Elliptic curves, ACM bundles and Ulrich bundles on prime Fano threefolds

Let $X$ be any smooth prime Fano threefold of degree $2g-2$ in $\mathbb{P}^{g+1}$, with $g \in \{3,\ldots,10,12\}$. We prove that for any integer $d$ satisfying $\left\lfloor \frac{g+3}{2} \right\rfloor \leq d \leq g+3$ the Hilbert scheme parametrizing smooth irreducible elliptic curves of degree $d$ in $X$ is nonempty and has a component of dimension $d$, which is furthermore reduced except for the case when $(g,d)=(4,3)$ and $X$ is contained in a singular quadric. Consequently, we deduce that the moduli space of rank--two slope--stable $ACM$ bundles $\mathcal{F}_d$ on $X$ such that $\det(\mathcal{F}_d)=\mathcal{O}_X(1)$, $c_2(\mathcal{F}_d)\cdot \mathcal{O}_X(1)=d$ and $h^0(\mathcal{F}_d(-1))=0$ is nonempty and has a component of dimension $2d-g-2$, which is furthermore reduced except for the case when $(g,d)=(4,3)$ and $X$ is contained in a singular quadric. This completes the classification of rank-two $ACM$ bundles on prime Fano threefolds. Secondly, we prove that for every $h \in \mathbb{Z}^+$ the moduli space of stable Ulrich bundles $\mathcal{E}$ of rank $2h$ and determinant $\mathcal{O}_X(3h)$ on $X$ is nonempty and has a reduced component of dimension $h^2(g+3)+1$; this result is optimal in the sense that there are no other Ulrich bundles occurring on $X$. This in particular shows that any prime Fano threefold is Ulrich wild.


Introduction
Let X ⊂ P m be a smooth irreducible projective variety.A vector bundle E on X is said to be arithmetically Cohen-Macaulay (ACM for short) if h i (E(t)) = 0 for all t ∈ Z and all 1 i dim(X) − 1.A vector bundle E on X is said to be an Ulrich bundle if h i (E(−p)) = 0 for all i 0 and all 1 p dim(X).An Ulrich bundle is always ACM (cf., e.g., [Be1,(3.1)]and semistable (cf.[CM, Thm.2.9(a)]).
The problem of studying and classifying ACM or Ulrich bundles is in general a hard and interesting one, and in the last years there has been a lot of activity in this direction (see, e.g., [Be1,CH,Co,CMP] for surveys).There is however only a short list of varieties for which ACM or Ulrich bundles are completely classified.For works regarding ACM and Ulrich bundles on Fano threefolds, mostly of low ranks, we refer to [AC,Be1,Be2,Be3,Be4,BF1,BF2,BF3,CH,CKL,CM,CFaM1,CFaM2,CFaM3,CFiM,CFK,FP,LMS,LP,MT].
In this paper we consider the case of smooth prime Fano threefolds of degree 2g − 2 in P g+1 , for which one has g ∈ {3, . . ., 10, 12} (see [IP]), and we study ACM bundles of rank 2 and Ulrich bundles of any possible rank on them.
Rank-two bundles on threefolds are connected to curves in the threefolds by the famous Hartshorne-Serre correspondence (cf.Proposition 3.1).The bundles that we will be interested in are related to elliptic curves.Our first main result is: Theorem 1.1.Let X be a smooth prime Fano threefold of degree 2g − 2 in P g+1 .For any g+3 2 d g +3, the Hilbert scheme parametrizing smooth elliptic curves of degree d in X has an irreducible component of dimension d.There is a reduced such component except precisely when (g, d) = (4, 3) and X is contained in a singular quadric, in which case the Hilbert scheme is irreducible and nonreduced with 4-dimensional tangent space at every point.
Note that the Hilbert scheme in question is empty if d < g+3 2 (cf.Corollary 3.9).As a consequence, we prove the following: Theorem 1.2.Let X be a smooth prime Fano threefold of degree 2g − 2 in P g+1 .For any g+3 2 d g + 3 there exists a slope-stable ACM bundle F d on X such that The moduli space of such bundles has an irreducible component of dimension 2d − g − 2, which is reduced except precisely when (g, d) = (4, 3) and X is contained in a singular quadric, in which case the moduli space consists of a single point with a one-dimensional tangent space.
Note that the moduli spaces in the theorem are empty when d is outside the given range (cf.Remark 3.4 and Corollary 3.9).
For d > g+3 2 the latter result is an improvement of [BF1,Thm. 3.1], since our result avoids the assumption that X be ordinary (that is, X contains a line ℓ with normal bundle N ℓ/X ≃ O ℓ ⊕ O ℓ (−1)).For d = g+3 2 our statement is contained in [BF1,, except that we avoid again the assumption that X be ordinary when (g, d) = (3, 3); also note that the nonemptiness statement of the moduli spaces in question when d = g+3 2 has until now only been known by a case-by-case treatment in various papers (see [BF1,§3.1.1]and references therein), and that our proof via the existence of elliptic curves as in Theorem 1.1 provides an alternative, unified and simpler proof of this fact scattered in literature.In particular, our proof is independent of the one in [BF1].
The main motivation behind Theorem 1.2 is that it completes the classification of rank-two ACM bundles on smooth prime Fano threefolds.Indeed, since the property of being ACM is independent of twists, one may restrict the study of such bundles to the ones with determinants O X (m) for m ∈ {0, 1}.Combining [BF1,Thm. p. 114] and Theorem 1.2 one obtains: A smooth prime Fano threefold X of degree 2g − 2 in P g+1 carries a rank-two ACM bundle F such that c 1 (F) = [O X (m)] with m ∈ {0, 1} if and only if d := c 2 (F) • O X (1) satisfies: The new contribution is the abolition of the assumption that X be ordinary when d g 2 + 2 in the "if" part.The proof of Theorem 1.1 occupies §4.The idea of the proof is conceptually simple: We exploit Mukai's description of the K3 hyperplane sections of prime Fano threefolds and the surjectivity of the period map of K3 surfaces to prove that there always exists a codimension-one family of hyperplane sections containing a smooth elliptic curve of degree g+3 2 .We then construct the desired elliptic curves of higher degree by smoothing the union of an elliptic curve and a line intersecting transversally in one point.Despite the simple idea of the proof, there are several technical details to be fixed, and most of this work is carried out in §3.The proof of Theorem 1.2 follows as a rather direct consequence of Theorem 1.1 and is carried out in §5.
Our second main result concerns Ulrich bundles: Theorem 1.3.Let X be a smooth prime Fano threefold of degree 2g − 2 in P g+1 .Then, for every h ∈ Z + , there exists a stable1 Ulrich bundle E of rank 2h and determinant O X (3h) on X.The moduli space of such bundles has a reduced and irreducible component of dimension h2 (g + 3) + 1.
