Projectivity of Bridgeland Moduli Spaces on Del Pezzo Surfaces of Picard Rank 2
We prove that, for a natural class of Bridgeland stability conditions on and the blow-up of at a point, the moduli spaces of Bridgeland semistable objects are projective. Our technique is to find suitable regions of stability conditions with hearts that are (after “rotation”) equivalent to representations of a quiver. The helix and tilting theory is well-behaved on Del Pezzo surfaces and we conjecture that this program (begun in [ABCH13]) runs successfully for all Del Pezzo surfaces, and the analogous Bridgeland moduli spaces are projective.
- 1 Introduction
- 2 Useful definitions
- 3 Discussion of the strategy
- 4 Proof of the conjecture for
- 5 Proof of the conjecture for
- A Mutations defined by height functions
- B Calculations
Let be a smooth projective complex surface and be its associated bounded derived category. A Bridgeland stability condition gives a notion of stability for objects of and it is interesting to study the moduli space of -semistable objects of Chern character .
Unlike Mumford and Gieseker stability, Bridgeland stability conditions are not a priori tied to Geometric Invariant Theory (GIT), and consequently the structure of Bridgeland moduli spaces is not fully understood in general, e.g., are they projective? A general approach to this question is taken in [BM14] where Bayer and Macrì construct nef divisors on Bridgeland moduli spaces and show that for K3 surfaces, the divisors are ample.
Another method to deduce structure for these moduli spaces is to find particular stability conditions that have ties to GIT, and exploit this connection. For example, the first author and Bertram [AB13] define a class of stability condition defined by a choice of general and ample divisor in . When has Picard rank 1, the space of these stability conditions is parameterized by real variables, with , and for large Bridgeland stability is equivalent to Gieseker stability (e.g. [ABCH13, Proposition 6.2]) and thus the corresponding Bridgeland moduli spaces have a GIT interpretation and are projective.
For a principally polarized abelian surface, Maciocia and Meachan [MM13] use this connection to deduce projectivity for other Bridgeland stability conditions by “sliding down the wall”: for invariants corresponding to twisted ideal sheaves, they show that one can move a stability condition (taken from a 1-parameter family of stability conditions) so that is arbitrarily small without crossing any walls for those invariants (the walls are nested semi-circles centered on the -axis). They conclude projectivity by relating stability conditions with small to those with large via a Fourier-Mukai transform.
For , the first author, Bertram, Coskun, and Huizenga [ABCH13] find stability conditions with small where Bridgeland stability for complexes of sheaves is equivalent to King’s -stability for representations of a quiver [Kin94]. By a result of King, it follows that the associated Bridgeland moduli spaces are projective. To extend this structure to other stability conditions, they show that for invariants satisfying the Bogomolov inequality, one can “slide down the wall” and see the original Bridgeland moduli space as isomorphic to one associated to a stability condition with small .
The quivers involved in [ABCH13] are associated to certain exceptional collections of objects in . These exceptional collections exist on other surfaces, e.g., Del Pezzo surfaces, but the “quiver regions” associated to the most natural exceptional collections are too small. Furthermore, determining new exceptional collections is difficult, as the action of the Artin braid group on the set of exceptional collections for does not apply when , which is the case for all other Del Pezzos.
However, Bridgeland and Stern [BS10] describe an operation (called mutation defined by a height function) which produces new exceptional collections in this more general setting (and matches the action of the braid group when ). This operation is applicable for any smooth Fano variety, but works in an ideal way on Del Pezzo surfaces. We look to extend the techniques of [ABCH13] to the other Del Pezzo surfaces.
The program of [ABCH13] can be carried out on any Del Pezzo surface , yielding the projectivity of the spaces .
