a smooth K3 surface, and L an ample, globally generatedline bundle on S with L = 2 g −
2. Let KC g be the moduli space of pairs ( S, C ) such that( S, O S ( C )) ∈ K g . Then one has a forgetful morphism(5.1) c g : KC g → M g and clearly the question is when is c g dominant. Since dim KC g = g + 19 and dim M g = 3 g − g ≤
11. Now, on the one hand, Theorem 5.1 shows that c g isdominant if g ≤ g = 11. On the other hand, a clean connection with Wahl’s theorem, wasmade by Ciliberto, Harris and Miranda (later proved also by Voisin). Theorem 5.2. [CHM, V]
Let C be a general curve of genus g = 10 or g ≥ . Then Φ ω C is surjective. In particular C , in its canonical embedding, is not extendable. The second part (see also [HM, Cor. page 26]) follows by Proposition 3.3 and Zak-L’vovsky’stheorem.Once the dominance of c g is settled, the next question in order is the study of its nonemptyfibers, or, in other words, the study of in how many ways can a hyperplane section of a K3surface extend. A speculation about this was already made in [CM1], where some evidence waspresented to the idea that the corank of the Wahl map, in that case, for g = 11 or g ≥ Definition 5.3. A prime K3 surface of genus g is a smooth K3 surface S ⊂ P g of degree2 g − S ) is generated by the hyperplane bundle. We denote by H g the uniquecomponent of the Hilbert scheme of prime K3 surfaces of genus g .The idea of [CLM1] is to degenerate a prime K3 surface to the union of two rational normalscrolls and then to degenerate the latter to a union of planes whose hyperplane section is asuitable graph curve having corank one Wahl map. The result obtained is the following heorem 5.4. (corank one theorem [CLM1, Thm. 4]) Suppose that g = 11 or g ≥ . Let S be a K3 surface represented by a general point in H g and let C be a general hyperplane section of S . Then corkΦ ω C = 1 . As a matter of fact, when g ≥
17, the same result holds for any smooth section of S (see[CLM2, Thm. 2.15 and Rmk. 2.23]).Together with some other computations of coranks (see [CLM1]), there are two immediateconsequences.The first one is about the map c g (see (5.1)). Theorem 5.5. [CLM1, Thm. 5](i) If ≤ g ≤ and g = 11 , then c g is dominant and the general fiber is irreducible; (ii) codim Im c = 1 ; (iii) codim Im c = 2 ; (iv) If g = 11 or g ≥ , then c g is birational onto its image; (v) If g = 11 or g ≥ , then a general canonical curve that is hyperplane section of a primeK3 surface, lies on a unique one, up to projective transformations. The fact that c g is birational onto its image was later reproved by Mukai [M7, M8], Arbarello,Bruno and Sernesi [ABS1] and Feyzbakhsh [Fe1, Fe1-c, Fe2]. This is part of Mukai’s beautifulprogram of reconstructing the K3 surface from a moduli space of sheaves on the hyperplanesection. We will not be concerned about these matters here.Returning now to the degeneration method explained after Definition 5.3, it allows also toprove that H ( N C ( − C of genus g ≥ Definition 5.6. A prime Fano threefold of genus g is a smooth anticanonically embeddedthreefold X ⊂ P g +1 of degree 2 g − X ) is generated by the hyperplane bundle.We denote by V g the Hilbert scheme of prime Fano threefolds of genus g .Then we have the following results, reproving in a quite simple way important classificationresults of Fano threefolds [I1, I2, M5, M6]. Theorem 5.7. [CLM1, Thm. 6 and 7](i) If g = 11 or g ≥ , then there is no prime Fano threefold of genus g ; (ii) If ≤ g ≤ or g = 12 , then V g is irreducible and the examples of Fano and Iskovskihfill out all of V g . Concerning the last statement, the meaning is that just the knowledge of an example forevery g allows to conclude that they fill out the Hilbert scheme. The main idea behind (ii)(and also behind Theorem 5.9 below) being that a prime Fano threefold is projectively Cohen-Macaulay, hence (see [Pi, page 46]) flatly degenerates to the cone over its hyperplane section S .Now knowledge of the cohomology of the normal bundle of S allows to prove that this cone isa smooth point of the Hilbert scheme. Then to obtain irreducibility one uses the fact that suchK3 surfaces S are general in H g .Another important observation in [CLM1, Table 2] is that, if 6 ≤ g ≤
10 or g = 12, and C is a general hyperplane section of a general K g , then corkΦ ω C >
1, thussuggesting that the curve might be precisely (corkΦ ω C )-extendable.Also this turned out to be true and related to another milestone in the study of Fano varieties:Fano threefolds of index greater than 1 and Mukai varieties.We will state, for simplicity of exposition, only the results about Picard rank 1 and g ≥ efinition 5.8. For r ≥ H r,g be the component of the Hilbert scheme whose generalelements are obtained by embedding prime K3 surfaces of genus g via the r -th multiple of theprimitive class.We denote by V n,r,g the Hilbert scheme of smooth nondegenerate varieties X ⊂ P N of dimen-sion n ≥
3, such that ρ ( X ) = 1 and whose general surface section is a K3 surface representedby a point in H r,g . For n ≥
4, a
Mukai variety is a variety X with ρ ( X ) = 1 represented by apoint in V n, ,g .With similar computations of coranks and cohomology of the normal bundle, the followingclassification results were obtained in [CLM2] (note also [CLM2-c], even though the classificationresults are not affected).The first one is for Fano threefolds of index r ≥
2. Again, as the classification was knownbefore, the novelty is (ii).
Theorem 5.9. [CLM2, Thm. 3.2](i) If ( r, g )
6∈ { (2 , , (2 , , (2 , , (2 , , (3 , , (4 , } , then there is no Fano threefold in V ,r,g ; (ii) If ( r, g ) ∈ { (2 , , (2 , , (2 , , (2 , , (3 , , (4 , } , then V ,r,g is irreducible and the ex-amples of Fano and Iskovskih form a dense open subset of smooth points of V ,r,g . Similarly, for Mukai varieties, classified by Mukai himself [M6], we have (for the definition of n ( g ) see [CLM2, Table 3.14]) the following, where again (ii) was new. Theorem 5.10. [CLM2, Thm. 3.15](i) If ( r, g )
6∈ { (1 , s ) , ≤ s ≤ , (2 , } or if ( r, g ) ∈ { (1 , s ) , ≤ s ≤ , (2 , } and n > n ( g ) , then there is no Mukai variety in V n, ,g ; (ii) If ( r, g ) ∈ { (1 , s ) , ≤ s ≤ , (2 , } and ≤ n ≤ n ( g ) , then V n, ,g is irreducible and theexamples of Mukai form a dense open subset of smooth points of V n, ,g . So much for general canonical curves! What about extendability of any canonical curve?One way to measure how far a curve is from being general is via its Clifford index, that wenow recall.