This result is optimal in the sense that a prime Fano threefold X does not carry any Ulrich bundles of odd rank by [CFK,Cor. 3.7] and any Ulrich bundle of rank 2h on X is forced to have determinant O X (3h) by [CFK,Thm. 3.5].
Theorem 1.3 in particular shows that any smooth prime Fano threefold of degree 2g − 2 in P g+1 is Ulrich wild 2 .Only very few cases of varieties carrying stable Ulrich bundles of infinitely many ranks are hitherto known.
The idea of the proof of Theorem 1.3 is also conceptually quite simple.Rank-two Ulrich bundles are obtained twisting by O X (1) the rank-two ACM bundles in Theorem 1.2 with d = g+3.Then Ulrich bundles of all even ranks r 4 are obtained by induction as deformations of extensions of rank r − 2 Ulrich bundles with rank 2 Ulrich bundles.The proof is carried out in §6.
Finally, in §7, taking profit from a correspondence we found in [CFK,Thm. 3.5] between rank two Ulrich bundles and certain curves on prime Fano threefolds, we propose an application to the moduli space of curves of genus 5g with theta-characteristics with g + 2 independent global sections, for 3 g 5.
Throughout the paper we work over the field of complex numbers.
Acknowledgements: C. Ciliberto and F. Flamini are members of GNSAGA of the Istituto Nazionale di Alta Matematica "F.Severi" and acknowledge support from the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006. A. L. Knutsen acknowledges support from the Meltzer Research Fund, the Trond Mohn Foundation Project "Pure Mathematics in Norway" and grant 261756 of the Research Council of Norway.

Basics on prime Fano threefolds
].These threefolds exist only for g ∈ {3, . . ., 10, 12} (see [IP]) and g is called the genus of the threefold.Any such X is projectively normal.Moreover, its general curve section is a canonical curve of genus g and its general hyperplane section is a K3 surface S of genus g, which satisfies Pic(S) ≃ Z[O S (1)] by Noether-Lefschetz (cf.[Vo,Thm. 3.33]).
Prime Fano threefolds of genus g = 3, 4, 5 are complete intersections: for g = 3 they are quartic hypersurfaces in P 4 , for g = 4 they are complete intersections of type (2, 3) in P 5 and for g = 5 they are complete intersections of type (2, 2, 2) in P 6 .For g 6 the description of prime Fano threefolds is more complicated (see [IP]).If g = 4, we will say that X is in case (⋆) 4 if the unique quadric containing X is singular.
The Hilbert scheme L(X) of lines in a prime Fano threefold X is one-dimensional (cf., e.g., [Is2,Cor. 1]), and the lines in X sweep out a surface that we will denote by R 1 (X).The normal bundle of any line ℓ in X satisfies As recalled in the introduction, X is said to be ordinary 3 if it contains a line ℓ with normal bundle We will make use of the following result: Proposition 2.1.Let X be a prime Fano threefold of degree 2g − 2 in P g+1 .One of the following occurs: (I) R 1 (X) has a component that is not a cone; or (II) R 1 (X) consists of 40 irreducible components that are all cones, and general pairs of lines in different components do not intersect.The latter case 4 occurs only for g = 3.
We also record here the following information that will be necessary for us.
Lemma 2.2.Let X be a prime Fano threefold of degree 2g − 2 in P g+1 .Then its dual variety X * is a hypersurface.As a consequence we have: (i) a general tangent hyperplane to X at a general point x ∈ X is simply tangent only at x; (ii) the general member of any g-dimensional family of singular hyperplane sections of X has a rational double point at a general point of X.
Proof.If X * is not a hypersurface, then the general tangent hyperplane to X at a general point x ∈ X is tangent along a linear subspace of positive dimension containing x (see [Ru,Thm. 1.5.10]),contradicting the fact that dim(L(X)) = 1.Assertion (i) follows again from [Ru,Thm. 1.5.10] and assertion (ii) follows from (i).

Elliptic curves and rank-two bundles
Throughout this section X will denote an arbitrary prime Fano threefold of degree 2g − 2 in P g+1 .We gather some useful results concerning rank-two vector bundles and elliptic curves on X.The starting point is the following: 3 Only two instances of non-ordinary prime Fano threefolds are known: the Fermat quartic for g = 3 and an example of Mukai-Umemura for g = 12 [MU].It is furthermore known that a general prime Fano threefold of any genus is ordinary. 4The Fermat quartic is an example, and it is not known whether there are others (see again [HT]).
Proposition 3.1.(i) Any local complete intersection curve E d ⊂ X of degree d with trivial canonical bundle gives rise to an exact sequence (ii) Conversely, if F d is a vector bundle on X satisfying (4), then it is slope-stable and any non-zero section s d ∈ H 0 (F d ) vanishes along a local complete intersection curve E d of degree d with trivial canonical bundle and there exists a sequence like (3).Moreover, non-proportional non-zero sections vanish along distinct curves.
(iii) If, in (ii), we furthermore have h 0 (F d ) 2, then we have a natural map where s 1 , s 2 is any basis for V and Z(s 1 ∧ s 2 ) is the zero scheme of s 1 ∧ s 2 .The surface Z(s 1 ∧ s 2 ) contains the pencil P(V ) of curves that are zero schemes Z(us 1 + vs 2 ) of sections of the form us 1 + vs 2 for all (u : v) ∈ P 1 and it is singular at least where s 1 and s 2 both vanish.
Proof.The correspondence in (i)-(ii) between the curve E d and the vector bundle F d with section s d is well-known, by the Hartshorne-Serre correspondence (see, e.g., [Ha1,Ha2,Ar]).Properties (4) can be deduced from (3); moreover, the facts that h 0 (F d (−1)) = 0 and and Pic(X) ≃ Z[O X (1)] imply that any section of F d vanishes along a curve.Finally, slope-stability of F d is an immediate consequence of the facts that Pic(X) ≃ Z[O X (1)], rk(F d ) = 2, det(F d ) ≃ O X (1) and h 0 (F d (−1)) = 0. Assertion (iii) is standard.