For the Del Pezzo surfaces of Picard rank 2, and , we find stability conditions that, after an operation called “rotation,” have their hearts equivalent to a category of quiver representations. These hearts are generated by (shifts of) line bundles for and (shifts of) line bundles and a torsion sheaf for . Understanding the Bridgeland stability of these objects is essential to finding these “quiver stability conditions,” and while the stability of line bundles is fully understood for these surfaces [AM14], a full characterization of their stability is not known when the surface has Picard rank greater than 2.
Following [ABCH13], Bridgeland stability for these quiver stability conditions is equivalent to King’s stability of representations of a quiver and we deduce projectivity for the associated moduli spaces of Bridgeland semi-stable objects. To extend this structure to moduli spaces associated to other stability conditions we “slide down the wall,” using a result of Maciocia on the nestedness of walls in certain vertical planes in the space of stability conditions.
Our main theorem is the following.
The conjecture is true for and , which are Del Pezzo surfaces of Picard Rank . In particular, every moduli space is equivalent to a moduli space of representations of a quiver, and is therefore projective.
2 Useful definitions
In this section, we recall definitions and tools that we need in the rest of the paper.
2.1 Bridgeland stability conditions on surfaces
For a smooth projective complex surface, [AB13] define a natural class of Bridgeland stability conditions that depend on a choice of ample and general divisor. We denote the set of such Bridgeland stability conditions (the “div” stands for “divisor”). We omit a general introduction to Bridgeland stability conditions and instead point out where the [AB13] construction meets the requirements “as we go.” For full generality, the interested reader should see [Bri07].
Let be a smooth projective surface. Given two -divisors with ample, we define a stability condition . To do so, we must specify a heart of a bounded -structure on and a function (called the central charge) satisfying certain positivity, filtration, and non-degenerate conditions.
Our heart is generated by torsion sheaves, Mumford -stable sheaves of “high slope,” and shifts of Mumford -stable sheaves of “low slope,” where the Mumford -slope is
Specifically, let be the tilt of the standard -structure on at defined by where
is the full subcategory closed under extensions generated by torsion sheaves and -stable sheaves with .
is the full subcategory closed under extensions generated by -stable sheaves with .
Now define by It is equal to
The central charge satisfies the following positivity property: for we have and if then . This property allows us to define stability for an object using the “Bridgeland slope” function
(Note that if then , and if then .) We say that is -stable (resp. -semistable) if for all nontrivial in we have (resp. ). We extend the notion of stability to all objects of the derived category: is -(semi)stable if some shift and is -(semi)stable.
There exist Harder-Narasimhan filtrations of objects with respect to -semistable objects, defined analogously to Harder-Narasimhan filtrations of coherent sheaves with respect to Mumford -semistable sheaves. By [AB13, Corollary 2.1] and [Tod13, Sections 3.6 & 3.7], is a full, numerical stability condition on .
2.2 Exceptional collections and associated hearts
Here we describe the heart and quiver associated to a full, strong exceptional collection, as well as an operation that (in certain circumstances) yields new collections from others. We use instead of to denote a particular exceptional object since in Section 5 we use for the exceptional divisor of .
An exceptional object is one with and for all . For a Del Pezzo surface , Kuleshov and Orlov show that any exceptional object is either a Mumford -stable locally-free sheaf or a torsion sheaf of the form with an irreducible rational curve satisfying and an integer (see, e.g. [GK04] or [BS10, Theorem 8.1]). Note that the Del Pezzo condition is not necessary for much of the following discussion (we will make a note where it is used).
An exceptional collection is a sequence such that each is an exceptional object and implies . An exceptional collection is called full if the smallest full triangulated subcategory of containing is itself, and is called strong if for we have unless .
Bridgeland and Stern show that a full strong exceptional collection yields a heart (of a bounded -structure) that is equivalent to the module category of a quiver algebra, which in turn is equivalent to the category of finite-dimensional representations of a quiver (possibly with relations) [BS10, Theorem 2.4]. Furthermore, the heart is the smallest full extension-closed subcategory of containing the dual collection to (for this reason we often denote by ). The objects are defined by , where is the left mutation of through defined by the canonical evaluation triangle:
The quiver associated to the heart has a vertex associated to each . The number of arrows from vertex to vertex can be obtained using either irreducible hom’s between objects in or extensions of objects in :
The objects correspond to the simple representation over the vertex.