Definition 5.11.
Let C be a smooth irreducible curve of genus g ≥
4. The Clifford index of C is Cliff( C ) = min { deg L − r ( L ) , L a line bundle on C such that h ( L ) ≥ , h ( L ) ≥ } . Recall that Cliff( C ) = 0 if and only if C is hyperelliptic; Cliff( C ) = 1 if and only if C istrigonal or isomorphic to a plane quintic; Cliff( C ) = 2 if and only if C is tetragonal or isomorphicto a plane sextic. A general curve of genus g has Clifford index ⌊ g − ⌋ .Already Beauville and M´erindol, in their proof of Wahl’s theorem, observed that if a smoothirreducible curve C sits on a K3 surface S , then the surjectivity of Φ ω C implies the splitting ofthe normal bundle sequence 0 → T C → T S | C → N C/S → . This introduced the idea, exploited by Voisin in [V], that the elements of (Coker Φ ω C ) ∨ shouldbe interpreted as ribbons, or infinitesimal surfaces, embedded in P g and extending a canonicalcurve C . We will return to this later.In 1996, Wahl proposed a possible converse to his theorem (Theorem 1.5), that, as we willsee, it is very much connected with Voisin’s idea and further developments. Theorem 5.12. [W4, Thm. 7.1] et C ⊂ P g − be a canonical curve of genus g ≥ and Cliff( C ) ≥ . Suppose that C satisfies (5.2) H ( I C/ P g − ( k )) = 0 for every k ≥ . Then C is extendable if and only if Φ ω C is not surjective. Concerning the condition (5.2), Wahl proved that it is satisfied by a general canonical curve,while it is not satisfied by a general tetragonal curve of genus g ≥
8. He also conjectured that(5.2) holds for every canonical curve C with Cliff( C ) ≥
3. This conjecture was then proved, inalmost all cases, in a beautiful paper by Arbarello, Bruno and Sernesi.
Theorem 5.13. [ABS2, Thm. 1.3]
Let C ⊂ P g − be a canonical curve of genus g ≥ and Cliff( C ) ≥ . Then H ( I C/ P g − ( k )) = 0 for every k ≥ . Besides playing a crucial role in what follows, this theorem, together with Wahl’s theoremand Theorem 5.12, gives
Corollary 5.14. [ABS2, Cor. 1.4]
Let C ⊂ P g − be a canonical curve of genus g ≥ and Cliff( C ) ≥ . Then C is extendableif and only if Φ ω C is not surjective. In the same paper Arbarello, Bruno and Sernesi, also proved another important result. Wefirst recall
Definition 5.15.
A smooth irreducible curve C is called a Brill–Noether–Petri curve if themultiplication map H ( L ) ⊗ H ( K C − L ) → H ( K C ) is injective for every line bundle L on C .Following an idea of Mukai, Voisin suggested in [V] to study the relation between the ex-tendability of Brill–Noether–Petri curves to K3 surfaces and the non-surjectivity of Φ ω C . Wahlconjectured in [W4] that for g ≥ Theorem 5.16. [ABS2, Thm. 1.1]
Let C be a Brill–Noether–Petri curve of genus g ≥ . Then C lies on a polarised K3 surface,or on a limit thereof, if and only if its Wahl map is not surjective. This circle of ideas and results was closed, as of now, by several beautiful theorems provedin [CDS]. The paper is full of very interesting results, we will describe only some of them.The first one is that, conversely to Zak-L’vovsky’s theorem, the condition α ( X ) ≥ k issufficient for the k -extendability of most canonical curves and K3 surfaces. This was obtainedby a generalization of the method of Voisin and Wahl of integrating ribbons, as follows. Notethat the vanishing in Theorem 5.13 is fundamental for this method to work. Theorem 5.17. [CDS, Thm. 2.1]
Let C be a smooth curve of genus g with Cliff( C ) ≥ . Consider the following two statements: (i) corkΦ ω C ≥ k + 1 ; (ii) There exists an arithmetically Gorenstein normal variety Y ⊂ P g + k , not a cone, with dim( Y ) = k + 2 , ω Y = O Y ( − k ) , which has a canonical image of C as a section witha ( g − -dimensional linear subspace of P g + k (in particular, C ⊂ P g − is ( k + 1) -extendable).If g ≥ , then (i) implies (ii). Conversely, if g ≥ and the canonical image of C is ahyperplane section of a smooth K3 surface in P g , then (ii) implies (i). As for the corank of the Wahl map, the following result is proved by the same authors. heorem 5.18. [CDS, Cor. 2.10] Let C be a smooth curve of genus g > with Cliff( C ) ≥ . If the canonical model of C is ahyperplane section of a K3 surface S , possibly with ADE singularities, then corkΦ ω C = 1 . It is an intriguing open question to find examples of higher corank.
Question 5.19. [CDS, Question 2.14]Does there exist any Brill-Noether general curve of genus g ≥ , g = 12, such that corkΦ ω C >
1? As explained in [CDS, Rmk. 2.13], all canonical curves in smooth Fano threefolds with Picardnumber greater than 1 are Brill-Noether special.Before stating the next result let us define
Definition 5.20.
