The desingularization of the theta divisor of a cubic threefold as a moduli space

We show that the moduli space $\overline{M}_X(v)$ of Gieseker stable sheaves on a smooth cubic threefold $X$ with Chern character $v = (3,-H,-H^2/2,H^3/6)$ is smooth and of dimension four. Moreover, the Abel-Jacobi map to the intermediate Jacobian of $X$ maps it birationally onto the theta divisor $\Theta$, contracting only a copy of $X \subset \overline{M}_X(v)$ to the singular point $0 \in \Theta$. We use this result to give a new proof of a categorical version of the Torelli theorem for cubic threefolds, which says that $X$ can be recovered from its Kuznetsov component $\operatorname{Ku}(X) \subset \mathrm{D}^{\mathrm{b}}(X)$. Similarly, this leads to a new proof of the description of the singularity of the theta divisor, and thus of the classical Torelli theorem for cubic threefolds, i.e., that $X$ can be recovered from its intermediate Jacobian.

Moduli spaces of sheaves provide examples of algebraic varieties with an interesting and rich geometry and they have been widely studied in the past few decades.In particular, there are many strong results regarding moduli spaces on surfaces, while the situation on threefolds is less understood.We refer to Huybrechts and Lehn [23] for a more detailed account of the theory, which has been revolutionized by the introduction of stability conditions on triangulated categories by Bridgeland [12].
Perhaps the main player of the seminal paper by Clemens and Griffiths [14] on the geometry of cubic threefolds is the theta divisor ‚ of its intermediate Jacobian J.X /.Various authors have studied parametrizations of the theta divisor by moduli spaces of sheaves; see Artebani, Kloosterman and Pacini [3], Beauville [9] and Iliev [24].
In this paper, we find a new, and in a sense most efficient, parametrization of this type: a smooth four-dimensional moduli space of stable sheaves isomorphic to the desingularization of the theta divisor.
Let X P 4 be a smooth cubic threefold over C and H the hyperplane section.Let M X .v/be the moduli space of Gieseker-semistable sheaves on X with Chern character v WD 3; H; 1 2 H 2 ; 1 6 H 3 .
Theorem 7.1 The moduli space M X .v/ is smooth and irreducible of dimension 4.More precisely, it is the blowup of ‚ in its unique singular point.The exceptional divisor is isomorphic to the cubic threefold X itself , and parametrizes non-locally-free sheaves in M X .v/.

Moduli space in the Kuznetsov component
The original motivation for our analysis of the moduli space M X .v/comes from the study of moduli spaces of stable objects in a full triangulated subcategory Ku.X / D b .X / called the Kuznetsov component.
It is defined through the semiorthogonal decomposition D b .X / D hKu.X /; O X ; O X .H /i: See Kuznetsov [25] for details on the decomposition and on the Kuznetsov component.
Stability conditions on Ku.X / have been constructed in Bernardara, Macrì, Mehrotra and Stellari [11] and Bayer, Lahoz, Macrì and Stellari [5].These stability conditions are Serre-invariant, which roughly means that stability of an object is preserved by the action of the Serre functor of Ku.X /; see Section 8 for the precise definition.This property allows us to study stability of objects irrespective of the specific construction of stability conditions.
The class v in Theorem 7.1 is chosen as the class of the projection K P of a skyscraper sheaf O P for a point P 2 X , which is defined by the short exact sequence 0 !K P !O ˚4 !I P .1/!0: Geometry & Topology, Volume 28 (2024) The desingularization of the theta divisor of a cubic threefold as a moduli space 129 These are the non-locally-free torsion-free slope-stable sheaves appearing in Theorem 7.1, and we show that they are also stable as objects of Ku.X / with respect to any Serre-invariant stability condition.Hence, the moduli space M .v/ of -stable objects in Ku.X / of Chern character v contains X , yet its expected dimension is four.This was our first clue that this moduli space is of interest.Indeed, our next result says that the moduli spaces M .v/and M X .v/agree entirely.
Theorem 1.1 (Theorem 8.7 and Proposition 8.10) Let be an arbitrary Serre-invariant stability condition on Ku.X /.Then the moduli space M .v/ is isomorphic to the moduli space M X .v/.
To summarize, we project the structure sheaf of a point into the Kuznetsov component and take its moduli space.It obviously contains X but is bigger.It is the resolution of the theta divisor, with X as the exceptional divisor.Thus, we recover X from Ku.X / or from the intermediate Jacobian, ie we obtain new proofs of both the categorical and classical Torelli theorem for cubic threefolds: Theorem 1.2 (Corollary 7.6 and Theorem 8.1) Let X 1 and X 2 be smooth cubic threefolds.The following are equivalent: (i) X 1 and X 2 are isomorphic.

Proof ideas
The proof of Theorem 7.1 relies on two classical ingredients.Firstly, we use the fact that any irreducible theta divisor is normal, due to Ein and Lazarsfeld [16].Secondly, we use a characterization of the theta divisor of the intermediate Jacobian in terms of twisted cubics; see Proposition 2.2.This was proved by Beauville in [9], but it can also be deduced from the description of ‚ as differences of lines in Clemens and Griffiths [14]; see Remark 2.3.
The strategy to prove Theorem 7.1 is to vary the notion of stability and reach a detailed description of the objects that belong to the moduli space M X .v/through wall-crossing.Since X has Picard rank one, Gieseker stability cannot be varied.This is where the derived category comes into play in the form of tilt-stability introduced in Bridgeland [13] for K3 surfaces, and then further generalized to other surfaces and threefolds in Arcara and Bertram [2] and Bayer, Macrì and Toda [7].In fact, we give a complete description of the wall and chamber structure; see Section 6.Once a set-theoretic description of M X .v/has been reached, we use standard deformation theory arguments in Corollary 6.9 to deduce that it is smooth and of dimension four.
To prove Theorem 8.7, we first show the claim for the specific stability condition constructed in Bayer, Lahoz, Macrì and Stellari [5] which are Serre-invariant by Pertusi and Yang [35].We then prove in a completely separate argument that our moduli space is independent of the choice of Serre-invariant stability conditions .The essential ingredient in this last argument is the weak Mukai lemma from [35].