Given a full exceptional collection one may generate the helix using the rule . (This generates a helix of type in the notation of [BS10].) The helix is said to be geometric if implies unless . If is a geometric helix then each “thread” is a full strong exceptional collection.
Bridgeland and Stern define an operation on exceptional collections (resp. helices) called mutation defined by a height function for , which constructs a new exceptional collection (resp. helix ) from a given one (resp. ) and choice of object (resp. . If is geometric then so is and it is shown that, if is a Del Pezzo surface, then for any choice of object there is an associated height function. We do not formally define this operation here, but direct the interested reader to Appendix A for the precise definition as well as an interpretation using the quiver associated to a thread containing .
Because the quiver algebra associated to is the left tilt of that associated to ([BS10, Proposition 7.3]) and since for Del Pezzos the mutation by a height function operation is determined solely by the choice of object (see Proposition A.1), we refer to a mutation defined by a height function for as a left tilt at .
3 Discussion of the strategy
Let us now discuss the specifics of our strategy.
From Section 2.2, a full, strong exceptional collection with dual collection yields a heart that is generated by extensions of the objects , and equivalent to finite-dimensional representations of a quiver. The exceptional collection is “Ext” in the sense of [Mac07, Definition 3.10] and so by [Mac07, Lemma 3.16], if and then .
A choice of invariants corresponds to a choice of dimension vector and following the proof of [ABCH13, Proposition 8.1] we see that -stability is equivalent to King’s -stability [Kin94]. Thus the moduli space of Bridgeland semistable objects is projective when semistable objects are considered (and if only stable objects are considered, the space is quasi-projective).
We want to use the above observations to deduce that is projective for all and (Bogomolov) . The first issue with this strategy is that there is no with . This is because all dual collections we consider have objects where is a sheaf, and these objects cannot belong to any by definition. However, there are stability conditions such that, after a gentle operation called “rotation” (which does not affect stability, but does affect what shift of certain objects belong in the heart), the rotated stability condition has .
A rotation is defined as follows: Given and , a rotation by yields the Bridgeland stability condition where and is the subcategory closed under extensions generated by the objects
such that is -semistable and ,
such that is -semistable and .
In particular, if is a -semistable object with arg , then is “replaced” by in .
We emphasize that rotation does not affect stability: is -(semi)stable iff is -(semi)stable (recall that is -(semi)stable iff is -(semi)stable for all ).
In Sections 4 and 5 we find regions associated to a dual collection (where after a rotation by , we have ). We call a quiver region and any a quiver stability condition. We find by determining the stability conditions such that each object (or shift of an object) in is -semistable and where the objects have the correct Bridgeland slopes relative to each other so that after rotating we have .
After finding a quiver region , any choice of line bundle yields the quiver region associated to the exceptional collection . Our discussion above shows that for any quiver stability condition and choice of invariants , we have projective. To extend this projectivity to other stability conditions, we “slide down the wall” in the following sense.
In [Mac14], Maciocia presents real 3-dimensional slices of determined by a choice of an ample -divisor and a divisor orthogonal to . When has Picard rank 2, these slices are determined solely by , and are defined as
We parametrize with the coordinates where the -plane is a parametrization of in .
To determine -stability of objects and relative Bridgeland slopes, we must consider walls
Maciocia shows that there are disjoint vertical111By “vertical” we mean defined by an equation involving only and . planes depending on a real parameter such that , and such that for any object whose Chern characters satisfy the Bogomolov inequality, the walls are nested semi-circles (or vertical lines) [Mac14, Proposition 2.6].