The Clifford index Cliff(
S, L ) of a polarized K3 surface (
S, L ) is the Cliffordindex of any smooth curve C ∈ | L | .Note that this does not depend on the choice of C by [GL]. Now if C is a smooth hyperplanesection of a K3 surface S ⊂ P g , then Cliff( C ) = Cliff( S, L ), and in [CDS, Cor. 2.8], as aconsequence of the proof of Theorem 5.23 below, the authors prove thatcorkΦ ω C = h ( T S ( − . Then Theorem 5.17 gives
Theorem 5.21. [CDS, Thm. 2.18]
Let ( S, L ) ∈ K g be a polarized K3 surface with Cliff(
S, L ) ≥ . Consider the following twostatements: (i) h ( T S ⊗ L ∨ ) ≥ k ; (ii) There exists an arithmetically Gorenstein normal variety X ⊂ P g + k , not a cone, with dim( X ) = k + 2 , ω X = O X ( − k ) , having the image of S by the linear system | L | as asection with a g -dimensional linear subspace of P g + k .If g ≥ , then (i) implies (ii). Conversely, if g ≥ , then (ii) implies (i). On the other hand, extendability of K3 surfaces, is possible only for bounded genus, usingresults of Prokhorov [Pr1] and Karzhemanov [Ka1, Ka2]:
Theorem 5.22. [CDS, 2.9]
Let S ⊂ P g be a K3 surface possibly with ADE singularities. If g = 35 or g ≥ , then S isnot extendable. This bound is sharp [Z, Pr1]. Moreover the example of genus 37 is a K3 surface that isextendable but not smoothly extendable.As for the map c g (see (5.1)), Ciliberto, Dedieu and Sernesi prove that the fibers are smoothover curves with Clifford index at least 3 and g ≥ Theorem 5.23. [CDS, Thm. 2.6]
Let ( S, C ) ∈ KC g with g ≥ and Cliff( C ) ≥ . Then dim Ker dc g ( S,C ) = dim c − g ( C ) = corkΦ ω C − . A similar theorem [CDS, Thm. 2.19] holds for the analogous map for Fano threefolds and K3surfaces. See also [CD1, CD2] for the map c g in the non-primitive case.We end this section by recalling what happens, and what is known, for curves C withCliff( C ) ≤ heorem 5.24. [CM1, W1, W3, Br1, Br2, Br3] Let C be a smooth irreducible curve of genus g . (i) If C is hyperelliptic of genus g ≥ , then corkΦ ω C = 3 g − (in fact this characterizeshyperelliptic curves); (ii) If C is trigonal and g ≥ , then corkΦ ω C = g + 5 ; (iii) If C is a plane curve of degree at least , then corkΦ ω C = 10 ; (iv) If C is a general tetragonal curve and g ≥ , then corkΦ ω C = 9 .Remark . In [CDS, Ex. 9.7] (see also [W4]) a 10-extension of the canonical embedding ofany smooth plane curve of degree d ≥ d ≥
7, then the curve is precisely (corkΦ ω C )-extendable. On the otherhand, if d = 5 ,
6, then Zak-L’vovsky’s theorem does not apply since H ( N C/ P g − ( − = 0(see [CM1]) and α ( C ) = corkΦ ω C = 10 > g −
1. As it turns out, these curves actually are( α ( C ) + 1)-extendable (see [CD2, Rmk. 3.7] and the appendix).As far as we know, it is not known if trigonal or tetragonal curves are precisely (corkΦ ω C )-extendable.6. Extendability of Enriques surfaces and Enriques-Fano threefolds
For Enriques surfaces we also have that they are not extendable if the sectional genus is largeenough. It is an application of Theorem 8.1.
Theorem 6.1. [KLM]
Let S ⊂ P r be an Enriques surface of sectional genus g ≥ . Then S is not extendable. A more precise result for g = 15 and 17 is proved in [KLM, Prop. 12.1]. Actually a completelist of line bundles associated to possibly extendable very ample linear systems on an Enriquessurface is available to the authors. See also below for more precise results.As for extendability of Enriques surfaces let us define Definition 6.2. An Enriques-Fano threefold is an irreducible three-dimensional variety X ⊂ P N having a hyperplane section S that is a smooth Enriques surface, and such that X is not a coneover S . We will say that X has genus g if g is the genus of its general curve section.In analogy with Fano threefolds, we obtain a genus bound. The bound was also proved, withcompletely different methods, by Prokhorov [Pr2] (see also [Ka1, Ka3]). Moreover he producedan example of genus 17. For possible genera see also [Bay, Sa] for cyclic quotient terminalsingularities and [KLM, Prop. 13.1] for an example with other singularities. Theorem 6.3. [KLM, Thm. 1.5]
Any Enriques-Fano threefold has genus g ≤ . Both the study of extendability of Enriques surfaces and the moduli map analogous to theone on K3 surfaces (see (5.1)), were recently vastly extended in [CDGK]. We will recall onlysome of the many results contained in that paper, referring the reader to [CDGK] for a moredetailed version.
Definition 6.4.
Let S be an Enriques surface and let L be a line bundle on S such that L > φ ( L ) = min { E · L : E ∈ N S ( S ) , E = 0 , E > } . Let E g,φ be the moduli space of pairs ( S, L ) where S is an Enriques surface, L is an ample linebundle with L = 2 g − φ ( L ) = φ . Let EC g,φ be the moduli space of triples ( S, L, C ) where(
S, L ) ∈ E g,φ and C ∈ | L | is a smooth irreducible curve. Let R g be the moduli space of Prymcurves, that is of pairs ( C, η ) with C a smooth irreducible curve of genus g and η a non-zero2-torsion element of Pic ( C ). ote that E g,φ has in general many irreducible components. There is a classification in lowgenus in [CDGK, §
2] and more generally in [Kn].We have the diagram EC g,φc g,φ " " ❋❋❋❋❋❋❋❋ χ g,φ / / R gf g (cid:15) (cid:15) M g where χ g,φ ( S, L, C ) = (
C, K S | C ), f g is the degree 2 g − c g,φ = f g ◦ χ g,φ .Then we have Theorem 6.5. [CDGK, Thm. 1, 2, 3](i) If φ ≥ , then χ g,φ is generically injective on any irreducible component of EC g,φ , withthe exception of 10 components where the dimension of the general fiber is given in thelist in [CDGK, Thm. 1] ; (ii) χ g, is generically finite on all irreducible components of EC g, when g ≥ . For g ≤ the dimension of a general fiber of χ g, on the various irreducible components of EC g, is given in the list in [CDGK, Thm. 2] ; (iii) The dimension of a general fiber of χ g, and of c g, is max { − g, } . Hence c g, dominates the hyperelliptic locus if g ≤ and is generically finite if g ≥ . This has the following nice consequence
Corollary 6.6. [CDGK, Cor. 1.1]
A general curve of genus , , and lies on an Enriques surface, whereas a general curve ofgenus or ≥ does not. A general hyperelliptic curve of genus g lies on an Enriques surface ifand only if g ≤ . The authors then went on to study extendability of Enriques surfaces (related to the fiberdimension above), proving
Theorem 6.7. [CDGK, Cor. 1.2]
Let S ⊂ P r be an Enriques surface not containing any smooth rational curve. If S is -extendable, then ( S, O S (1)) belongs to the following list (see [CDGK] for precise definitions) E ( IV ) + , , E ( II ) + , , E ( II )13 , , E ( II )10 , , E +9 , , E ( II )9 , , E , . Furthermore, the members of this list are all at most -extendable, except for members of E ( II )10 , ,which are at most 2–extendable, and of E +9 , , which are at most -extendable. See also [CDGK, Rmk. 6.1] for nodal Enriques surfaces.7.