Related work
In the recent paper [1], Altavilla, Petković and Rota studied moduli spaces of some torsion sheaves in the Kuznetsov components of Fano threefolds with Picard rank one and index two.In the case of cubic threefolds they study M .OES 2 .K P // (S is the Serre functor on Ku.X /), but do not obtain our detailed geometric description.A key difference is that in their case the moduli space in the Kuznetsov component is different from the moduli space of Gieseker-semistable sheaves.
Classical Torelli is the implication (iii) D ) (i) in Theorem 1.2, which was first proved in Clemens and Griffiths [14].The implication (ii) D ) (iii) was first established in Bernardara, Macrì, Mehrotra and Stellari [11,Theorem 1.1], where it was shown that the Fano variety of lines F.X / can be recovered from Ku.X / as a moduli space of stable objects.Thus, one obtains the intermediate Jacobian J.X / as the Albanese variety of F.X /.A more recent argument for (ii) D ) (iii) can be deduced from Perry's categorical construction of intermediate Jacobians [34,Section 5.3], when the equivalence is given by a Fourier-Mukai kernel on X 1 X 2 .Instead, our paper gives a very direct geometric argument for (ii) D ) (i), as well as a variant of the proof of classical Torelli via the description of the singularity of theta divisor implied by Theorem 7.1.
Since this article originally appeared on the arXiv, Feyzbakhsh and Pertusi [17] and Zhang [40] proved uniqueness of Serre-invariant stability conditions on Ku.X /.Proposition 8.10 in the last section could now be obtained as an immediate corollary.
The desingularization of the theta divisor of a cubic threefold as a moduli space 131 2 Cubic threefolds and intermediate Jacobians Let X P 4 be a smooth cubic threefold.In their celebrated article [14], Clemens and Griffiths introduced the intermediate Jacobian of X .It is the complex torus defined as J.X / WD H 2;1 .X / _ =H 3 .X; Z/ D H 1 . 2X / _ =H 3 .X; Z/: It turns out that J.X / is a principally polarized abelian variety of dimension five.
Let fZ b g b2B be a family of 1-cycles over a variety B. The choice of a basepoint b 0 2 B leads to an Abel-Jacobi map ‰ B W B ! J.X / as follows.For any b 2 B the cycle Z b Z b 0 has degree 0, ie it is homologically trivial, and can be written as the boundary @ for a 3-chain .The integral R is an element in H 1;2 .X / _ whose class in J.X / is the image of the Abel-Jacobi map.By [19,Theorem 2.20] the map ‰ B is algebraic along the smooth locus of B.
If Z b D C is a smooth curve, then the induced morphism on tangent spaces has been described by Welters; see [39,Section 2].Recall that the tangent space of the Hilbert scheme at C is naturally given by H 0 .N C =X /, where N C =X is the normal bundle.The tangent space of J.X / at any point is given by H The following diagram is commutative and has exact rows and columns: Proof This is mostly [39, Lemma 2.8] and the preceding construction of the morphisms.The map Recall that the Lefschetz hyperplane theorem says that the hyperplane section H 2 Pic.X / generates the Picard group.One can use twisted cubics to characterize the theta divisor of J.X /.A proof of the following result can be found in [9,Proposition 5.2].Let T be the open locus of smooth twisted cubics in the Hilbert scheme of X , and let T be its closure.

Proposition 2.2
The Abel-Jacobi map ' W T !J.X / with basepoint of class H 2 is algebraic.Its image is a theta divisor ‚ J.X / and its generic fiber is isomorphic to P 2 .
Remark 2.3 Proposition 2.2 can be deduced from the description of ‚ as differences of lines as well.
We give a rough sketch of the argument here.
Let F be the Fano variety of lines on X .According to [14] the morphism F F ! J.X / that maps .L; L 0 / 7 !OEL OEL 0 is generically a 6-to-1 cover of ‚.
Since a twisted cubic C X lies in a unique cubic surface Y X , the morphism T !J.X / factors via the moduli space F of pairs .D; Y /, where Y is a cubic surface and D is the divisor class of a twisted cubic.The generic fiber of the morphism T !F is given by P If D is the class of a twisted cubic on a smooth cubic surface, then D H 2 can be written as the difference of two lines on a cubic surface.Therefore, the Abel-Jacobi morphism maps onto ‚.Moreover, there are precisely six ways to write D H 2 as the difference of two lines.Together with the fact that F F ! J.X / is generically a 6-to-1 cover of ‚, we get that F ! ‚ has degree 1.
Lemma 2.4 Let P 1 Š C X P 4 be a twisted cubic.Then In particular, the Hilbert scheme T is smooth of dimension six.
Proof We have a short exact sequence Since Ext

Divisors on hyperplane sections
We need to understand the singularities that can occur on hyperplane sections of X .Proposition 3.1 Any cubic hyperplane section Y D V \ X P 4 is normal and integral.
Proof Since hypersurfaces satisfy condition S2, by Serre's condition [10, Section 031S], it is enough to show that Y has isolated singularities.Assume for contradiction that Y contains a curve C of singular points.Let F and x be the defining equations of X and V , respectively.Then @F=@x is a homogeneous degree 2 polynomial and hence vanishes somewhere along C .At such a point, all partial derivatives of F vanish, hence it is a singular point of X , a contradiction.
In order to deal with singular hyperplane sections, we need to recall the relation between Weil divisors and rank-one reflexive sheaves on integral normal varieties.This is very similar to the standard relation between line bundles and Cartier divisors.We refer to [10, Tag 0EBK] or [36] for proofs of the following facts.They can also be found in [22]

Notions of stability
In this section, we recall a number of notions of stability for sheaves.Let X be a smooth projective threefold, and let H be an ample divisor on X .While slope-stability suffices to construct moduli spaces of vector bundles on curves, a refinement is necessary in higher dimensions.
Definition 4.2 We define a preorder on the polynomial ring ROEm as follows.
(i) For all nonzero f 2 ROEm, we have f 0.
and let a f and a g be the leading coefficients of f and g, respectively.Then f g if and only if f .m/=af Ä g.m/=a g for all m 0.
(iv) If f; g 2 ROEm with f g and g f , we write f g.
For any E 2 Coh.X /, we denote its Hilbert polynomial and the terms ˛i.E/ by The sheaf E is Gieseker-(semi)stable if for all nontrivial proper subsheaves F E, the inequality P .F; m/ ./P .E; m/ holds.
Note that for 2-Gieseker-semistability we could have equivalently asked P 2 .F; m/ P 2 .E; m/, but for 2-Gieseker-stability, P 2 .F; m/ P 2 .E; m/ is a stronger condition that is almost never fulfilled for all such subsheaves.These notions imply each other as follows: slope-stable The intermediate notion of 2-Gieseker stability is not classical and will just appear in the technical parts of our arguments.
Due to [18; 30; 31; 37] there exists a projective moduli space M X .v/parametrizing S-equivalence classes of Gieseker-semistable sheaves with Chern character v.Here two semistable sheaves are called S-equivalent if they have the same stable factors, up to order and isomorphism, in their Jordan-Hölder filtrations: Based on Bridgeland stability on surfaces, the notion of tilt stability was introduced in [7].It is not quite a Bridgeland stability condition, but it turns out to suffice for our purposes.The basic idea is to change the category in which subobjects are taken when defining stability.This is done via the theory of tilting introduced in [20].As before, let X be a smooth projective threefold with an ample divisor H . Definition 4.6 For any ˇ2 R, we define two full additive subcategories of Coh.X /: Note that Hom.T; F / D 0 for all T 2 T ˇ.X / and F 2 F ˇ.X /, by semistability.It is well known that the category Coh ˇ.X / is abelian.A sequence of morphisms X / is a short exact sequence if and only if the induced sequence To simplify notation, we define for any E 2 D b .X / its twisted Chern character ch ˇ.E/ WD ch.E/ e ˇH .Note that when ˇ2 Z, this is nothing but ch.E ˝OX .ˇH //.Definition 4.7 For ˛> 0, ˇ2 R and E 2 Coh ˇ.X /, we define a slope function where again division by zero needs to be interpreted as C1.Analogously to slope-stability, an object If it is clear from context, we will sometimes abuse notation and write tilt-(semi)stable instead of ˛;ˇ-(semi)stable.Note that by definition, any E 2 Coh ˇ.X / satisfies H 2 ch 1 .E/ 0. Therefore, this function plays the same role in Coh ˇ.X / as the rank does in Coh.X /.
As previously, Harder-Narasimhan filtrations exist.However, note that a version of Jordan-Hölder filtrations exists, but the stable factors are not unique up to order.
The notion of 2-Gieseker stability occurs as a limit of tilt stability as follows.
The statement in [13] Most applications of tilt stability come from varying .˛;ˇ/ and determining what that means for the stability of a given set of objects.We visualize the parameter space of tilt stability, .˛;ˇ/ 2 R 2 with ˛> 0, as the upper half-plane via i ˛C ˇ.For a given class v 2 K 0 .X /, it turns out that there is a locally finite wall and chamber structure such that stability only changes as we cross a wall.These walls are either semicircles with center on the ˇ-axis or vertical lines; see Figures 1 and 2. In the following, we recall what this means formally.
For v 2 K 0 .X / we write ch.v/, .v/,˛;ˇ.v/ and .v/ to mean the appropriate versions where E is replaced by v. Definition 4.11 For v; w 2 K 0 .X /, we define The set W .v; w/ is a numerical wall if W .v; w/ ¤ ∅ and W .v; w/ ¤ R >0 R, ie if it is a proper nontrivial subset of the upper half-plane.
Numerical walls in tilt stability have a rather simple structure, as shown in [28]: Theorem 4.12 (nested wall theorem) Let v 2 K 0 .X / with .v/0. (i) A numerical wall for v is either a semicircle centered along the ˇ-axis, or a vertical line parallel to the ˛-axis in the upper half-plane.
Ď Geometry & Topology, Volume 28 (2024) (ii) If ch 0 .v/¤ 0, then there is a unique numerical vertical wall for v given by ˇD .v/.The remaining numerical walls for v are split into two sets of nested semicircles, whose apexes lie on the hyperbola ˛;ˇ.v/ D 0. In particular, no two distinct walls intersect.
(iii) If ch 0 .v/D 0 and H 2 ch 1 .v/¤ 0, then every numerical wall for v is a semicircle, whose apex lies on the ray ˇD .
The following is a well-known consequence of the fact that walls do not intersect.