Remark 3.2.Assume that in case (iii) of Proposition 3.1 the zero schemes E i := Z(s i ), i = 1, 2, are smooth, irreducible, distinct, elliptic curves.Let Z be the intersection scheme E 1 ∩ E 2 .Then the surface S := Z(s 1 ∧ s 2 ) is singular along Z.More precisely, suppose that E 1 and E 2 have a contact of order k at a point p, so that Z locally at p is a curvilinear scheme of length k.We can introduce local coordinates (x, y, z) on X around p so that E 1 and E 2 have local equations x = y = 0 and x − y = y + z k = 0, respectively.Then S has equation xz k + y 2 = 0 around p, so that S has a double point at p and actually it has k double points infinitely near to p along E 1 or, equivalently, along E 2 , or, if we prefer, along Z.Therefore, on a desingularization S ′ of S the strict transforms E ′ 1 and E ′ 2 of E 1 and E 2 are isomorphic to E 1 and E 2 respectively, and belong to a base point free pencil, hence E ′ 2 1 = E ′ 2 2 = 0. Denote by H d (X) the Hilbert scheme parametrizing local complete intersection curves in X of degree d with trivial canonical bundle and by H d (X) the sublocus parametrizing smooth irreducible (elliptic) curves.Recall that the tangent space of ), by standard theory of Hilbert schemes.
Denote by M d (X) the moduli space of vector bundles on X satisfying (4), which are all slope-stable by Proposition 3.1(ii).By the same proposition there is a morphism The image consists of the vector bundles having a section (by Proposition 3.1(ii)), and the fiber over a point [F d ] is P(H 0 (X, F d )).
Remark 3.6.By (7) twisted by O X (1) and linear normality of X we see that E d spans a P d−1−h 1 (J E d /X (1)) in its embedding in P g+1 .Hence, E d is projectively normal if and only if h 1 (J E d /X (1)) = 0 (which can only happen for d g + 2) and nondegenerate if and only if h 1 (J E d /X (1)) = d − g − 2 (which can only happen for d g + 2).
Then the following conditions are equivalent: there is a section of N E d /X vanishing along a scheme of length d.
Proof.We have h ) by Lemma 3.5(i), proving that (i) and (iii) are equivalent.Moreover, for any x ∈ E d , we have h 0 (N E d /X (−x)) d − 2 > 0, whence there is a nonzero section s : O E d → N E d /X vanishing at x. Letting Z denote its scheme of zeros, and saturating, we obtain If deg(Z) < d, we thus have h 1 (N E d /X ) = 0.This proves that (i) implies (v).Assume (v) and let Z be the length d scheme along which a section s of This proves that (v) implies (iv).Clearly, (iv) implies (ii), and (ii) implies (i).
We denote by S d (X) the locus of hyperplane sections of X of the form Z(s 1 ∧ s 2 ) for (non-proportional) s 1 , s 2 ∈ H 0 (F d ) for some [F d ] ∈ M d (X).Note that each member of S d (X) contains a pencil of elements from H d (X) by Proposition 3.1(iii).The following result is crucial in the proof of Theorem 1.1.
(iii) the general S in any component of S d (X) has at worst a rational double point, which is a general point of X, and contains a pencil of members of H d (X), all zero loci of non-zero sections of the same element of M d (X), having at most a single base point at the rational double point of S; that through each intersection point there passes a unique line ℓ contained in X (hence this line intersects E d transversally in one point), and with the property that there is a hyperplane section of X containing sally) in at least two points through which there pass two non-intersecting lines.
Proof.Lemma 3.5(v), and this also holds for a general [F d ] ∈ M d (X) by semicontinuity.Using Lemma 3.3(i) we thus get whence p is surjective by Proposition 3.1(ii).For any component M ′ of M d (X), one has, by standard deformation theory of vector bundles and Lemmas 3.3(ii) and 3.5(vi), Let us pretend that M d (X) is a fine moduli space (if not one argues similarly at a local level).Then we can consider the variety G with a morphism π : 9).Together with (10), this implies We can define a universal version of the map w as in ( 5): By definition of S d (X), the image of the map w is S d (X).Note that every component of S d (X) has dimension at most g, as Pic(S) ≃ Z[O S (1)] for the general S ∈ |O X (1)|.We claim that (13) w is generically finite on any component of G.
To prove this, we argue by contradiction and assume that w is not generically finite.Let [S] be general in the image of w.Then there is a positive-dimensional family of pairs ) and S has a positive-dimensional family of pencils P(V ) as in Proposition 3.1(iii).Accordingly, S has a family E of dimension at least 2 of curves, the general one E being a smooth irreducible elliptic curve moving in a pencil.By Remark 3.2, the strict transform E ′ of E on the minimal desingularization of S satisfies E ′2 = 0, which contradicts the fact that dim(E) 2. We have thus proved (13).
From ( 13) it follows that dim(S ′ ) = dim(G ′ ) = g for any component S ′ of S d (X) and G ′ of G (by ( 11)), and as a consequence, also that for any component , which finishes the proof of (i).At the same time, (9) shows that h 1 (F d ) = 0, and Lemma 3.5(v) yields that h 0 (J which finishes the proof of (ii).
(iii) If the general member S of any component of S d (X) is singular, then by (i) and Lemma 2.2(ii), it has a rational double point at a general point of X.Then (iii) follows from Proposition 3.1(iii).
(iv) Let R be an irreducible component of R 1 (X) and S be a general member of a component S ′ of S d (X).Note that, since the possible rational double point of S is a general point of X, it does not lie on R.
We claim that C := S ∩R is reduced.Suppose this is not the case; then the hyperplane Π containing S would be tangent to R and, since dim(S ′ ) = g, it would be the general hyperplane tangent to R so the dual variety R * of R would be a hypersurface.On the other hand, since the curve cut out by Π on R is non-reduced, this would imply that the general tangent hyperplane to R is tangent along a curve, whence the dual variety R * of R would not be a hypersurface, a contradiction.
Next we claim that the family of curves Since E d moves in a pencil on S and has no base point on C by (iii), we see that E d passes through the general point of R, proving (iv).
(v) Since through the general point of R there passes only one line, and S contains at most finitely many lines (because its resolution of singularities is a K3 surface by (iii)), we can by (iv) assume that the locus (vi) If we are in case (II) of Proposition 2.1, the result follows from (v) and the fact that lines in different components of R 1 (X) do not intersect.
Assume therefore that we are in case (I) of Proposition 2.1, so that we can pick a component R of R 1 (X) that is not a cone.By assumption there is a reduced component of H d (X), which we denote by H ′ , and we let [ and occurs at points through which there passes a unique line in X, which each intersects E d in only that point and transversally (in particular the lines are distinct).It follows that E d ∩ R consists of md 3 points.