Extendability of curves in other embeddings
We will collect a sample of nonextendability results for curves. We will not treat, in thissection, the case of canonical curves, as it was the object of a specific section.Perhaps the first significant one is the non extendability of elliptic normal curves of degreeat least 10, as it follows by del Pezzo [DP] and Nagata’s [N] classification of surfaces of degree d in P d . Another important result is Castelnuovo-Enriques’s theorem [Ca, En], that proves thatfor a curve of degree large enough with respect to the genus, the only smooth extensions canbe ruled surfaces (see [CR, H] for a modern proof; see also [HM]).There are of course many other results in this direction. Our choice is to mention the onesthat are relevant to the approach via Zak-L’vovsky’s theorem and Gaussian maps.One important distinction to be made is between the case of any curve, and the case of a general curve, that is a curve with general moduli. or a given curve we have the following results, according to the Clifford index of C . Theorem 7.1. [KL, Cor. 2.10] , [BEL, Thm. 2] Let C ⊂ P r be a smooth irreducible nondegenerate linearly normal curve of genus g ≥ anddegree d . Then C is not extendable if (i) C is trigonal, g ≥ and d ≥ max { g − , g + 7 } ; (ii) C is a plane quintic and d ≥ ; (iii) Cliff( C ) = 2 and d ≥ g − ; (iv) Cliff( C ) ≥ and d ≥ g + 1 − C ) . For curves with general moduli we have
Theorem 7.2. [L, Cor. 1.7]
Let C ⊂ P r be a curve of genus g with general moduli and degree d . Then C is not extendableif (i) d ≥ g + 15 for ≤ g ≤ ; (ii) d ≥ g + 13 for ≤ g ≤ ; (iii) d ≥ g + 10 for g ≥ . For general embeddings of curves with general moduli we have
Theorem 7.3. [L, Cor. 1.7]
Let C ⊂ P r be a general linearly normal degree d embedding a curve of genus g with generalmoduli. Then C is not extendable if (i) d ≥ g + 14 for ≤ g ≤ ; (ii) d ≥ g + 12 for ≤ g ≤ ; (iii) d ≥ g + 9 for g ≥ . Another possibility is to study extendability of curves in terms of general Brill-Noether em-beddings.
Theorem 7.4. [L, Thm. 1.9]
Let d, g, r be integers such that ρ ( d, g, r ) ≥ , d < g + r, r ≥ or r = 9 . If C is a curvewith general moduli and L is a general line bundle in W rd ( C ) , then C ⊂ P r = P H ( L ) is notextendable. Extendability of other surfaces
In this section we will outline some results about extendability of surfaces that are neither K Theorem 8.1. [KLM, Thm. 1.1]
Let S ⊂ P r be a smooth irreducible linearly normal surface and let H be its hyperplane bundle.Assume there is a base-point free and big line bundle D on S with H ( H − D ) = 0 and suchthat the general element D ∈ | D | is not rational and satisfies (i) the Gaussian maps Φ H D ,ω D ( D ) is surjective; (ii) the multiplication maps µ V D ,ω D and µ V D ,ω D ( D ) are surjectivewhere V D := Im { H ( S, H − D ) → H ( D, ( H − D ) | D ) } . Then α ( S ) ≤ corkΦ H D ,ω D . We give first a general result, application of Theorem 8.1, that holds on any surface, oncegiven a suitable embedding (for the definition of m ( L ) see [KLM]). heorem 8.2. [KLM, Cor. 3.3] Let S ⊂ P V be a smooth irreducible surface with V ⊆ H ( mL + D ) , where L is a base-pointfree, big, nonhyperelliptic line bundle on S with L · ( L + K S ) ≥ and D ≥ is a divisor.Suppose that S is regular or linearly normal and that m is such that H (( m − L + D ) = 0 and m > max { m ( L ) − ( L · D ) /L , ( L · K S + 2 − L · D ) /L + 1 } . Then S is not extendable. See also [KLM, Cor. 3.5] for a better result in the case of adjoint embeddings.We now list what is known in terms of Kodaira dimension. We will concentrate on minimalsurfaces.8.1.
Surfaces of Kodaira dimension −∞ . For embeddings of P the non-extendability was settled by Scorza [Sc1]. See also Proposition4.1.As explained in [BS, § X is P d -bundle over a smooth variety and X is anample divisor on a locally complete intersection variety Y , then in almost all cases, includingthe case of surfaces, Y is a P d +1 -bundle over the same base and X belongs to the tautologicallinear system. This shows that l.c.i. extensions of P d -bundles are P d +1 -bundles.On the other hand, as far as we know, not so many results are known, in general, aboutextendability of ruled surfaces.A result was proved in [DB, Thm. 3.3.22] for rational ruled surfaces, as an application ofTheorem 8.1. We believe that this can be extended to other ruled surfaces.8.2. Surfaces of Kodaira dimension . The cases of K3 and Enriques surfaces have been treated above. Abelian and bi-ellipticsurfaces are not extendable by Proposition 4.1.8.3.
Surfaces of Kodaira dimension . The only results that we are aware of are the ones in [LMS], that hold especially for Weier-strass fibrations.8.4.
Surfaces of general type.
We are not aware of results, except the one below, that applies, for example, for pluricanonicalembeddings.
Theorem 8.3. [KLM, Cor. 1.2]
Let S ⊂ P V be a minimal surface of general type with base-point free and nonhyperellipticcanonical bundle and V ⊂ H ( mK S + D )) , where D ≥ and either D is nef or D is reducedand K S is ample. Suppose that S is regular or linearly normal and that m ≥ K Y = 2;7 if K Y = 3;6 if K Y = 4 and the general curve in | K Y | is trigonal or if K Y = 5 and the general curve in | K Y | is a plane quintic;5 if either the general curve in | K Y | has Clifford index 2 or5 ≤ K Y ≤ | K Y | is trigonal;4 otherwise . Then S is not extendable. References [AB] E. Arbarello, A. Bruno.
Rank-two vector bundles on Halphen surfaces and the Gauss-Wahl map forDu Val curves . J. ´Ec. polytech. Math. (2017), 257-285. 10[ABFS] E. Arbarello, A. Bruno, G. Farkas, G. Sacc`a. Explicit Brill-Noether-Petri general curves . Comment.Math. Helv. (2016), no. 3, 477-491. 10 ABS1] E. Arbarello, A. Bruno, E. Sernesi.
Mukai’s program for curves on a K3 surface . Algebr. Geom. (2014), no. 5, 532-557. 8[ABS2] E. Arbarello, A. Bruno, E. Sernesi. On hyperplane sections of K3 surfaces . Algebr. Geom. (2017),no. 5, 562-596. 10[AC] E. Arbarello, M. Cornalba. On a conjecture of Petri . Comment. Math. Helv. (1981), no. 1, 1-38. 4[ACG] E. Arbarello, M. Cornalba, P. A. Griffiths. Geometry of algebraic curves. Volume II. With a contributionby Joseph Daniel Harris . Grundlehren der Mathematischen Wissenschaften [Fundamental Principlesof Mathematical Sciences], . Springer, Heidelberg, 2011. xxx+963 pp 4[AS] E. Arbarello, E. Sernesi.
Petri’s approach to the study of the ideal associated to a special divisor . Invent.Math. (1978), no. 2, 99-119. 4[Bad1] L. B˘adescu. Infinitesimal deformations of negative weights and hyperplane sections . In: Algebraicgeometry (L’Aquila, 1988), 1-22, Lecture Notes in Math., , Springer, Berlin, 1990. 3[Bad2] L. B˘adescu.