Corollary 4.13
Let The following proposition seems to be well known to experts, but we could find no proof in the literature.
Proposition 4.18 Let E 2 Coh.X / be torsion-free.Then EOE1 is tilt-stable along the vertical wall ˇD .E/ if and only if E is slope-stable and reflexive.In particular, slope-stable reflexive sheaves do not get destabilized along the vertical wall.
Proof If E is slope-unstable, then E 6 2 Coh .E/ .X /.Assume that E is strictly slope-semistable.Then there is a short exact sequence of slope-semistable sheaves Taking a shift by one, this becomes a short exact sequence in Coh .E/ .X / with ˛; .E/ .F OE1/ D ˛; .E/ .GOE1/.
Assume that E is not reflexive, but slope-stable.Then we have a short exact sequence in Coh .E/ .X / given by 0 where T is a nontrivial sheaf supported in dimension less than or equal to one.However, this sequence makes EOE1 strictly tilt-semistable along ˇD .E/.
Assume conversely that E is a slope-semistable reflexive sheaf.Then it is an object in Coh .E/ .X / of maximal phase, and in particular tilt-semistable.If it is strictly semistable, then it admits a short exact sequence where F , GOE1, H 1 .F /OE1 and H 0 .F / are also of maximal phase.In particular, H 1 .F / and G are torsion-free and slope-semistable of slope .E/, and H 0 .F / has support of dimension at most one.
Consider the long exact sequence Geometry & Topology, Volume 28 (2024) Since we assume that E is strictly stable, this is a contradiction unless H 1 .F / D 0. Taking duals we get an exact sequence Since F is supported in dimension less than or equal to one, this implies Ext 1 .F; O X / D 0 and G _ Š E _ .
Hence, E ¨G D G __ D E __ , a contradiction to E being reflexive.
From now on, we assume X P 4 is a smooth cubic threefold.In the later sections, we need the following result of [26, Proposition 3.2], which improves the Bogomolov inequality in the case of a Fano threefold of Picard rank one.Be aware that our notation differs from Li's.
In the case of cubic threefolds, direct sums of line bundles can be detected among semistable sheaves or objects by their Chern characters, as follows.We first claim that the only slope-stable reflexive sheaf of class .r;0; 0; eH / D r 3e is positive, and so E admits a morphism from O X or a morphism to O X .As both are reflexive and slope-stable of slope 0, this shows E Š O X .Now consider an object E as in case (i).Then EOE1 is ˛;0 -semistable.By Proposition 4.18, its Jordan-Hölder factors are either of the form F OE1 for a slope-stable reflexive sheaf F with ch.F / D .rF ; 0; d F H 2 ; e F H 3 /, or a torsion sheaf supported in dimension Ä 1.In fact, Proposition 4.15 shows d F D 0 in the former case, and thus, F D O X by the previous case, and that the torsion sheaves are supported in dimension zero.As 3e is the total length of the torsion sheaves, we get e Ä 0. If e D 0, all factors are isomorphic to O X OE1 and the claim follows from Ext 1 .O X ; O X / D 0.
In case (ii), we again consider a Jordan-Hölder filtration with respect to ˛;0 -stability.Let E i ,! E iC1 be the first filtration step where the quotient E iC1 =E i is a zero-dimensional torsion sheaf T , should one exist.Then E i D O X OE1 ˚k for some k > 0. Since Ext 1 .T; O X OE1/ D H 1 .T / _ D 0, we have E iC1 D E i ˚T , and so T is a subobject of E. This contradicts stability of E for ˇ> 0. Thus, E D O X OE1 ˚r , as claimed.

Construction of sheaves
In this section, we introduce the sheaves that make up our moduli space M X .v/.It turns out that all of them are at least reflexive, and the generic one is a vector bundle.From now on X P 4 is an arbitrary smooth cubic threefold.
Let Y X be an arbitrary hyperplane section, D be an effective Weil divisor on Y, and V H 0 .O Y .D// be a nontrivial subspace.Then we define E D;V 2 D b .X / to be the cone of the induced morphism Therefore, ch Ä1 .E D;V / D .dimV; H / is primitive and it is enough to show that E D;V is slope-semistable.If not, let F E D;V be the slope-semistable subsheaf in the Harder-Narasimhan filtration of E D;V .Then .F / > .E D;V / and the quotient E D;V =F is torsionfree.Since F is also a subsheaf of O X ˝V , we must have .F / D 0. Let ch.F / D .r;0; dH 2 ; eH 3 /.The quotient .O X ˝V /=F satisfies ch..O X ˝V /=F / D .dimV r; 0; dH 2 ; eH 3 /.By the snake lemma this quotient is either torsion-free or has a torsion subsheaf purely supported on Y.However, if it is not torsion-free, then its torsion-free quotient would destabilize O X ˝V , a contradiction.As a torsion-free quotient of O X ˝V with slope zero, .O X ˝V /=F has to be slope-semistable as well.
The classical Bogomolov inequalities H .F / 0 and H ..O X ˝V /=F / 0 imply d D 0. Applying Proposition 4.20 to both F and .O X ˝V /=F implies e D 0, and finally, F D O ˚r X .However, by construction, E D;V has no global sections, a contradiction.
To see that E D;V is reflexive it suffices to show that Ext q .E D;V ; O X / D 0 for q 2 and Ext 1 .E D;V ; O X / is supported in dimension zero.If additionally Ext 1 .E D;V ; O X / D 0, then E D;V is a vector bundle.
Clearly, Ext q .O X ˝V; O X / D 0 for q ¤ 0. Because O Y .D/ is a rank-one reflexive sheaf on the codimension one subvariety Y, the quotient .O X ˝V /=E D;V O Y .D/ is purely supported on Y.We can use [23,Proposition 1.1.10]to see that Ext q ..O X ˝V /=E D;V ; O X / D 0 for all q ¤ 1; 2, and Ext 2 ..O X ˝V /=E D;V ; O X / is supported in dimension zero.The long exact sequence obtained from dualizing the short exact sequence (1) 0 implies the required vanishings.
If additionally