Any lies on at most one of these nested semi-circles, say . Since moving along crosses no other walls , for we have -(semi)stable iff -(semi)stable. More generally, since the walls are determined by the invariants of the objects involved, for we have .
Thus, if intersects a quiver region, then for some quiver stability condition and hence is projective. For each of and we find a quiver region such that the quiver regions cover the entire -plane for each . Since each is in a plane, we obtain the projectivity of for all Bogomolov and and thus prove Conjecture 1 in these cases.
We also use the fact that walls in are nested semi-circles for simplification: To understand the geometry of a wall in it suffices to consider its intersection with the -plane.
4 Proof of the conjecture for
In this section, we determine a suitable222Suitable in the sense that the entire -plane of is covered by tensoring with line bundles. quiver region in for and use the region to conclude projectivity for the Bridgeland moduli spaces where the invariants satisfy the Bogomolov inequality (following the discussion of Section 3).
We shall denote by the line bundle with the natural projections from to the two copies of , respectively. Also, we shall denote by and the divisors corresponding to and , respectively. These are the generators of the cone of effective curves, and every other divisor can be written as for some . Note that , and .
A Bridgeland stability condition is determined by real divisor classes and , with ample. By the Nakai-Moishezon criterion, is ample iff .
For a fixed pair, the corresponding Bridgeland stability condition (we now drop the subscript ) has heart generated by -stable sheaves of slope , torsion sheaves, and objects of the from , where is a -stable sheaf of slope .
The central charge is defined as
where , and .
The slices defined in Section 3 are given here by . We identify the Bridgeland stability condition where with the coordinates . Often we need only consider the and coordinates, and we project to the the -plane.
4.1 A suitable exceptional collection
The natural exceptional collection
on is full and strong (and in fact generates a geometric helix), but does not have a large enough quiver region for our purposes, so we find another exceptional collection. The quiver associated to is
[yshift=0.7ex]r \arrow[yshift=-0.7ex]r \arrow[xshift=0.7ex]d \arrow[xshift=-0.7ex]d & O_S(1,0)∙ \arrow[xshift=0.7ex]d \arrow[xshift=-0.7ex]dO_S(0,1)∙\arrow
[yshift=0.7ex]r \arrow[yshift=-0.7ex]r & O_S(1,1)∙
and we use this information to left tilt333For details on performing the left tilt operation, see Proposition A.1. at and obtain the exceptional collection
Straightforward calculations yield444See Appendix B.2 for the calculations.
where is a sheaf of rank 3, , and . The dual collection to is is555See Appendix B.3 for the calculations.
Note that is indeed an “Ext” exceptional collection in the sense of [Mac07, Definition 3.10]. The heart is naturally equivalent to finite-dimensional (contravariant) representations of the quiver
[yshift=0.7ex]dr \arrow[yshift=-0.7ex]dr \arrow[yshift=-1.4ex]dr \arrowdr
& G∙ \arrow[yshift=0.7ex]r \arrow[yshift=-0.7ex]r \arrow[xshift=-0.7ex]d \arrow[xshift=0.7ex]d & O_S(1,0)∙
This quiver remains constant for any tensor of and by line bundles (with the labels above the vertices adjusted appropriately). We now locate the quiver region associated to .
4.2 The associated quiver region
It follows from [AM14, Theorem 1.1] that all objects appearing in (and their shifts) are -stable for all . Therefore, to find the associated quiver region, we need only find the stability conditions that can be rotated so the new heart contains , , , and . We prove the following, where we restrict to a slice
The quiver region associated to
is the region strictly inside both of the ellipsoidal walls and .
By the definition of the hearts associated to , if is a line bundle on then either or is in . Also, if is -stable, then after rotating with , we have either or in . Thus, to rotate to and have we must have
To ensure that rotating does not force we must also have
For any such , either or is in . If , then we must rotate so that . But then to ensure (as above) that rotating does not force we must have
Similarly, if , then we must rotate so that . To ensure that rotating does not force we must have
We now restrict to a particular slice and determine the region consisting of the that satisfy one of these two sets of conditions.