On a result of Zak-L’vovsky . In: Projective geometry with applications, 57-73, LectureNotes in Pure and Appl. Math., , Dekker, New York, 1994. 1, 3[Bal] E. Ballico.
Extending infinitely many times arithmetically Cohen-Macaulay and Gorenstein subvarietiesof projective spaces . Preprint arXiv:2102.06457. 6[Bar] W. Barth.
Submanifolds of low codimension in projective space . In: Proceedings of the InternationalCongress of Mathematicians (Vancouver, B.C., 1974), Vol. 1, pp. 409-413. Canad. Math. Congress,Montreal, Que., 1975. 6[Bay] L. Bayle.
Classification des vari´et´es complexes projectives de dimension trois dont une section hyper-plane g´en´erale est une surface d’Enriques . J. Reine Angew. Math. (1994), 9-63. 12[Br1] J. N. Brawner.
The Gaussian map Φ K for curves with special linear series . Thesis (Ph.D.)-The Uni-versity of North Carolina at Chapel Hill. 1992. 73 pp. 11, 12[Br2] J. N. Brawner. The Gaussian-Wahl map for trigonal curves . Proc. Amer. Math. Soc. (1995), no.5, 1357-1361. 12[Br3] J. N. Brawner.
Tetragonal curves, scrolls, and K3 surfaces . Trans. Amer. Math. Soc. (1997), no.8, 3075-3091. 11, 12[BEL] A. Bertram, L. Ein, R. Lazarsfeld.
Surjectivity of Gaussian maps for line bundles of large degree oncurves . In: Algebraic geometry (Chicago, IL, 1989), 15-25, Lecture Notes in Math., , Springer,Berlin, 1991. 14[BF] E. Ballico, C. Fontanari.
Gaussian maps, the Zak map and projective extensions of singular varieties .Results Math. (2003), no. 1-2, 29-34. 3[BM] A. Beauville, J. Y. M´erindol. Sections hyperplanes des surfaces K3 . Duke Math. J. (1987), no. 4,873-878. 2[BS] M. C. Beltrametti, A. J. Sommese. The adjunction theory of complex projective varieties . De GruyterExpositions in Mathematics, . Walter de Gruyter & Co., Berlin, 1995. 1, 3, 6, 15[BV] W. Barth, A. Van de Ven. A decomposability criterion for algebraic 2-bundles on projective spaces .Invent. Math. (1974), 91-106. 6[Ca] G. Castelnuovo. Massima dimensione dei sistemi lineari di curve piane di dato genere . Ann. Mat. (2) (1890), 119-128. 13[Ci] C. Ciliberto. Sul grado dei generatori dell’anello canonico di una superficie di tipo generale . Rend.Sem. Mat. Univ. Politec. Torino (1983), no. 3, 83-111. 4[Co] I. Coand˘a. A simple proof of Tyurin’s Babylonian tower theorem . Comm. Algebra (2012), no. 12,4668–4672 6[CD1] C. Ciliberto, T. Dedieu. Double covers and extensions . Preprint arXiv:2008.03109. 8, 11[CD2] C. Ciliberto, T. Dedieu.
K3 curves with index k >
1. Preprint arXiv:2012.10642. 11, 12[CDGK] C. Ciliberto, T. Dedieu, C. Galati, A. L. Knutsen.
Moduli of curves on Enriques surfaces . Adv. Math. (2020), 107010, 42 pp. 5, 12, 13[CDS] C. Ciliberto, T. Dedieu, E. Sernesi.
Wahl maps and extensions of canonical curves and K3 surfaces .J. Reine Angew. Math. (2020), 219-245. 10, 11, 12[CHM] C. Ciliberto, J. Harris, R. Miranda.
On the surjectivity of the Wahl map . Duke Math. J. (1988),no. 3, 829-858. 7[CLM1] C. Ciliberto, A. Lopez, R. Miranda. Projective degenerations of K3 surfaces, Gaussian maps, and Fanothreefolds . Invent. Math. (1993), no. 3, 641-667. 7, 8[CLM2] C. Ciliberto, A. Lopez, R. Miranda.
Classification of varieties with canonical curve section via Gaussianmaps on canonical curves . Amer. J. Math. (1998), no. 1, 1-21. 8, 9[CLM2-c] C. Ciliberto, A. Lopez, R. Miranda.
Corrigendum to: Classification of varieties with canonical curvesection via Gaussian maps on canonical curves [Amer. J. Math. 120(1998), no. 1, 1-21] . To appearon Amer. J. Math.. 9 CM1] C. Ciliberto, R. Miranda.
Gaussian maps for certain families of canonical curves . In: Complex pro-jective geometry (Trieste, 1989/Bergen, 1989), 106-127, London Math. Soc. Lecture Note Ser., ,Cambridge Univ. Press, Cambridge, 1992. 7, 12[CM2] C. Ciliberto, R. Miranda.
On the Gaussian map for canonical curves of low genus . Duke Math. J. (1990), no. 2, 417-443. 4[CR] C. Ciliberto, F. Russo. Varieties with minimal secant degree and linear systems of maximal dimensionon surfaces . Adv. Math. (2006), no. 1, 1-50. 13[DB] L. Di Biagio.
Some issues about the extendability of projective surfaces . Roma Tre UndergraduateThesis, 2004. Available at http://ricerca.mat.uniroma3.it/users/lopez/Tesi-DiBiagio.pdf
On varieties cut out by hyperplanes into Grassmann varieties of arbitrary indexes .Rend. Istit. Mat. Univ. Trieste (1981), no. 1-2, 51-57. 6[DP] P. del Pezzo. Sulle superficie dell’ n mo ordine immerse nello spazio di n dimensioni . Rend. del CircoloMatematico di Palermo (1887), 241-271. 13[En] F. Enriques. Sulla massima dimensione dei sistemi lineari di dato genere appartenenti a una superficiealgebrica . Atti Reale Acc. Sci. Torino (1894) 275-296. 13[Ep1] D. H. J. Epema. Surfaces with canonical hyperplane sections . CWI Tract 1. Stichting MathematischCentrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1984. 5[Ep2] D. H. J. Epema.
Surfaces with canonical hyperplane sections . Nederl. Akad. Wetensch. Indag. Math. , (1983), 173-184. 5[Fe1] S. Feyzbakhsh. Mukai’s program (reconstructing a K3 surface from a curve) via wall-crossing . J. ReineAngew. Math. (2020), 101-137. 8[Fe1-c] S. Feyzbakhsh.
Erratum to Mukai’s program (reconstructing a K3 surface from a curve) via wall-crossing (J. reine angew. Math. 765 (2020), 101–137) . J. Reine Angew. Math. (2020), 183. 8[Fe2] S. Feyzbakhsh.
Mukai’s Program (reconstructing a K3 surface from a curve) via wall-crossing, II .Preprint arXiv:2006.08410. 8[Fl] H. Flenner.