Variation of stability
In this section, we investigate semistable sheaves with Chern character v WD 3; H; 1 2 H 2 ; 1 6 H 3 : The main goal is to use wall-crossing to prove the following theorem, which gives a set-theoretic description of the moduli space M X .v/.At this wall, the short exact sequences ( 2) and ( 1) become destabilizing short exact sequences in Coh ˇ.X / in the form Moreover, we can show that every object gets destabilized, and the destabilizing short exact sequence must be of one of these types; see Lemma 6.8.

Classification of some torsion sheaves
In this section, we prove the following proposition.

Proposition 6.2
The wall W of equation (3) is the unique actual wall in tilt stability for objects G with Chern character ch.G/ D 0; H; 1 2 H 2 ; 1 6 H 3 .
(i) Above W the moduli space of tilt-semistable objects is the moduli space of Gieseker-semistable sheaves, and contains precisely the following two types of sheaves G: (ii) Below W the moduli space of tilt-semistable objects contains precisely the following two types of objects G: (a) the unique nontrivial extensions We start by dealing with slightly more general objects without fixing ch 3 .
Geometry & Topology, Volume 28 (2024) Lemma 6.3 The wall W of equation (3) is the unique actual wall in tilt stability for objects G with Chern character ch Ä2 .G/ D 0; H; 1 2 H 2 .If G is strictly semistable along W, then any Jordan-Hölder filtration of G is given by either where Z X is a zero-dimensional subscheme of length 1  6 H 3 ch 3 .G/.
Proof All walls for 0; H; , and in particular, r is odd.We will deal with the case r < 0. If r > 0, then B has negative rank and one simply has to exchange the roles of A and B in the following argument.
For ˛; Therefore, W is a wall for G and by Lemma 6.3, the destabilizing sequence is This implies G D O Y .H / for some Y 2 jH j and ch 3 .G/ D 1 6 H 3 .
Proof of Proposition 6.2 Assume that G is strictly tilt-semistable along W. Then Lemma 6.3 splits our problem into two cases.
Firstly, assume that G fits into a nonsplitting short exact sequence for a point P 2 X .Then clearly G D I P =Y .H / for some Y 2 jH j.This object is tilt-stable above W, and tilt-unstable below W by precisely this sequence.
Secondly, assume that G fits into a nonsplitting short exact sequence for some P 2 X .By Serre duality, Ext 1 .I P .H /; O X OE1/ D h 1 .I P .H // D 1 and hence, there is a unique G for each P 2 X .Clearly, this object is tilt-unstable above W. Assume it is also tilt-unstable below W. Then there is a short exact sequence 0 !A ! G ! B ! 0 destabilizing G below the wall.However, G is strictly semistable at W, and by Lemma 6.3, this implies B D O X OE1.However, that means the short exact sequence (5) splits, a contradiction.
Lastly, assume that G is ˛;ˇ-stable for all .˛;ˇ/.By Proposition 4.17, D.G/ lies in a distinguished triangle where T is a torsion sheaf supported in dimension zero and z G. Since hom.O X ; T OE i / D 0 for i > 0, The triangle (6) shows that there is a nontrivial morphism O X !D.G/.Dualizing this morphism leads to a nontrivial morphism G ! O X OE1.However, this is in contradiction to the assumption that G is stable along W.
G is a sheaf, so Ext q .G; O X / D 0 for q > 1.Thus, [23,Proposition 1.1.10]implies that G is reflexive and supported on a hyperplane section Y 2 jH j.This means G D O Y .D/ for some Weil divisor D on Y.