Denote by H • the dense open subset of H ′ parametrizing curves not contained in R and by Sym 2 (R) • the dense open subset of Sym 2 (R) consisting of all elements x + y such that x = y and there passes a unique line in R through both x and y and these lines are distinct.Consider the incidence variety Since dim(H • ) = d by (i) and the fibers of p d are finite, we have dim • such that the unique lines ℓ x and ℓ y through x and y, respectively (which are distinct by definition of Sym 2 (R) • ) intersect.If through a point of R there passes a unique line, this intersects only finitely many other lines on R (because R is not a cone).Hence either Z d is empty or dim(Z d ) = 2.In the former case we have finished.Thus assume that dim(Z d ) = 2.We want to prove that the image of q d does not lie inside Z d .
We remark that for x + y ∈ im(q d ), the fiber q −1 d (x + y) is isomorphic to H • (x, y), the locus of curves in H • passing through x and y.The tangent space of the fiber at a point (where we have used ( 14)).
Consider now the incidence variety , whence, since the general fiber is reduced, we have ( 16) Comparing ( 15) and ( 16), we see that Therefore, N E d /X has a nonzero section vanishing at md points, whence Lemma 3.7 yields h 0 (N E d /X ) > d = dim(H ′ ), contradicting the hypothesis that H ′ is reduced.
We finish the section with a result that we will use later: 6)).Assume first that there is a non-zero section 6 Indeed, if A and B are curves in a smooth threefold X intersecting transversally in one point p, then one locally has J A/X |B ≃ OB ⊕ Op.To prove this, introduce coordinates (x, y, z) around p, such that the ideal of B is (x, y) and the ideal of A is (y, z).
which we have to view as a C[[z]]-module.As such we see that y generates its torsion (because zy = 0), and z generates its free part.
to E d and saturating (and recalling Lemma 3.5(i)) one obtains 5), and S and S ′ contain the pencils P(V ) (containing E d ) and P(V ′ ) whose general members are smooth elliptic curves of degree d.By Proposition 3.8(iii), the pencils have at most one base point, and by the case we treated above, we may assume that they are base point free.
Let C := S ∩ S ′ .We claim that C is reduced.If g = 3, we may specialise V ′ to a general 2-dimensional subspace containing s d ; in this case S ∩ S ′ = E 3 + ℓ, with ℓ a line, which is reduced.If g = 4 and C is not reduced, then the intersection of the two hyperplanes containing S and S ′ would be a P 3 in P 5 that is tangent to X along a component of C, against Zak's theorem on tangencies (see [Ru,Thm. 3.2.3]).
Possibly Proposition 4.1.Let X ⊂ P g+1 be a prime Fano threefold of degree 2g−2.Then there is a codimension-one family of hyperplane sections of X containing smooth elliptic curves of degree d g , whereas a general hyperplane section does not contain such curves.
Proof.Let g ∈ {3, . . ., 10, 12}.By [Kn,Thm. 3.2] (which is a consequence of the surjectivity of the period map for K3 surfaces) there exists a smooth K3 surface S ⊂ P g containing a smooth elliptic curve E (whence E 2 = 0) satisfying deg(E) = d g and such that (S, O S (1)) is a so-called BN -general polarized K3 surface, as defined by Mukai in [Mu,Def. 3.8] 7 .From [Mu,Thms. 4.4 and 5.5] it follows that S can be realized as a hyperplane section of a prime Fano threefold X of degree 2g − 2 in P g+1 .The general hyperplane section of X does however not contain any smooth elliptic curves, as its Picard group is generated by its hyperplane section (cf.[Vo,Thm. 3.33]).
Consider now • the Hilbert scheme F g parametrizing (smooth) prime Fano threefolds of degree 2g − 2 in P g+1 , • the closure K g of the Hilbert scheme parametrizing smooth BN -general K3 surfaces of degree 2g − 2 in P g+1 , • the closure E g of the Hilbert scheme parametrizing smooth elliptic curves of degree d g in P g+1 , and S ⊂ X, • the natural projections α : I g → K g and β : It is well-known that both K g and F g are irreducible and that K * g is a divisor in K g .Set k := dim(K g ) and f := dim(F g ).
For any [X] ∈ F g the fiber β −1 ([X]) is isomorphic to the complete linear system |O X (1)| ≃ P g+1 .Hence dim(I g ) = f + g + 1, whence the general fiber of p has dimension f + g + 1 − k.Thus, special fibers have dimension at least f ) is surjective.It follows that any prime Fano threefold X ⊂ P g+1 contains a codimension one family of hyperplane sections containing a curve E that is a member of E g , whereas a general hyperplane section does not contain such a curve.
Let (E, S) be a general pair consisting of a hyperplane section S of X and a curve E in E g contained in S. We may write E = E ′ + R, with E ′ minimal with respect to the property that p a (E ′ ) = 1; thus R must consist of smooth rational curves and E ′ is of degree d ′ d g .By Lemma 2.2, the surface S has at worst a rational double point (at a general point of X).By considering the desingularization of S, one sees that E ′ is linearly equivalent to a smooth, irreducible elliptic curve on S. By Corollary 3.9, we must have d ′ = d g , so that E = E ′ .This proves that S contains a smooth elliptic curve of degree d g .
Remark 4.2.A similar reasoning using [Kn,Thm. 3.2] proves that a general prime Fano threefold X ⊂ P g+1 contains smooth elliptic curves of any degree d d g , but it fails to guarantee that all such threefolds contain such curves.
Remark 4.3.Something can be said about the structure of the Hilbert schemes H dg (X).
(i) Let X ⊂ P 4 be a smooth quartic threefold.There is a map H 3 (X) → L(X), sending a cubic E to the unique line residual intersection of X with the plane spanned by E. The fibers are isomorphic to P 2 .
(ii) Let X ⊂ P 5 be a smooth prime Fano threefold of degree 6.Then X is contained in a unique quadric Q X , which is either smooth or a cone with vertex a point, and is cut out by a cubic there.We have dim(H 3 (X)) = 3 by Propositions 4.1 and 3.8.Any cubic E in H 3 (X) spans a plane, which is contained in Q X , and distinct cubics clearly span distinct planes.If Q X is smooth, it contains two families of planes both parametrised by a P 3 , whence H 3 (X) has two irreducible components.If Q X is singular (that is, X is as in (⋆) 4 ), it contains only one family of planes, whence H 3 (X) is irreducible.