Babylonian tower theorems on the punctured spectrum . Math. Ann. (1985), no. 1,153-160. 6[Fu1] T. Fujita.
Classification theories of polarized varieties . London Mathematical Society Lecture NoteSeries, . Cambridge University Press, Cambridge, 1990. xiv+205 pp. 6[Fu2] T. Fujita.
Impossibility criterion of being an ample divisor . J. Math. Soc. Japan (1982), no. 2,355-363. 3, 6[GL] M. Green, R. Lazarsfeld. Special divisors on curves on a K3 surface . Invent. Math. (1987), no. 2,357-370. 11[Ka1] I. V. Karzhemanov. On Fano threefolds with canonical Gorenstein singularities . Mat. Sb. (2009),no. 8, 111-146; translation in Sb. Math. (2009), no. 7-8, 1215-1246. 11, 12[Ka2] I. V. Karzhemanov.
Fano threefolds with canonical Gorenstein singularities and big degree . Math. Ann. (2015), no. 3-4, 1107-1142. 11[Ka3] I. V. Karzhemanov.
On some Fano-Enriques threefolds . Adv. Geom. (2011), no. 1, 117-129. 12[Kn] A. L. Knutsen. On moduli spaces of polarized Enriques surfaces . J. Math. Pures Appl. (9) (2020),106-136. 13[KL] A. L. Knutsen, A. F. Lopez.
Surjectivity of Gaussian maps for curves on Enriques surfaces . Adv.Geom. (2007), no. 2, 215-247. 14[KLM] A. L. Knutsen, A. F. Lopez, R. Mu˜noz. On the extendability of projective surfaces and a genus boundfor Enriques-Fano threefolds . J. Differential Geom. (2011), no. 3, 485-518. 12, 14, 15[I1] V. A. Iskovskih. Fano threefolds. I . Izv. Akad. Nauk SSSR Ser. Mat. (1977), no. 3, 516-562, 717. 8[I2] V. A. Iskovskih. Fano threefolds. II . Izv. Akad. Nauk SSSR Ser. Mat. (1978), no. 3, 506-549. 8[H] R.cHartshorne. Curves with high self-intersection on algebraic surfaces . Inst. Hautes ´Etudes Sci. Publ.Math. No. (1969), 111-125. 13[HM] J. Harris, D. Mumford. On the Kodaira dimension of the moduli space of curves . With an appendixby William Fulton. Invent. Math. (1982), no. 1, 23-88. 7, 13[L] A. F. Lopez. Surjectivity of Gaussian maps on curves in P r with general moduli . J. Algebraic Geom. (1996), no. 4, 609-631. 14[LMS] A. F. Lopez, R. Mu˜noz, J. C. Sierra. On the extendability of elliptic surfaces of rank two and higher .Ann. Inst. Fourier (Grenoble) (2009), no. 1, 311-346. 15[LPS1] A. Lanteri, M. Palleschi, A. J. Sommese. Double covers of P n as very ample divisors . Nagoya Math. J. (1995), 1-32. 6[LPS2] A. Lanteri, M. Palleschi, A. J. Sommese. On triple covers of P n as very ample divisors . In: Classificationof algebraic varieties (L’Aquila, 1992), 277-292, Contemp. Math., , Amer. Math. Soc., Providence,RI, 1994. 6 LPS3] A. Lanteri, M. Palleschi, A. J. Sommese.
Del Pezzo surfaces as hyperplane sections . J. Math. Soc.Japan (1997), no. 3, 501-529. 6[Lv] S. L’vovsky. Extensions of projective varieties and deformations. I, II . Michigan Math. J. (1992),no. 1, 41-51, 65-70. 1[M1] S. Mukai. Curves and Grassmannians . In: Algebraic geometry and related topics (Inchon, 1992), 19-40,Conf. Proc. Lecture Notes Algebraic Geom., I, Int. Press, Cambridge, MA, 1993. 7[M2] S. Mukai.
Curves and symmetric spaces. I . Amer. J. Math. (1995), no. 6, 1627.1644. 7[M3] S. Mukai.
Curves and symmetric spaces, II . Ann. of Math. (2) (2010), no. 3, 1539-1558. 7[M4] S. Mukai.
Curves, K3 surfaces and Fano 3-folds of genus ≤
10. Algebraic geometry and commutativealgebra, Vol. I, 357-377, Kinokuniya, Tokyo, 1988. 7[M5] S. Mukai.
Fano 3-folds . In: Complex projective geometry (Trieste, 1989/Bergen, 1989), 255-263, Lon-don Math. Soc. Lecture Note Ser., , Cambridge Univ. Press, Cambridge, 1992. 8[M6] S. Mukai.
Biregular classification of Fano 3-folds and Fano manifolds of coindex 3 . Proc. Nat. Acad.Sci. U.S.A. (1989), no. 9, 3000-3002. 8, 9[M7] S. Mukai. Curves and K3 surfaces of genus eleven . In: Moduli of vector bundles (Sanda, 1994; Kyoto,1994), 189-197, Lecture Notes in Pure and Appl. Math., , Dekker, New York, 1996. 8[M8] S. Mukai.
Non-abelian Brill-Noether theory and Fano 3-folds . Sugaku Expositions (2001), no. 2,125-153. 8[MM] S. Mori, S. Mukai. The uniruledness of the moduli space of curves of genus 11 . Algebraic geometry(Tokyo/Kyoto, 1982), 334-353, Lecture Notes in Math., , Springer, Berlin, 1983. 7[MS] S. Mori, H. Sumihiro.
On Hartshorne’s conjecture . J. Math. Kyoto Univ. (1978), no. 3, 523-533. 2,4[N] M. Nagata. On rational surfaces. I. Irreducible curves of arithmetic genus 0 or 1 . Mem. Coll. Sci.Univ. Kyoto Ser. A. Math. (1960), 351-370. 13[Pi] H. C. Pinkham. Deformations of algebraic varieties with G m action . Ast´erisque, No. . Soci´et´eMath´ematique de France, Paris, 1974. i+131 pp. 8[Pr1] Yu. G. Prokhorov. The degree of Fano threefolds with canonical Gorenstein singularities . Mat. Sb. (2005), no. 1, 81-122; translation in Sb. Math. (2005), no. 1-2, 77-114. 11[Pr2] Yu. G. Prokhorov.
On Fano-Enriques varieties . Mat. Sb. (2007), no. 4, 117-134; translation in Sb.Math. (2007), no. 3-4, 559-574. 12[R1] F. Russo.