Set-theoretic description of the moduli space
We now prepare the proof of Theorem 6.1.Lemma 6.5 There are no walls along ˇD 1 for tilt-semistable objects E with Chern character ch Ä2 .E/ D 3; H; 1 2 H 2 .
Proof Assume there is such a wall induced by a short exact sequence We first show that ext 2 .E; E/ D 0. Since .F / D 0. Thus, either F or the quotient E=F have infinite tilt-slope, a contradiction.This means E is ˛; 1=2 -stable for all ˛> 0. By Proposition 4.18, the object EOE1 is tilt-stable for ˇD 0 and ˛ 0. Since H 3 ch 1 .EOE1/ D H 3 , the same type of argument as above shows that there cannot be any wall along ˇD 0. Hence, E. 2H /OE1 is .By Lemma 6.5, we have that E is ˛; 1 -stable for any ˛> 0. One can easily compute 6 H 3 Ä 0, as claimed.Lastly, assume that a slope-stable sheaf E of Chern character v is not reflexive.We have a short exact sequence Since E __ is also slope-stable, and both H ch 2 .E/ and H ch 3 .E/ are maximal, one gets ch.E/ D ch.E __ /.This is only possible if T D 0.
To prove Theorem 6.1, we start in the large volume limit.Lemma 6.7 Take ˇ> Next, we move down from the large volume limit and investigate walls for objects of class v.Note that all walls to the right of the vertical wall must intersect ˇD 1 3 .
Lemma 6.8 The wall W of equation (3) is the unique actual wall for objects with Chern character v to the right of the vertical wall.There are no tilt-semistable objects below W. Any tilt-semistable z E with Chern character v fits into one of the following two cases: where D is a Weil divisor on hyperplane section Y 2 jH j.
(ii) z E fits into a short exact sequence Proof Let z E be a tilt-semistable object with Chern character v. Let W 0 be a wall strictly above W induced by a short exact sequence 0 !F ! z E !G ! 0. Then the wall W 0 contains points .˛;0/ with ˛> 0. In particular, Clearly, any morphism z Let r WD hom.z E; O X OE1/ 3. We get a short exact sequence of tilt-semistable objects along W given by For r D 4, we get ch.G.H // D 1; 0; 0; 1 3 H 3 and so G D I P .H / for some P 2 X .
If r D 3, then ch.G/ D 0; H; 1 2 H 2 ; 1 6 H 3 .Assume G is not of the form O Y .D/ for some Weil divisor D on a hyperplane section Y 2 jH j.Then Proposition 6.2 implies that G has to be strictly semistable along our wall W. Since z E is tilt-semistable above the wall, we know Hom.O X OE1; E/ D 0. Therefore, Lemma 6.3 shows that there is a short exact sequence for a point P 2 X .But then there is an inclusion I P .H / ,! z E and we are in the second case.
Proof of Theorem 6.1 Let D be a Weil divisor on a hyperplane section Y 2 jH j with ch.O Y .D// D 0; H; 1 2 H 2 ; 1 6 H 3 .By Proposition 6.2, the sheaf O Y .D/ is tilt-stable for all ˛> 0 and ˇ2 R. A straightforward computation shows that the numerical wall W .O Y .D/; O X .2H /OE1/ is nonempty, and therefore, We pick a three-dimensional subspace V h 0 .O Y .D// to get an object E D;V 2 D b .X / as in Section 5.By Lemma 5.1, the sheaf E D;V D H 1 .E D;V / is slope-stable and reflexive.If H 0 .E D;V / ¤ 0, then E D;V has a Chern character in contradiction to Proposition 6.6.This shows that O Y .D/ is globally generated.
Since E D;V is slope-stable, we know h 0 .E D;V / D 0 and h 3 .E D;V / D hom.E D;V ; O X .2H // D 0. Moreover, as in the proof of Proposition 6.6 we get The long exact sequence obtained from taking sheaf cohomology of ; which shows that C is of degree 3 with .K C / D 1, ie a twisted cubic.This completes the proof of part (i).
For part (ii), we already showed in Corollary 5.2 that K P is slope-stable for any P 2 X .Conversely, if E is slope-stable, we can immediately conclude by Lemma 6.8.
As a consequence we can already infer that our moduli space M X .v/ is smooth.Corollary 6.9 Every Gieseker-semistable sheaf E with ch.E/ D 3; H; and M X .v/M X .v/ is the open locus of Gieseker-semistable vector bundles.The aim of this section is to prove the following theorem.
Theorem 7.1 The moduli space M X .v/ is smooth and irreducible of dimension 4.Moreover, there is an Abel-Jacobi morphism ‰ W M X .v/!J.X / sending E 7 !z c 2 .E/ H 2 , whose image is a theta divisor ‚ in the intermediate Jacobian J.X /.The theta divisor has a unique singular point, and M X .v/ is the blowup of ‚ in this point.The exceptional divisor is isomorphic to the cubic threefold X itself.
We have already shown that M X .v/ is smooth of dimension 4 in Corollary 6.9.By Proposition 2.2, the image of ' W T !J.X / is ‚ J.X /, where T is the open locus of smooth twisted cubics in the Hilbert scheme of X , and T is its closure.By Theorem 2.6, we know that ‚ is normal.
Proposition 7.2 There is a surjective map ' 0 W T !M X .v/that sends a twisted cubic C to the vector bundle E C .The map 'j T W T !J.X / factors through ' 0 : Proof Let C be a twisted cubic in X .Then it lies in a unique hyperplane section Y.There is a short exact sequence where T is a sheaf supported on C with rank one.
Proof We interpret X as the moduli spaces of twisted ideal sheaves I P .H / for all P 2 X .By definition of K P , we have a canonical short exact sequence (7) 0 The appropriate version in families, considered below, induces the morphism i .It is injective, as P is the unique point where K P is not locally free by Corollary 5.2.
Applying Hom. ; K P / to (7), we get an isomorphism Ext To determine the normal bundle, we need a relative version of the previous arguments to determine the cokernel of this embedding as a line bundle on X .The universal family inducing i is given by the sheaf K on X X fitting into the short exact sequence where p W X X !X is the projection to the first factor.The pullback of the tangent bundle via i is i T M X .v/D H We can finish the proof of Theorem 7.1 with the following lemma.
Lemma 7.5 The formal neighborhood of 0 2 ‚ is isomorphic to the vertex of the affine cone over X P 4 .Moreover, we have an isomorphism M X .v/D Bl 0 .‚/.Thus, X is the union of all rational curves on M X .v/,and the unique divisor contracted by any morphism to a complex abelian variety.
Proof The first two claims are scheme-theoretic enhancements of the set-theoretic statements in the previous lemma, which hold for any contraction of a divisor with ample conormal bundle to a point.We will only sketch the arguments.
Since the normal bundle of X M X .v/ is antiample, by Artin's contractibility criterion [4, Corollary 6.12] there is a contraction ‰ 0 W M X .v/!N to an algebraic space N of finite type over C that is an isomorphism away from X , and contracts X to a point 0 2 N .Moreover, by Artin's construction in [4, Theorem 6.2], the formal neighborhood of 0 2 N is given by the affinization of the formal neighborhood of X M X .v/.More precisely, if I is the ideal of X , then it is given by Spec ie the completion of the vertex of the affine cone over X .Since the image of every infinitesimal neighborhood of X under ‰ is affine, it factors via its affinization.Taking the limit, we see that ‰ factors via ‰ 0 both in the formal neighborhood of X , and in its complement.Hence (eg by [4, Theorem 3.1]) we get an induced morphism j W N !‚ factoring ‰.As j is bijective on points and has normal target, it is an isomorphism.
For the last claim, note that X is uniruled, hence the union U of all rational curves in M X .v/contains X .If there was any other rational curve C not contained in X , then ‰ W C ! ‚ is a nonconstant map from a rational to an abelian variety, a contradiction.Corollary 7.6 If X 1 and X 2 are smooth projective threefolds with J.X 1 / D J.X 2 / as principally polarized abelian varieties, then X 1 D X 2 .
Proof As in the classical argument, this is an immediate consequence of the description of the singularity of the theta divisor in Lemma 7.5.

Kuznetsov component
The bounded derived category of a cubic threefold X admits a semiorthogonal decomposition D b .X / D hKu.X /; O X ; O X .1/i;whose nontrivial part Ku.X / is called the Kuznetsov component.The goal of this section is to give a new proof of the following theorem.Lemma 8.4 There is an embedding M X .v/,! M .˛/.OEI ` C OES.I `// from the moduli space M X .v/for v D ch.I `/ C ch.S.I `// D 3; H; 1 2 H 2 ; 1 6 H 3 to M .˛/.OEI ` C OES.I `//, which parametrizes .˛/-semistableobjects in Ku.X / of class OEI ` C OES.I `/ 2 N .Ku.X //.Proof According to Lemma 6.5 there is no wall for objects of Chern character v to the left of the vertical wall.Thus, E is ˛; 1=2 -stable for any ˛> 0. Since 0 ˛; 1=2 is just a rotation of ˛; 1=2 , we obtain that E is 0 ˛; 1=2 -stable.By Theorem 6.1(ii), the sheaf E 2 Ku.X / lies in the Kuznetsov component.Thus, E is .˛/-stable.Note that the object E could be destabilized by objects with Z 0 ˛; 1=2 D 0 after rotation.But we know that these are all sheaves supported in dimension zero and would not be in Ku.X / and therefore, E is stable after restriction to Ku.X /.
Proof of Theorem 8.1 As a cubic threefold has free Picard group of rank one, the first implication is obvious.As for the second implication, assume there is an exact equivalence ˆW Ku.X 1 / !Ku.X 2 /.Lemma 8.3 implies that, up to composing with a power of the Serre functor of Ku.X 1 / and shift functor, we may assume OEˆ .K P / D OEK P 0 for points P and P 0 in X 1 and X 2 , respectively.Take an S-invariant stability condition on Ku.X 1 /.Theorem 8.7 and Proposition 8.10 imply that (16) M X 1 .v/Š M .Ku.X 1 /; OEK P / Š M ' .Ku.X 2 /; OEK P 0 /: Since the Serre functor commutes with autoequivalences, ' is an S-invariant stability condition on Ku.X 2 /.Thus, Theorem 8.7 gives M ' .Ku.X 2 /; OEK P 0 / Š M X 2 .v/: Combining this with (16) gives M X 1 .v/Š M X 2 .v/.By Lemma 7.5, we know X 1 and X 2 are the unique exceptional divisors of M X 1 .v/and M X 2 .v/which get contracted by any map to a complex abelian variety.Thus, X 1 Š X 2 .