(iii) Let X ⊂ P 6 be a smooth prime Fano threefold of degree 8, which is the complete intersection of three quadrics.Any smooth elliptic normal quartic E ⊂ X is a complete intersection of two quadrics in its span E = P 3 .Since X is intersection of three quadrics in P 6 , then E is contained in one of the quadrics containing X, which is therefore singular.Let H be the Hesse curve, namely the curve in the net of quadrics |I X/P 6 (2)| parametrizing singular quadrics.Then any point h ∈ H corresponds to a quadric Q h having at most a double line as singular locus, and it is easy to see that quadrics Q h having a double line are isolated in H. Therefore, the maximal dimension for linear spaces contained in Q h , for any h ∈ H, is 3.This gives a 2 : 1 cover f : H −→ H defined as follows: for h ∈ H general, f −1 (h) consists of two points corresponding to the two distinct rulings of P 3 s contained in Q h .Thus f ramifies exactly at points h ∈ H corresponding to quadrics Q h with a double line.The map f induces a map g : P −→ H, where P is a P 3 -bundle over H such that, for any h ′ ∈ H, the fiber g −1 (h ′ ) parametrizes the ruling of P 3 s corresponding to h ′ and which is contained in Q f (h ′ ) .Any elliptic normal quartic E ⊂ X determines a point in P; conversely, any point in P gives rise to a P 3 which cuts out on X a normal elliptic quartic curve E in X.Thus H 4 (X) is isomorphic to P.
(iv) For g 6 it is shown in [BF1,Thm. 3.2] that M dg (X) is irreducible, whence also H dg (X) is irreducible by Proposition 3.8(i).We will however not use this.
Let [E dg ] ∈ H dg (X) be general in any component of H dg (X).Then h 1 (N E dg /X ) = 0.
Proof.We first note that since d 3 = 3, the lemma is proved for g = 3 by Lemma 3.10.We may therefore assume that g 4.
Let H ′ be an irreducible component of H dg (X) and let M ′ := p(H ′ ) (cf. ( 6) and recall that p is surjective by Proposition 3.8(i)).Denote by S ′ the locus in S dg (X) of hyperplane sections of X of the form Z(s 1 ∧ s 2 ) for (non-proportional) s 1 , s 2 ∈ H 0 (F dg ) for some [F dg ] ∈ M ′ .By Proposition 3.8(iii), the general member S of S ′ has at most a rational double point and contains a pencil of curves that are all zero loci of sections of a member F dg of M ′ (which we may take to be general), having at most a single base point at the rational double point.We write the pencil as |E dg |, where we may take [E dg ] ∈ H ′ to be general.We henceforth write E = E dg and F = F dg for simplicity.
Assume first that we are in the case where |E| has a base point.Let E ′ ∈ |E| be a smooth curve different from E and let s ′ ∈ H 0 (F) be such that E ′ := Z(s ′ ).Then E and E ′ intersect transversally in one point x.A local computation as in the footnote of page 10 shows that to E and saturating (and recalling Lemma 3.5(i)) one therefore obtains For the rest of the proof we may therefore assume that |E| is base point free on S, in particular, we may assume that E does not pass through the possible rational double point of S. In particular, the divisor ∆ := O S (1)(−E) is Cartier on S. Since h 0 (J E/X (1)) g + 2 − d g 3 by Lemma 3.5(iii), the short exact sequence yields that h 0 (∆) 2, whence ∆ can be represented by an effective nonzero divisor.
Claim 4.5.h 1 (N E/X ) = 0 if and only if there is a section of F whose zero locus is ∆.This can only occur for g = 4 or 5.
Thus the lemma is proved except for the cases g = 4 and 5, which we now treat.One computes ∆ 2 = 0, whence p a (∆) = 1, and deg(∆) = d g .As in the proof of Proposition 4.1 we may write ∆ = ∆ ′ + R with ∆ ′ minimal with respect to the property that p a (∆ ′ ) = 1, and one may again check that ∆ ′ can be represented by a smooth curve.Thus, Corollary 3.9 yields that ∆ = ∆ ′ ; in other words, we may assume that |∆| is a pencil whose general member is a smooth elliptic curve of degree d g .
We now treat the cases g = 4 and g = 5 separately.First we examine the case g = 4, where d g = 3.We prove that if h 1 (N E/X ) = 0, then X ⊂ P 5 is contained in a quadric cone.We argue by contradiction and assume that X is contained in a smooth quadric Q. Recalling the map (5), we may write [S] = w([V ]), where V is a general member of Grass(2, H 0 (F)) = Grass(2, 4).Then the pencil |E|, whose general member is a smooth plane cubic, equals P(V ).The surface S is contained in a hyperplane π S of P 5 and it is there a complete intersection of type (2, 3).The unique quadric Q S containing S is the intersection Q S = Q ∩ π S .Since S contains a pencil of plane cubics, the quadric Q S contains a pencil of planes, thus it is singular.Since Q is smooth, Q S must be of rank 4, hence it is a cone with vertex a point, and therefore it contains a unique other pencil of planes that cut out on S the pencil |∆|, with |E| and |∆| distinct.By the claim, there is another V ′ in Grass(2, H 0 (F)) such that w([V ′ ]) = S and P(V ′ ) = |∆|, and then w Let us now interpret the map w in (5) as a generically 2 : 1 map w : Grass(2, H 0 (F)) −→ (P 5 ) * We have that w(Grass(2, H 0 (F))) = Q * .The above argument shows that w cannot have any ramification point, since it would correspond to a hyperplane section of Q of rank 3.This is absurd since Q * , which is itself a smooth quadric in P 5 , is simply connected.This gives a contradiction that proves that Q must be a quadric cone.
Next we consider the case g = 5, where d g = 4.We assume that h 1 (N E/X ) = 0 and want to reach a contradiction.As we saw above, the surface S is endowed with two distinct pencils |E| and |∆| of elliptic quartic curves.Then E • ∆ = 4 and, by the claim, E and ∆ are both zero schemes of two sections s and s ′ of F. We can consider W = s, s ′ and [S ′ ] = w([W ]); then S ′ is a surface with 4 singular points at the intersection of E and ∆, which we can assume to be general in |E| and |∆|.The four singular points in question, forming a scheme Z, lie in a plane α, because on both E and ∆ they are a hyperplane section.Let us consider the projection π of S ′ from α to a plane β.The surface S ′ has a pencil P(W ) of elliptic quartic curves that contains both E and ∆.The curves of this pencil, all containing the scheme Z, are contracted to points by the projection π and therefore the image of S ′ under this projection is a curve.A general hyperplane section of S ′ containing E consists of E plus another elliptic quartic curve E ′ that, as E, passes through Z. Hence E ′ is also contracted by π, thus E ′ belongs to P(W ) as well as E. This implies that O S ′ (1) ∼ 2E ∼ 2E ′ , whence π(S ′ ) is a conic.This proves that S ′ sits on a quadric Q ′ that is a cone with vertex the plane α.The quadric Q ′ is the hyperplane section of a quadric Q in P 6 containing X, and Q must be singular with vertex a line.