Lines on projective varieties and applications . Rend. Circ. Mat. Palermo (2) (2012), no.1, 47-64. 5, 6[R2] F. Russo. On the geometry of some special projective varieties . Lecture Notes of the Unione MatematicaItaliana, . Springer, Cham; Unione Matematica Italiana, Bologna, 2016. 5, 6[Te] A. Terracini. Alcune questioni sugli spazi tangenti e osculatori ad una variet`a. Nota I . Atti Accad.Torino, , 1913. 6[Tj] A. N. Tjurin. Finite-dimensional bundles on infinite varieties . Izv. Akad. Nauk SSSR Ser. Mat. (1976), no. 6, 1248-1268, 1439. 6[Sa] T. Sano. On classifications of non-Gorenstein Q -Fano 3-folds of Fano index 1 . J. Math. Soc. Japan (1995), no. 2, 369-380. 12[Sc1] G. Scorza. Sopra una certa classe di variet`a razionali . Rend. Circ. Mat. Palermo , (1909) 400-401.6, 15[Sc2] G. Scorza. Sulle variet`a di Segre . Atti Accad. Sci. Torino , 119-31 (1910). 6[Sc3] G. Scorza. Opere Scelte . Vol. I, pp. 376-386. Edizioni Cremonese, Napoli (1960). 6[Seg] C. Segre.
Sulle variet`a normali a tre dimensioni composte da serie semplici di piani . Atti Accad. Sci.Torino , 95-115 (1885). 6[Ser] F. Serrano. The adjunction mappings and hyperelliptic divisors on a surface . J. Reine Angew. Math. (1987), 90-109. 6[So] A. J. Sommese.
On manifolds that cannot be ample divisors . Math. Ann. (1976), no. 1, 55-72. 3, 6[SV] A. J. Sommese, A. Van de Ven.
On the adjunction mapping . Math. Ann. (1987), no. 1-4, 593-603.6[V] C. Voisin.
Sur l’application de Wahl des courbes satisfaisant la condition de Brill-Noether-Petri . ActaMath. (1992), no. 3-4, 249-272. 7, 9, 10[W1] J. M. Wahl.
The Jacobian algebra of a graded Gorenstein singularity . Duke Math. J. (1987), no. 4,843-871. 2, 12[W2] J. M. Wahl. A cohomological characterization of P n . Invent. Math. (1983), no. 2, 315-322. 2, 4[W3] J. M. Wahl. Gaussian maps on algebraic curves . J. Differential Geom. (1990), no. 1, 77-98. 12[W4] J. M. Wahl. On cohomology of the square of an ideal sheaf . J. Algebraic Geom. (1997), no. 3, 481-511.9, 10, 12[W5] J. M. Wahl. Deformations of quasihomogeneous surface singularities . Math. Ann. (1988), no. 1,105-128. 4 Z] F. L. Zak.
Some properties of dual varieties and their applications in projective geometry . In: Algebraicgeometry (Chicago, IL, 1989), 273-280, Lecture Notes in Math., , Springer, Berlin, 1991. 1, 11
Appendix A. Extendability of canonical models of plane quintics by THOMAS DEDIEU
Let C be a smooth plane curve of degree d ≥
5, which we most often consider in its canonicalembedding in P g − , g = ( d − d − α ( C ) = 10. If d ≥ C has Clifford index strictly larger than 2, hence h ( N C/ P g − ( − k )) = 0 for all k > C is precisely 10-extendable; auniversal extension of C is described in [5, § C arerational. If d = 5 (resp. 6), then C has Clifford index 1 (resp. 2), h ( N C/ P g − ( − h ( N C/ P g − ( − k )) = 0 for all k >
2, see [8] and the references therein. The expectation inthis case is that there should exist a 12 (resp. 10) dimensional family of non-isomorphic surfaceextensions of C , and inside this a 2 (resp. 0) dimensional family of surfaces in which the firstinfinitesimal neighbourhood of C is trivial, see [3, § d = 6 is studied in details in[3] and [4, § P branched over C is a K C ; this K P (1 , K C . In this appendixI show that a similar situation holds in the case d = 5. Proposition A.1.
Let C be a smooth plane quintic, in its canonical embedding C ⊂ P . A.1.1.
There is a -dimensional family of non-isomorphic K extensions of C . A.1.2.
There is a -dimensional family of non-isomorphic K extensions of C in which the firstinfinitesimal neighbourhood of C is trivial. A.1.3.
There is a -extension of C , which is an arithmetically Gorenstein normal Fano varietyof dimension and index . This tells us in particular that while the (surjective) moduli map c : KC → M has generalfibre of dimension 10, its general fibre over the locus of plane quintics has dimension (at least)12.Our story begins with the following construction of Ide [7, § Construction A.2.
Let f be a degree 5 homogeneous polynomial in x = ( x , x , x ) defining C ⊂ P , and ℓ a linear functional in x defining a line intersecting C transversely. Then thedegree 5 weighted hypersurface S in Y = P (1 ,
2) defined by ℓ ( x ) y + f ( x ) = 0 , in homogeneous coordinates ( x , y ) so that the x i have weight 1 and y has weight 2, is a K C is the complete intersection of S and the degree 2hypersurface Π defined by y = 0.A geometric construction of S is as follows. Let ε : P ′ → P be the blow-up at the five pointsof L ∩ C , and L ′ and C ′ be the proper transforms of L and C respectively. Let π : S ′ → P ′ bethe double cover branched over the smooth curve L ′ + C ′ . The pull-back π ∗ L ′ is a ( − S is obtained from S ′ by contracting it to an A double point.The latter contraction is realized by the map given by the complete linear system | − π ∗ K P ′ | .Arguing as in [3], one sees that the latter has dimension 6, has π ∗ L ′ as a fixed part, and thegeneral membre of its moving part is mapped birationally by ε ◦ π to a smooth quintic C passingthrough the 5 points of L ∩ C and otherwise everywhere tangent to C . Note that | − K P ′ | has L ′ as fixed part and the pull-back by ε of the linear system of conics as moving part, hence forall D ∈ | C ′ | its restriction to D is the canonical divisor K D . This implies that the image of themap given by | − π ∗ K P ′ | is a K S ⊂ P (with one ordinary double point), having C as hyperplane section. As a sideremark, note that | − π ∗ K P ′ | maps the pull-backs by π of the 5exceptional curves of ε to 5 lines passing through the node of S .The above description is conveniently complemented by adopting the following equation basedpoint of view. Weighted projective geometry A.3.