From the definition of the hearts , condition (1) states that for we must have to the left of the line and on or to the right of the lines and . Note that implies that these lines are both diagonal (in fact, negatively sloped), so that the notions “to the left” and “to the right” are sensible. (For the pictures below, we chose and .)
To understand condition (2), let us look at the walls and . We are working in , which means that we have fixed an ample , and we are considering stability conditions depending on and . We have that
The equation of the wall , which is given by , simplifies to
which is an ellipsoid in the space parametrized by . The region where is the region inside the ellipsoid. A similar calculation gives us the following equation for :
The intersections of the ellipsoidal walls with the -plane are the ellipses and , respectively. The line is tangent to both ellipses at , and the line [resp. ] is tangent to the ellipse of [resp. ] at [resp. ]. Note that the vertical planes in over these lines do not intersect the region inside the two ellipsoidal walls, so any in the region inside both walls automatically satisfies condition (1) (see Figure 3).
We now show that conditions (3) and (4) add no other restrictions. If then , and since and is -stable, we must have that , as needed in condition (3). Similarly, if , then . Moreover, since is -stable, and and has as subobjects, we must have and as needed in condition (4).
Therefore the quiver region is the intersection of the ellipsoidal regions bounded by the walls and , as we set out to show. ∎
The projection of the quiver region onto the -plane is a region containing a unit square with three corners cut off: . The analogous region associated to the quiver region is , where the indicates component-wise addition, and together the regions cover the entire -plane. Recalling the argument given in Section 3, we have shown the following.
Conjecture 1 holds for . In particular, if the invariants satisfy the Bogomolov inequality, then every moduli space is isomorphic to a moduli space where is in a quiver region , and hence is projective.
5 Proof of the conjecture for
In this section, we determine a suitable quiver region in for . The considerations are similar to those for , but with two exceptions: First, instead of a single quiver region, we find two quiver regions that together cover a “unit region,” and second, since the relevant hearts contain a torsion sheaf as one of the generators, the stability of those sheaves must be understood.
Let be the strict transform of the hyperplane class in , let be the exceptional divisor, and let . We then have , , , , and . The cone of effective curves on is the cone of non-negative linear combinations of and .
A Bridgeland stability condition is determined by real divisor classes and , with ample. By the Nakai-Moishezon criterion, is ample iff .
For a fixed pair, the corresponding Bridgeland stability condition has heart generated by -stable sheaves of slope , torsion sheaves, and objects of the from , where is a -stable sheaf of slope .
The central charge is defined as
where , and .
The slices defined in Section 3 are given here by . We identify the Bridgeland stability condition where with the coordinates . As before, we often project to the the -plane and only consider the and coordinates.
5.1 Finding two suitable exceptional collections
In [BS10], Bridgeland and Stern give an exceptional collection
which we rewrite as
The quiver for is
[yshift=0.7ex]r \arrow[yshift=-0.7ex]r \arrowdr & O_S(F)∙ \arrowd \arrowdr
& O_S(E+F)∙ \arrow[yshift=0.7ex]r \arrow[yshift=-0.7ex]r & O_S(E+2F)∙
and the dual collection to is
The quiver region associated to this exceptional collection turns out to be too small for our purposes, i.e., it does not cover a “unit region” in the -plane of . To cover a unit region, we combine the quiver regions from two exceptional collections.
The first of these exceptional collections is found replacing (which generates a geometric helix) with its left tilt at :
Straightforward calculations show that666See Appendix B.5 for the calculations.
where is a sheaf of rank 2 with and . The dual collection to is777See Appendix B.6 for the calculations.
The second exceptional collection is obtained from in three steps. We first move one position forward along the helix it generates to obtain the exceptional collection