List of symbols
X smooth cubic threefold in P 4 over C H the ample generator of Pic.X / Y a hyperplane section of X D b .X / bounded derived category of coherent sheaves on X Ku.X / the Kuznetsov component inside D b .X / CH .X / the Chow ring of X CH n .X / the numerical Chow ring of X , obtained as CH .X / modulo numerical equivalence H i .E/ the i th cohomology sheaf of a complex E 2 D b .X / H i .E/ the i th sheaf cohomology group of a complex E 2 D b .X / ch.E/ total Chern character of an object E 2 D b .X / up to numerical equivalence c.E/ total Chern class of an object E 2 D b .X / up to numerical equivalence
we will drop V , and just write E D and E D .Lemma 5.1 The sheaf E D;V is slope-stable and reflexive.If , additionally, H 0 .E D;V / D 0, then E D;V is a vector bundle.Proof The quotient .O X ˝V /=E D;V embeds into O Y .D/.Since Y is integral by Proposition 3.1, the sheaf .O X ˝V /=E D;V must be supported on Y.

(
a) G D I P =Y .H / for Y 2 jH j and P 2 Y , and (b) G D O Y .D/, where D is a Weil divisor on some Y 2 jH j.

( 4 )
0 !O X OE1 !G P !I P .H / !0 for points P 2 X , and (b) G D O Y .D/, where D is a Weil divisor on some Y 2 jH j.