As in the proof of Proposition 3.88 let us consider the universal grassmannian bundle G → M 4 (X), whose points correspond to pairs ([F], [V ]) with [F] ∈ M 4 (X) and [V ] ∈ Grass(2, H 0 (F)), and the map w : G → |O X (1)| as in (12).In the proof of Proposition 3.8 we proved that dim(G) = g = 5 and that w is generically finite.
As in the case g = 4, we have an involution ι : G G, such that for a general ), with P(V ) and P(V ′ ) two pencils of elliptic quartic curves on the surface defined by w( ).The fixed points of the involution ι form a non-zero divisor D in G.The image via the map w of a generic point ([F], [V ]) in a suitable component of D is a surface S with four singular points on a plane, with the unique pencil P(V ) of elliptic quartic curves passing through the four singular points.Such surfaces therefore lie, as we saw, on a quadric cone with vertex a plane.Accordingly, we have a family of quadrics containing X that are cones with vertex a line.This family must have at least dimension 1, because a unique quadric in P 6 with vertex a line gives rise only to a 3-dimensional family of hyperplane sections with vertex a plane.In conclusion, in the net N of quadrics that cut out X, there is at least a one-dimensional subfamily V of quadrics with vertex a line.Let [Q] ∈ V be a general point.Then N is spanned by the tangent line L to V at Q and by another general quadric Q ′ .Now the quadrics in the pencil L all contain the singular line ℓ of Q, so they cut out a complete intersection singular along ℓ.By intersecting with Q ′ we see that X must be singular at the intersection points of Q ′ with ℓ, a contradiction.Lemma 4.6.Let X ⊂ P 5 be a smooth prime Fano threefold of degree 6 as in case (⋆) 4 .Then H 3 (X) is nonreduced with 4-dimensional tangent space at every point.
Proof.By Lemma 3.7 and Remark 4.3(ii) it suffices to prove that h 1 (N E/X ) > 0 for general [E] ∈ H 3 (X).By definition, X is contained in a singular quadric Q X with vertex one point.From the exact sequence we see that it suffices to prove that h 1 (N E/Q X ) > 0.
Consider the incidence variety with the projection f : consists of two irreducible components of dimension 12, and both are P 9 -bundles over P 3 .Indeed, a plane cubic C sits in Q if and only if the plane its spans lies in Q, and Q contains two families of planes both parametrised by a P 3 .If Q is singular with vertex one point, then the fiber f −1 (Q) consists of only one irreducible component of dimension 12, which is again a P 9 -bundle over P 3 , since Q contains only one family of planes.Therefore f −1 (Q X ) is non-reduced.Since it is isomorphic to the component of the Hilbert scheme of Q X parametrising plane cubics, we see that this component is non-reduced.Hence h 1 (N E/Q X ) > 0, as wanted.
Proposition 4.1 guarantees that H dg (X) = ∅ on any prime Fano threefold X ⊂ P g+1 , and Proposition 3.8(i) yields that each component has dimension d g .By Lemma 4.4 and standard theory of Hilbert schemes, H dg (X) is smooth at a general point of each component, whence reduced, except in case (⋆) 4 , where H 3 (X) is irreducible of dimension 3 by Remark 4.3(ii), with 4-dimensional tangent space everywhere by Lemma 4.6.
This concludes the proof of Theorem 1.1 in the case d = d g .
4.2.The inductive step.Assume that d g d g + 2 and that Theorem 1.1 has been proved for the integer d.We will prove that it also holds for d + 1, that is, that H d+1 (X) has a reduced component of dimension d containing points representing elliptic normal curves if d + 1 g + 2 and nondegenerate curves if d + 1 = g + 3. To do so, it will be sufficient, by standard theory of Hilbert schemes and Remark 3.6, to prove the existence of a smooth elliptic curve E d+1 of degree d + 1 in X such that h 1 (N E d+1 /X ) = 0 and Denote by H d (X) ′ a component of H d (X) satisfying the conditions of Theorem 1.1.This means that dim(H d (X) ′ ) = d and, except for the case (⋆) 4 with d = 3, the scheme • The case d g + 1.By Proposition 3.8(v), the general curve E d in H d (X) ′ intersects R 1 (X) transversally in distinct points, and through each point there passes a (unique) line intersecting E d transversally in one point.Pick any of these lines ℓ and call the intersection point P .Set of E ′ is isomorphic to O P , since E ′ has the node at P as its only singularity.By local computations (see, e.g., the proof of [HH,Thm. 4.1]), we have the exact sequences Thus, from ( 17)-( 19), and recalling (2) and Lemma 3.5(i), we find Hence, by standard deformation theory, the Hilbert scheme of X has dimension at least d + 1 in a neighborhood U of [E ′ ].Since the sublocus of reducible curves of the form E d ∪ ℓ has dimension equal to dim(H d ) = d (by Proposition 3.8(i)), the general member of U must be a smooth elliptic curve.Thus, we have proved that H d+1 (X) is nonempty and has a component H d+1 (X) ′ containing elements of the form and ℓ a line intersecting E d transversally in one point.
We now prove that H d+1 (X) ′ contains elements E d+1 such that h 1 (N E d+1 /X ) = 0.This is automatically satisfied if d + 1 = g = 4 by Lemma 3.10.Therefore, we may assume that h 0 (N As we just saw, we may specialize a general E d+1 in H d+1 (X) ′ in a flat family to a curve E ′ := E d ∪ ℓ, with E d a general member of H d (X) ′ and ℓ a line intersecting E d transversally in one point P .Arguing by contradiction, using Lemma 3.7, we will assume that Then, by ( 17) and ( 19), we find Similarly, from ( 17) and ( 18), together with (2), we find Thus, both V ℓ and V E d are properly contained in V .This means that the limit of a general section s : Since N E ′ /X is locally free, as E ′ is nodal, whence a local complete intersection, coker(s ′ ) must be locally free, hence isomorphic to . Hence (18) yields χ(N ℓ/X ) = 2, in contradiction with (2).We have therefore proved that the general member We have left to prove that h 1 (J E d+1 /X (1)) = 0.This is satisfied if d g by Proposition 3.8(ii).If d = g + 1 we may as above specialize E g+2 in a flat family to By what we just proved, h 1 (J E g+1 /X (1)) = 0, whence h 0 (J E g+1 /X (1)) = 1 by Lemma 3.5(iii).Since, by Proposition 3.8(v), there is a hyperplane section of X containing E g+1 but not ℓ, we have h 0 (J E g+1 ∪ℓ/X (1)) = 0. Hence, by semicontinuity, h 0 (J E g+2 /X (1)) = 0 as well, so that h 1 (J E g+2 /X (1)) = 0 by Lemma 3.5(iii) again, as desired.
This concludes the proof of Theorem 1.1 in the case d g + 1.