For useful background on the subject, I recommend[9, Exercises V.1.3] and [6]. The weighted projective space Y = P (1 ,
2) has only one singularpoint, namely the coordinate point (0 : 0 : 0 : 1) at which it has a quotient singularity of type (1 , , v ( P )). It has dualizing sheaf O (5), which is not invertible. All (weighted) quintichypersurfaces in Y pass through the singular point (0 : 0 : 0 : 1), and the general such has anordinary double point there.By adjunction, the general quintic surface in Y has trivial (invertible) canonical sheaf, henceis a K O (2).Maintaining the notation of A.2, the curve C may be embedded in Y as the complete inter-section defined by(A.3.1) y = f ( x ) = 0 . On the other hand, the automorphisms of Y are given in homogeneous coordinates by(A.3.2) ( x : y ) ( A x : ay + Q ( x ))with A ∈ GL(3), a ∈ C ∗ , and Q ∈ H ( P (1 ) , O (2)), see for instance [1, § Y , which shows that it is isomorphic to a plane quintic curve.The sheaf O (2) is invertible on Y , and the associated complete linear system induces anembedding φ : Y → P with image a cone over a Veronese surface v ( P ) ⊂ P , with vertexa point. The map φ sends S to a K P passing through the vertex,and having C (in its canonical embedding) as a hyperplane section: in other words φ ( S ) is anextension of the canonical curve C . It coincides with the model of S in P given by | − π ∗ K P ′ | and described in A.2. The sheaf O Y (1) is not invertible on Y and neither is its restriction to S ;the associated complete linear series induces the rational map S P coinciding with ε ◦ π off the node. Proof of Proposition A.1.
As we have seen, we may consider C as the complete intersection in Y of two hypersurfaces of degrees 2 and 5 as in (A.3.1). The linear system of quintics containing C has dimension h ( Y, I C/Y (5)) − h ( Y, O Y (3)) = 13 , and its general element gives a K C ⊂ P . By the description in (A.3.2),The automorphisms of Y fixing the hypersurface y = 0 are all of the form ( x : y ) ( x : ay ),hence they form a 1-dimensional group. ¿From this we conclude that there is a 12-dimensionalfamily of mutually non-isomorphic K C , which proves A.1.1..Among these there is a 2-dimensional family of surfaces in which the first infinitesimal neigh-bourhood of C is trivial, namely those surfaces constructed as in A.2 by taking a double coverbranched over the disjoint union of C itself and a line: that the infinitesimal neighbourhood isindeed trivial in this case follows from the argument given in [2, p. 875]. We get a 2-dimensionalfamily by letting the line L move freely in P : the double cover of the blow-up remembers the5 points on C , hence two general lines give two non-isomorphic surfaces. This proves A.1.2..Eventually, to prove A.1.3. we use Totaro’s construction as in [4, § X the quintic hypersurface in P (1 , , ) given in homogeneous coordinates ( x : y : z ) by f ( x ) + G ( x , y ) z + · · · + G ( x , y ) z = 0 , here G , . . . , G form a basis of H ( Y, O (3)), and embed it in P with the complete linearsystem |O X (2) | , which is of the anticanonical series. That X ⊂ P is arithmetically Goren-stein and normal follows from the fact that it has canonical curves as linear sections as in [5, § (cid:3) Question A.4.
Does the family constructed in the above proof contain all K C ? In particular, is the dimension of the fibre of c over a plane quintic equal to 12? If theanswer is affirmative, then all K P (cid:0) ker( T Φ C ) ⊕ L k> H ( N C/ P ( − k )) (cid:1) , there is a uniquesurface extension of C in which it has the corresponding formal infinitesimal neighbourhood.This would imply that indeed all the surface extensions of C are gotten as quintics in Y having C as a degree 2 section, as in the proof.On the other hand, as was pointed out to me by Sernesi, the rational surface extensions ofcanonical plane curves constructed in [5, §
9] have one elliptic singularity which in the case ofquintics is smoothable. This suggests that these rational surface extensions may be smoothable,necessarily to K Remark
A.5 . As was the case for plane sextics [4, Rmk 3.7], canonical plane quintics pro-vide curves which are not complete intersections, for which the assumptions of either form ofTheorem 1.3 don’t hold, and which are extendable more than α times.Moreover, canonical plane quintics show that Zak’s claim [11, p. 278], that if an n -dimensional X ⊂ P N has α ( X ) < ( N − n − N +1) then it is at most α ( X )-extendable, cannot hold withoutan additional assumption. Remark
A.6 . We can play the same game with plane quartics and cubics as with sextics andquintics. Let ε : P ′ → P be the blow-up at the 8 (resp. 9) points of the transverse intersectionΓ ∩ C , with Γ a conic and C a smooth quartic (resp. E ∩ E , with E and E two smoothcubics), and π : S ′ → P ′ be the double cover branched over the proper transforms of Γ and C (resp. E and E ). In the former case, the linear system | − π ∗ K P ′ | maps S ′ to a quarticsurface S in P (1 ,
1) = P with an ordinary double point. In the latter case we get what Iwould describe as a virtual cubic surface in P (1 , | − π ∗ K P ′ | is then of theform | F | with F an elliptic curve, hence gives a map to P contracting S ′ onto a smooth conic. References [1] A. Al Amrani,
Classes d’id´eaux et groupe de Picard des fibr´es projectifs tordus , in
Proceedings of ResearchSymposium on K -Theory and its Applications (Ibadan, 1987) , vol. 2, 1989, 559–578. 20[2] A. Beauville and J.-Y. M´erindol, Sections hyperplanes des surfaces K
3, Duke Math. J. (1987), no. 4,873–878. 20[3] C. Ciliberto and T. Dedieu, Double covers and extensions , arXiv:2008.03109. 19[4] , K curves with index k >
1, arXiv:2012.10642. 19, 20, 21[5] C. Ciliberto, T. Dedieu and E. Sernesi,
Wahl maps and extensions of canonical curves and K surfaces , J.Reine Angew. Math. (2020), 219–245. 19, 21[6] A. R. Iano-Fletcher, Working with weighted complete intersections , in
Explicit birational geometry of 3-folds ,London Math. Soc. Lecture Note Ser., vol. 281, Cambridge Univ. Press, Cambridge, 2000, 101–173. 20[7] M. Ide,
Every curve of genus not greater than eight lies on a K surface , Nagoya Math. J. (2008),183–197. 19, 21[8] A. L. Knutsen, Global sections of twisted normal bundles of K3 surfaces and their hyperplane sections ,arXiv:1811.09805, to appear in Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 19[9] J. Koll´ar,
Rational curves on algebraic varieties , Ergebnisse der Mathematik und ihrer Grenzgebiete. 3.Folge. A Series of Modern Surveys in Mathematics, vol. 32, Springer-Verlag, Berlin, 1996. 20[10] J. Wahl,
Gaussian maps on algebraic curves , J. Differential Geom. (1990), no. 1, 77–98. 19[11] F. L. Zak, Some properties of dual varieties and their applications in projective geometry , in
Algebraicgeometry (Chicago, IL, 1989) , Lecture Notes in Math., vol. 1479, Springer, Berlin, 1991, 273–280. 21 ipartimento di Matematica e Fisica, Universit`a di Roma Tre, Largo San Leonardo Murialdo 1,00146, Roma, Italy. e-mail [email protected] Institut de Math´ematiques de Toulouse ; UMR5219. Universit´e de Toulouse ; CNRS. UPS IMT,F-31062 Toulouse Cedex 9, France. [email protected]@math.univ-toulouse.fr.