e
ch.E/ total Chern character of an object E 2 D b .X / up to rational equivalencez c.E/ total Chern class of an object E 2 D b .X / upto rational equivalence ch Äl .E/ .ch0 .E/; : : : ; ch l .E// e ch Äl .E/ .e ch 0 .E/; : : : ; e ch l .E// 1 Homological invariants of codimension 2 contact submanifolds LAURENT CÔTÉ and FRANÇOIS-SIMON FAUTEUX-CHAPLEAU 127 The desingularization of the theta divisor of a cubic threefold as a moduli space AREND BAYER, SJOERD VIKTOR BEENTJES, SOHEYLA FEYZBAKHSH, GEORG HEIN, DILETTA MARTINELLI, FATEMEH REZAEE and BENJAMIN SCHMIDT 161 Coarse-median preserving automorphisms ELIA FIORAVANTI 267 Classification results for expanding and shrinking gradient Kähler-Ricci solitons RONAN J CONLON, ALIX DERUELLE and SONG SUN 353 Embedding calculus and smooth structures BEN KNUDSEN and ALEXANDER KUPERS 393 Stable maps to Looijenga pairs PIERRICK BOUSSEAU, ANDREA BRINI and MICHEL VAN GARREL 1.O P 1 .3/;OP 1 .5//D0,wegetNC =P 4 D O P 1 .5/˚2˚OP 1 .3/.Next, we have a short exact sequence0 !N C =X !N C =P 4 D O P 1 .5/˚2˚OP 1 .3/!N X =P 4 ˝OC D O P 1 .9/! 0Thus, N C =X has degree 4 and can only be O P 1 .m/˚OP 1 .4m/forsome 1 Ä m Ä 5.The claim about the cohomology of N C =X holds for each of them.Aong the locus of smooth curves T T , the Abel-Jacobi morphism ' has differential of rank four.Proof Let C X be a smooth twisted cubic.Clearly, restriction mapsH 0 .O X .H // Š C 5 surjectively onto H 0 .O C .H // Š C 4 .By Lemma 2.4, we have h 0 .N C =P 4 .2H// D h 0 .O P 1 .1/˚2 ˚OP 1 .3//D 0. D ch 3 .E/ C H ch 2 .E/ C 2 3 H 2 ch 1 .E/ C 1 3 H 3 ch 0 .E/:Proof By Kodaira vanishing H i .O X / D 0 for i ¤ 0, and therefore, .O X / D 1.By the Hirzebruch-Riemann-Roch Theorem we get td 3 .X / D .O X / D 1 3 H 3 .Similarly, Kodaira vanishing implies H i .O X .H // D 0 for i ¤ 0. Again by Hirzebruch-Riemann-Roch, in more generality.Let Y be a normal integral projective variety.By Cl.Y / we denote the group of Weil divisors modulo rational equivalence.For two rank-one reflexive sheaves L 1 ; L 2 2 Coh.Y / we can define a new rank-one reflexive sheaf by .L 1 ˝L2 / __ .This defines a group law for rank-one reflexive sheaves on Y, where inverses are given by L 7 !L _ .For any effective prime divisor D one can define a rank-one reflexive sheaf O Y .D/ WD I _ D .This can be linearly extended to any divisor.The group of isomorphism classes of rank-one reflexive sheaves is isomorphic to Cl.Y / under the homomorphism D 7 !O Y .D/.Two sections s 1 ; s 2 2 H 0 .L/ define the same divisor if they satisfy s 1 D s 2 for some 2 C .
From the definition it follows immediately that if Pic.X / D Z H and E is slope-semistable with gcd ch 0 .E/; H 2 ch 1 .E/=H 3 D 1, then E is slope-stable.
is for K3 surfaces, but the same proof works in our setting.If ˇ> .E/ the situation is slightly more complicated.The following proposition is a combination of[6, Lemma 2.7] and [27, Proposition 3.1].Proposition 4.9 Take a ˛;ˇ-semistable object E 2 Coh ˇ.X /.If ˇ¤ .E/, then H 1 .E/ is a reflexive sheaf , and if ˇ .E/ and ˛ 0, then H 1 .E/ is a torsion-free slope-semistable sheaf and H 0 .E/ is supported in dimension less than or equal to one.
Proposition 4.16 Assume that an object E is destabilized by a semicircular wall induced by a subobject F ,! E or quotient E F with ch 0 .F / > ch 0 .E/ 0 or ch 0 .F / < ch 0 .E/ Ä 0. Then the radius of W .F; E/ satisfies .E/ 4.H 3 ch 0 .F //.H 3 ch 0 .F / H 3 ch 0 .E// : Tilt stability interacts nicely with the derived dual D. / WD RHom.; O X /OE1.Proposition 4.17 [7, Proposition 5.1.3]Suppose that E 2 Coh ˇ.X / is a ˛;ˇ-semistable object with ˛;ˇ.E/ ¤ 1.Then there is a ˛; ˇ-semistable object z E 2 Coh ˇ.X /, a torsion sheaf T supported in dimension zero, and a distinguished triangle [29,t sequence of ˛;ˇ-semistable objects in Coh ˇ0 .X / for some .˛0;ˇ0/ 2 W .F; E/.Then this is a short exact sequence of ˛;ˇ-semistable objects in Coh ˇ.X / for any .˛;ˇ/ 2 W .E; F /. Definition 4.14 Let v 2 K 0 .X /.A numerical wall W for v is called an actual wall for v if there is a short exact sequence0 !F !E !G !0 of ˛;ˇ-semistable objects in Coh ˇ.X / for one .˛;ˇ/ 2 W .F; E/ such that W D W .F; E/ and ch.E/ D v.The above corollary implies that this is a short exact sequence in Coh ˇ.X / for all .˛;ˇ/ 2 W .F; E/.Determining walls is the key technique in this paper.It will allow us to classify sheaves with certain Chern characters in terms of short exact sequences; see Theorem 6.1.Note that the condition W .F; E/ ¤ R >0 R implies ˛;ˇ.F / > ˛;ˇ.E/ on one side of such a wall.We say that the short exact sequence0 !F !E !G !0;or sometimes the wall W .F; E/, destabilizes E. Proposition 4.15 [6, Appendix A] If an actual wall is induced by a short exact sequence of tilt-semistable objects 0! F ! E !G !0, then H .F / C H .G/ Ä H .E/;and equality can only occur if either F or G is a sheaf supported in dimension zero.It turns out that walls of large radius can only be induced by subobjects of small rank.The following precise statement is close to[15, Proposition 8.3].A proof of this version can be found in[29, Lemma 2.4]for the case of nonnegative ranks.The case of nonpositive ranks has the exact same proof, with reversed signs.2Ä H 3/ is O X .Assume otherwise.By Proposition 4.18, such an E is also stable at the vertical wall ˇD 0, and thus, it is ˛;ˇ-stable for all ˛> 0 and ˇ2 R.Since 0;ˇ.E/ D 1 2 ˇ> 1 2 ˇ 1 D 0;ˇ.O X .2H/OE1/and both objects are stable for ˛ 1 and ˇ2 .2; 0/, we have Ext 2 .O X ; E/ D Hom.E; O X .2H /OE1/ D 0. Similarly, from ˛;ˇ-stability for ˛ 1 and ˇ2 .0;2/ we obtain Ext 2 .E; O X / D Hom.O X .2H/; EOE1/ D 0. However, at least one of .O X ; E/ D r C 3e or .E; O X [23a reflexive sheaf on the codimensionone subvariety Y, and we can use[23, Proposition 1.1.10]again to see that Ext 2 .O Y .D/; O X / D 0. The same long exact sequence as above now implies Ext 1 .E D;V ; O X / D 0. Note that we will use this lemma for the case ch.O Y .D// D 0; H; 1 2 H 2 ; 1 6 H 3 .It will turn out that in this case h 0 .O Y .D// D 3 for any such D (see Theorem 6.1) and we will choose V D H 0 .O Y .D//.Moreover, we will show that in that case H 0 .E D / D 0, ie O Y .D/ is globally generated; see Theorem 6.1.A straightforward computation shows that in this example ch.E D / D 3; H; 1 2 H 2 ; 1 6 H 3 .
Corollary 5.2 Let P 2 X .Then h 0 .I P .H // D 4 and the sheaf K P defined through the exact sequence(2) 0 !K P !O ˚4 X !I P .H / !0 satisfies ch.K P / D 3; H; 1 2 H 2 ; 1 6 H 3 .Moreover, K P is reflexive and slope-stable, and locally free except at P .Proof By choosing an embedding K P ,! O ˚3 X we get a short exact sequence 0 !K P !O ˚3 X !I P =Y .H / !0 for some hyperplane section Y.The statement then follows from Lemma 5.1 by choosing D D H and V D H 0 .I P =Y .H // H 0 .O Y .H //.From the defining short exact sequence (2) one immediately sees that K P is locally free away from P (as it is the kernel of a surjective map of vector bundles), and not locally free at P (as Ext 2 .O P ; K P / D Ext 1 .O P ; I P .H // ¤ 0).
Suppose that D is a Weil divisor on a (possibly singular) hyperplane section Y with ch.O Y .D// D 0; H; 1 2 H 2 ; 1 6 H 3 .Then O Y .D/ is globally generated , and h 0 .O Y .D// D 3. In particular, there exists a smooth twisted cubic C in Y of class D. (ii) A sheaf E with Chern character v is Gieseker-semistable if and only if it is either equal to the reflexive sheaf K P for a point P 2 X as in (2), or the vector bundle E D for a Weil divisor D on a hyperplane section Y X as in (1) with ch.O Y .D// D 0; H; 1 2 H 2 ; 1 6 H 3 .Note that since ch 1 .E/ D H , any Gieseker-semistable sheaf of class v is slope-stable.The argument will essentially boil down to a detailed analysis of the numerical wall W defined by If r D 3, then ch Ä2 .A/ D 3; H; 1 6 H 2 .This case is immediately ruled out by Theorem 4.19.If r D 1, then ch Ä2 .A/ D .1; 0; 0/, and by Proposition 4.20, we know A D O X OE1.Then ch.B/ D 1; H; 1 2 H 2 ; ch 3 .G/ .By Proposition 4.15, there is no semicircular wall for B, and by Proposition 4.8, the object B has to be a 2-Gieseker-stable sheaf.Since ch.B.H // D 1; 0; 0; ch 3 .G/ 1 6 H 3 , the remaining statement follows by applying Proposition 4.20 to B. H /. The next step is to gain further control over the third Chern character.Let G be a ˛;ˇ-semistable object with ch Ä2 .G/ D 0; H; 1 2 H 2 .Then ch 3 .G/ Ä 1 6 H 3 .If ch 3 .G/ D 1 6 H 3 and .˛;ˇ/ is above W, then G Š O Y .H / for some Y 2 jH j.