• The case d = g+2.By Proposition 3.8(v)-(vi), the general curve E g+1 in H g+1 (X) ′ intersects R 1 (X) in at least two points x and y through which there pass unique lines ℓ x and ℓ y and such that ℓ x ∩ ℓ y = ∅ and each intersects E g+1 transversally at one point.By the first part of the proof, E g+1 ∪ ℓ x deforms to a smooth elliptic curve E g+2 in H g+2 (X) having the property that h 1 (N E g+2 /X ) = 0. Thus, we find a component H g+2 (X) ′ of H g+2 (X) of dimension dim(H g+2 (X) ′ ) = g + 2 with general member E g+2 satisfying h 1 (N E g+2 /X ) = 0 and containing E g+1 ∪ ℓ x in its closure.The latter curve has the property that the line ℓ y intersects it transversally in one point.Since this is an open property among members of the closure of H g+2 (X) ′ , and the general member intersects finitely many lines, it also holds for the general one.This means that for the general [E g+2 ] ∈ H g+2 (X) ′ , there is a line ℓ intersecting E g+2 transversally in one point.The first part of the proof applied once more yields that E g+2 ∪ ℓ deforms to a smooth elliptic curve E g+3 in H g+3 (X) having the property that h 1 (N E g+3 /X ) = 0.
This concludes the proof of Theorem 1.1.6).We will prove M d (X) ′ and its general member satisfy the conditions of Theorem 1.2.
In the case where d = 3 and X is as in (⋆) 4 , we have that H 3 (X) is irreducible by Remark 4.3(ii), whence M 3 (X) = p(H 3 (X)) is also irreducible, and of dimension 0, by Proposition 3.8(i).Moreover, by Lemmas 3.7 and 4.4, we have h 1 (N E 3 /X ) = 1 in this case; therefore, as h 1 (F 3 ) = 0 by Proposition 3.8(ii), we have h 2 (F 3 ⊗ F * 3 ) = 1 by Lemma 3.5(vi).Using the remaining identities in Lemmas 3.3(ii) and 3.5(vi) and standard theory on moduli spaces of vector bundles, we find that the tangent space at the general point of M 3 (X) ′ has dimension In the remaining cases, Lemma 3.3(ii).By standard theory of moduli spaces of vector bundles (see, e.g., [CH,Prop. 2.10] We have left to prove that the general member for all n ∈ Z, we see that it suffices to prove that h 1 (F ′ d (n)) = 0 for all n ∈ Z. From ( 3) and ( 7) we see that for all n ∈ Z.Thus, as X is projectively normal, F d is ACM if and only if E d is projectively normal, which happens precisely when E d is an elliptic normal curve.We are therefore done by Theorem 1.1 in the case d g + 2.
This concludes the proof of Theorem 1.2.Note that we have also proved the following Corollary 5.2.Let X ⊂ P g−1 be a prime Fano threefold of degree 2g − 2. There exists a slope-stable rank-two Ulrich bundle with determinant O X (3) on X.The moduli space of such bundles has a reduced component of dimension g + 4 and its general member E satisfies Proof.This follows from the case d = g + 3 of Theorem 1.2 and Lemma 5.1, letting E := F g+3 (1).Recall that we have proved that h 2 (F g+3 ⊗ F * g+3 ) = 0; the rest of (20) follows from Lemmas 3.3(ii) and 3.5(vii) and semicontinuity.

Ulrich bundles of higher ranks on prime Fano threefolds
In this section we will inductively define families of vector bundles of all even ranks on any prime Fano threefold, whose general members will be slope-stable Ulrich bundles, and thus prove Theorem 1.3.= −(g + 3) − (h − 1)(g + 3) = −h(g + 3).
Proof.We prove this by induction on h, the case h = 1 being satisfied by the choice of U X (1).By Lemma 6.1(i)-(ii) we have, for general Hence, by definition, U X (h + 1) = ∅.Its members have ranks 2(h + 1) and determinant O X (3(h+1)).It is immediate that extensions of Ulrich bundles are still Ulrich bundles, so the general member E h+1 of U X (h+1) is an Ulrich bundle.It satisfies h j (E h+1 ⊗E * h+1 ) = 0 for j = 2, 3 by Lemma 6.1(iii).
To finish the proof of Theorem 1.3 we have left to prove that the general member of U X (h) is slope-stable and that U X (h) is smooth of dimension h 2 (g + 3) + 1 at its general point.We will again prove this by induction on h.First we need an auxiliary result.Lemma 6.3.Let F h+1 be a general member of U X (h + 1) ext , sitting in an extension like (21).Assume furthermore that E ′ 1 and E h are slope-stable.Let G be a destabilizing subsheaf of F h+1 .Then G * ≃ E ′ 1 * and (F h+1 /G) * ≃ E * h−1 .Proof.We note that (21) yields , where µ as usual denotes the slope of a sheaf with respect to O X (1).Assume that G is a destabilizing subsheaf of F h+1 , that is 0 < rk(G) < rk(F h+1 ) = 2(h + 1) and µ(G) 3(g − 1).Define Then we may put (21) into a commutative diagram with exact rows and columns: ) and E ′ 1 is slope-stable by assumption, we must have rk , where D ′ is an effective divisor supported on the codimension-one locus of the support of K ′ , we have Hence D ′ = 0, which means that K ′ is supported in codimension at least two.Thus, ext i (K ′ , O X ) = 0 for i 1, and it follows that where D ′′ is an effective divisor supported on the codimension-one locus of the support of Q ′ , we have Hence D ′′ = 0, which means that Q ′ is supported in codimension at least two.Thus, ext i (Q ′ , O X ) = 0 for i 1, and it follows that which gives a splitting of (21), a contradiction.We are therefore left with the case where rk(Q) > 0 and rk(K) > 0. Since E ′ 1 and E h are stable of slope 3(g − 1) by hypothesis, we have µ(K) 3(g − 1), with equality only if rk(K) = rk(E ′ 1 ), ( 24) If equality holds in both ( 24)-( 25), we would have ), a contradiction to G ⊂ F h+1 being a destabilizing sheaf.Hence, at least one of the inequalities in ( 24)-( 25) must be strict, yielding Proposition 6.4.For all h ∈ Z + the scheme U (h) is reduced of dimension h 2 (g + 3) + 1 and its general member is slope-stable.
Proof.We prove this by induction on h, the case h = 1 again being satisfied by the choice of U X (1).Assuming that we have proved the lemma for all integers i h, we prove it for h + 1.