13 .An object z E 2 Coh ˇ.X / of Chern character v is ˛;ˇ-semistable for ˛ 0 if and only if z E Š EOE1 for a slope-stable reflexive sheaf E.Proof Take a ˛;ˇ-semistable object z E of class v. Proposition 4.9 implies that H 1 .z E/ is a slope-stable reflexive sheaf and H 0 .z E/ is a torsion sheaf supported in dimension Ä 1.Therefore, and for each D there is a unique slope-stable sheaf E D D E D;V .Let U Y be the smooth locus of Y.By Proposition 3.1, we know that Y is normal, and therefore, Y nU has dimension zero.In particular, a general section of O Y .D/ leads to a curve completely contained in U .Since we work in characteristic 0, we can use a version of Bertini's theorem [21, Corollary III.10.9,Remark III.10.9.1, Remark III.10.9.2] on the open subset U to see that a general section cuts out a smooth curve C .By adjunction, ch.K C / D ch.O Y .H C D/ jD / D ch.O Y .H C D// ch.O Y .H // Proof Since .3;H / is primitive, we know that E is slope-stable.Therefore, hom.E; E/ D 1.Moreover, we must have Ext 3 .E; E/ D Hom.E; E. 2H // _ D 0. By Lemma 6.5, the sheaf E is ˛; 1 -stable for any ˛> 0. Proposition 6.6 shows that E. 2H / is reflexive, so its shift E. 2H /OE1 lies in the heart Coh ˇD 1 .X / and it is ˛; 1 -stable for any ˛> 0 by Lemma 6.8.Since Recall that M X .v/ is the moduli space of Gieseker-semistable sheaves with Chern character (7) P ; K P / Š Ext 2 .I P .H /; K P /.Next, we apply the functor Hom.I P .H /; / to(7)to show that the induced morphism on tangent spaces Ext 1 .I P .H /; I P .H // ,! Ext 2 .I P .H /; K P / D Ext 1 .K P ; K P / is an embedding.Since both X and M X .v/ ae smooth, the morphism is an embedding.
1.p Hom.K; K//.Since p Hom.p P 4 j X .H /; K/ D 0, we have an isomorphism H 1 .pHom.K; K// D H 2 .pHom.I .0;H/;K/:Thedifferentialdi of i fits into the four-term long exact sequence0 !T X D H 1 pHom.I .0;H/;I.0;H//di !H 2 p Hom.I .0;H/;K/!H 2 p Hom.I .0;H/;pP 4 j X .H // !H 2 p Hom.I .0;H/;I.0;H//!0:UsingGrothendieck duality and the projection formula, the third term becomes P 4 j X .H / ˝H1 .pI.0;H// _ D P 4 j X .H / ˝H0 .pO.0;H// _ D P 4 j X .2H/:Asimilarcomputationusing the short exact sequence I ,! O X O X O gives H 2 .pHom.I ; I / D X .2H/for the fourth term.Thus, the cokernel ofd i is isomorphic to N _ X =P 4 .2H/DOX .H /, as claimed.The morphism ‰ induces an isomorphism M X .v/! ‚ nf0g.Moreover, ‰ contracts the irreducible divisor M X .v/nMX .v/ to the zero point.In particular, ‚ is smooth away from 0.Proof By Lemma 5.1 and Corollary 5.2, the locus M X .v/nM X .v/coincideswith vector bundles E C associated to a twisted cubic C .By Lemma 2.5, the map 'j T has full rank four on tangent spaces.Thus, the commutative diagram in Proposition 7.2 implies that ‰j M X .v/hasfull rank four on tangent spaces.Since M X .v/ is smooth of dimension four, ‰j M X .v/mustbe injective on tangent spaces.In particular, the morphism ‰j M X .v/musthave finite fibers.Since 'j T has generically connected fibers by Proposition 2.2, the same holds for ‰j M X .v/.Since ‚ is normal, Zariski's main theorem implies that ‰j M X .v/ is an open embedding.Since ‚ is singular at the origin, we must have ‰.M X .v//‚f0g.By definition, z c 2 .K P / D H 2 and we get ‰.K P / D 0. Thus ‰ 1 .0/D M X .v/n M X .v/,and the image of M X .v/ is indeed ‚ n f0g by Proposition 2.2.
[35,ies that the stability condition .˛/ is S-invariant, ie S .˛/D.˛/zGOE2k lies in the heart A.˛/ for some k 2 Z.We know its image under the stability function Z.˛/ is equal to H 3 , so it has maximum phase in the heart A.˛/, which immediately implies GOE2k is 1=2 -semistable.We claim that GOE2k has no subobject Q 2 Coh 0 ˛; 1=2 with Z 0 ˛; 1=2 .Q/ D 0, so it is ˛; 1=2 -semistable.Assume for a contradiction that there is such a subobject Q.By the definition of Coh 0 ˛; 1=2 .X /, it is a sheaf supported in dimension zero.Thus, hom.O X ; Q/ ¤ 0. Since O X 2 Coh 0 ˛; 1=2 .X /, we have hom.O X ; .GOE2k=Q/OE 1/ D 0. Therefore, hom.O X ; GOE2k/ ¤ 0, which is not possible because GOE2k 2 Ku.X /.Finally, since GOE2k is ˛; 1=2 -semistable for 0 < ˛ 1, the claim follows by Proposition 6.2(ii).Since the class 2OEI ` OES.I `/ is primitive in N .Ku.X //, any .˛/-semistableobject of this class is .˛/-stableifwechoose˛sufficientlysmall.We now describe the image of the semistable objects G 2 M .˛/.2OEI` OES.I `// under the Serre functor S .If G D G P .H /, then by (4), we know there is a distinguished triangleO X OE1 !G P !I P .H / !O X OE2; which gives L O X .G P / D L O X .I P .H // D K P OE1, so(11)S.G P / D K P OE2:If G D O Y .D H /, then G.H / D O Y .D/ is of class 0; H; 1 2 H 2 ; 1 6 H 3 , and lies in a distinguished triangle D L O X .O Y .D//OE1 D L O X .E D OE1/OE1 D E D OE2:The moduli space M .˛/.OEI ` C OES.I `// is isomorphic to the moduli space M X .v/parametrizingGieseker-stablesheaves of class v.The next step is to show that we can replace .˛/byanyS-invariantstabilitycondition on Ku.X /.For cubic threefolds, we also have a weak version of the Mukai lemma for K3 surfaces.Lemma 8.9 (weak Mukai lemma[35, Lemma 5.11]) Let be an S-invariant stability condition.LetA !E !B be a triangle in Ku.X / such that hom.A; B/ D 0 and the -semistable factors of A have phase greater than or equal to the phase of the -semistable factors of B. Thendim C Ext 1 .A; A/ C dim C Ext 1 .B; B/ Ä dim C Ext1 .E; E/: Proposition 8.10 Let 1 and 2 be two S-invariant stability conditions on Ku.X /.An object E 2 Ku.X / of class OEI ` C OES.I `/ is 1 -stable if and only if it is 2 -stale.Proof By [35, Proposition 4.6], I `and S.I `/ are -stable with respect to any S-invariant stability condition.Thus, Lemma 8.8 implies that (13) ' .I `/ < ' .S.I `// < ' .I `/ C 2: Take a 1 -stable object E 2 Ku.X / of class OEI ` C OES.I `/.Since 1 is S-invariant, Lemma 8.8 gives ' Since E is 1 -stable, we get hom.E; E/ D 1, which gives hom.E; EOE1/ D .E; E/ C 1 D 4: Suppose now for a contradiction that E is 2 -unstable.There is a distinguished triangle of destabilizing objects F 1 !E ! F 2 ! F 1OE1 with respect to 2 .We may assume F 1 is 2 -semistable.Thus, Lemma 8.8 implies that (14) hom.F 1 ; F 1 OE1/ 2: Since the phase of F 1 is bigger than the phase of 2 -semistable factors of F 2 , we have (15) hom.F 1 ; F 2 / D 0: Thus, the weak Mukai lemma (Lemma 8.9) implies hom.F 1 ; F 1 OE1/ C hom.F 2 ; F 2 OE1/ Ä hom.E; EOE1/ D 4: By (14), we get hom.F 2 ; F 2 OE1/ Ä 2. If hom.F 2 ; F 2 OE1/ D 0 or 1, then all its 2 -semistable factors would satisfy the same property by the weak Mukai lemma (Lemma 8.9), which is not possible by Lemma 8.8.Therefore, hom.F 1 ; F 1 OE2/ D hom.F 2 ; F 2 OE1/ D 2; and [35, Lemma 5.12] implies that F 1 and F 2 are 2 -stable.This gives .F i ; F i / D 1 for i D 1; 2, so OEF i is either ˙OEI `, or ˙OES.I `/, or ˙.OES.I `/ OEI `/.Since there are only 2 stable factors and the object E is of class OEI ` C OES.I `/, the destabilizing objects must be of class OEI ` and OES.I `/.Thus, [35, Proposition 4.6] implies that the destabilizing objects are I `OE2k and S.I `0 /OE2k 0 for two lines `; `0 and integers k; k 0 2 Z.