Finally, we prove that E h+1 is slope-stable.Assume, to get a contradiction, that it is not.Then we may find a one-parameter family of bundles {E (t) } over the disc D such that E (t) is a general member of U X (h + 1) for t = 0 and E (0) lies in U X (h + 1) ext , and such that we have a destabilizing sequence / / 0 for t = 0, which we can take to be saturated, that is, such that F (t) is torsion free, whence so that F (t) and G (t) are (Ulrich) vector bundles (see [CH,Thm. 2.9] or [Be1,(3.2)]).
This concludes the proof of Theorem 1.3.
7. An application to curves with certain theta-characteristics In this section we make some remarks about rank 2 Ulrich bundles on prime Fano threefolds X of genus g and degree 2g − 2 in P g+1 .By [CFK,Ex. 3.8] any such X carries a rank 2 Ulrich bundle E if and only if det(E) = O X (3) and there is a smooth, irreducible, non-degenerate, linearly and quadratically normal curve C in X such that More precisely, given a rank 2 Ulrich bundle E on X and a general section s ∈ H 0 (E), the curve C is the zero locus of s, and conversely, if there is such a curve C ⊂ X, there is a rank 2 Ulrich bundle E on X and a non-zero section s ∈ H 0 (E), such that C is the zero locus of s.
Let F g be the Hilbert scheme parametrizing (smooth) prime Fano threefolds X of degree 2g − 2 in P g+1 , and C g be the union of the components of the Hilbert scheme containing points parametrizing curves C in P g+1 as above.Consider the incidence correspondence I g = {(X, C) ∈ F g × C g : C ⊂ X}.
Let φ g : I g −→ C g be the projection to the second factor and consider the moduli map where, as usual, M p denotes the moduli space of smooth curves of genus p.The image of µ g is contained in the locus M g+1 5g of curves C having a theta-characteristic θ with dim(|θ|) = g +1.According to a result by J. Harris (see [Har,Cor. 1.11]), any irreducible component of M g+1 5g in M 5g has expected codimension (30) (g + 2)(g + 1) 2 and the actual codimension is at most this.
We want to discuss in this section the following question: is µ g dominant onto an irreducible component of M g+1 5g ?First we prove the following: Proposition 7.1.For g 6 the image of µ g has codimension larger than (30) and therefore µ g is not dominant onto any irreducible component of M g+1 5g , or equivalently, the general curve in any component of M g+1 5g does not sit on a prime Fano threefold X of degree 2g − 2 in P g+1 as a projectively Cohen-Macaulay semicanonical curve.
Next we complete the picture by proving that: Proposition 7.2.For 3 g 5 the map µ g is dominant onto a union of irreducible components of M g+1 5g , each of which is uniruled and of the expected codimension (30).Proof.We prove the proposition only in case g = 5.The cases 3 g 4, being similar, can be left to the reader.
One has dim(F 5 ) = 75 hence by (31) we have dim(I 5 ) = 99.Next we claim that: (a) φ 5 is birational onto its image; (b) any component of the image of φ 5 is a component of the sublocus of C 5 parametrizing curves whose hyperplane section is a theta-characteristic; (c) the fibers of the map µ 5 are finite, modulo the action of PGL(7, C).To prove (a), note that, given C, the quadrics of P 6 cut out on it the complete canonical series, which has dimension g(C) − 1 = 24.Since dim(|O P 6 (2)|) = 27, one has dim(|I C/P 6 (2)|) = 2, hence C sits on a unique prime Fano threefold X of degree 8 in P 6 .
To prove (b), denote by K any component of the sublocus of H 5 parametrizing curves whose hyperplane section is a theta-characteristic and which contains a component of im(φ 5 ).If C is a general element of K, by semi-continuity C is still projectively Cohen-Macaulay, hence one sees, as above, that dim(|I C/P 6 (2)|) = 2, so C sits on a prime Fano threefold X of degree 8 in P 6 , thus C in im(φ 5 ) as wanted.
Finally, to prove (c), suppose that a fiber of µ 5 has a positive dimensional component C. Every element C in C is endowed with its hyperplane section bundle O C (1), which is a theta-characteristic θ with h 0 (θ) = g + 2. Fix C and take a general element C ′ in C. Since C and C ′ are in the same fiber of µ 5 , they are isomorphic.The isomorphism that maps C ′ to C has to map the theta-characteristic O C ′ (1) to a very ample thetacharacteristic θ of C with h 0 (θ) = g + 2. Since C does not have a continuous family of theta-characteristics, the theta-characteristic θ has to be independent of C ′ in the fiber C which shows that all curves in C are projectively equivalent.
By Finally, each component of im(µ 5 ) is uniruled because so is I 5 .
smooth at x and there passes a unique line ℓ through x and ℓ ⊂ S} is the whole C but finitely many points.Since the possible base point of the pencil of curves |E d | on S lies outside R, whence off C, the general member E d of the pencil intersects C in points of C • and transversally.Thus the intersection E d ∩R is transverse, and through each intersection point there passes a unique line ℓ that is not contained in S. Since ℓ • S = ℓ • O X (1) = 1, we have that ℓ ∩ E d is transverse and consists of only one point.This proves (v).
replacing E d by a general member of |E d |, the intersection Z := E d ∩ C is transversal and consists of d distinct points, along the smooth locus of C. Pick any x ∈ Z and pick the unique curve E ′ = Z(s ′ ), for s ′ ∈ H 0 (F), in the pencil P(V ′ ) passing through x.Then E d and E ′ can only intersect along C, and if the intersection consists of fewer than d points, then the lemma is proved, by what we said above.Hence, we may assume that E ∩ E ′ = Z.In particular, the surface S := Z(s ∧ s ′ ) has d nodes at Z. Since C is smooth at Z, it cannot lie on S, as it would be a Cartier divisor on it, because C ⊂ S ∩ S and equality holds as both sides have degree 2g − 2. But then 2g − 2 = C • S 2d = 2g, a contradiction, as desired.4. Proof of Theorem 1.1 We will use induction on d, starting with the case d = d g := g+3 2 , recalling that there are no smooth elliptic curves of degree d in X if d < d g by Corollary 3.9.4.1.The base case d = d g .The key result is the following: deg(C) = 5g − 1, g(C) = 5g, ω C = O C (2), whence O C (1) is a theta-characteristic on C with h 0 (O C (1)) = g + 2.It is easy to check that such curves are automatically projectively Cohen-Macaulay.
R}, with projections f d : I d → H • and g d : I d → R. Since the fibers of f d are finite, we have, as above, dim(I d ) = d and I d is reduced.By (iv), the map g d is dominant, whence for general x ∈ R, the fiber g −1 d (x) is reduced and isomorphic to H • (x), the locus of curves in H • passing through x, and has dimension d − 2. The tangent space of the fiber at a general point [