Gaeta resolutions and strange duality
over rational surfaces
Abstract.
Over the projective plane and at most two-step blowups of Hirzebruch surfaces, where there are strong full exceptional sequences of line bundles, we obtain foundational results about Gaeta resolutions of coherent sheaves by these line bundles. Under appropriate conditions, we show the locus of semistable sheaves not admitting Gaeta resolutions has codimension at least 2. We then study Le Potier’s strange duality conjecture. Over these surfaces, for two orthogonal numerical classes where one has rank one and the other has sufficiently positive first Chern class, we show that the strange morphism is injective. The main step in the proof is to use Gaeta resolutions to show that certain relevant Quot schemes are finite and reduced, allowing them to be enumerated using the authors’ previous paper.
Key words and phrases:
Exceptional sequence; Gaeta resolution; Moduli of sheaves; Strange duality; Quot scheme2020 Mathematics Subject Classification:
Primary: 14D20, 14F06; Secondary: 14F08, 14J261. Introduction
In the moduli theory of sheaves over complex algebraic surfaces, there is a famous conjecture by Le Potier [LP05], called the strange duality conjecture, which relates the global sections of two determinant line bundles on certain pairs of moduli spaces of sheaves. Known results over rational surfaces are mostly in the cases where one of the moduli spaces parametrizes pure dimension 1 sheaves, see e.g. [Dan02, Abe10, Yua21]. On other surfaces, the conjecture requires different formulations, see e.g. [BMOY17]. In an attempt to provide a unified treatment of the conjecture over rational surfaces, the first author, Bertram, and Johnson [BGJ16] proposed to use Grothendieck’s Quot schemes [Gro61], following Marian and Oprea’s ideas [MO07a, MO07b] over curves. A key tool in the study of Quot schemes over in [BGJ16] is Gaeta resolutions of coherent sheaves in terms of the strong full exceptional sequence of line bundles .
This leads us to the study of exceptional sequences and Gaeta resolutions over rational surfaces. A strong full exceptional sequence, if it exists, completely captures the derived category in an explicit way, by theorems of Baer [Bae88] and Bondal [Bon89]. While coherent sheaves can always be resolved by locally free sheaves by the Hilbert Syzygy Theorem, we can bring the resolution under better control if the locally free sheaves are taken from an exceptional sequence. Resolutions built from such exceptional sequences, called Gaeta resolutions, have been applied toward a variety of problems in the study of sheaves on rational surfaces [Dre86, LP98, CH18, CH20, CH21]. Though the existence of strong full exceptional sequences of line bundles in general is an open question, the answer is affirmative over a rational surface that can be obtained from a Hirzebruch surface by blowing up at most two sets of points [HP11], which we call a two-step blowup of .
Over or a two-step blowup of , we choose a particular strong full exceptional sequence, determine when a sheaf admits Gaeta resolutions, and study general properties of such sheaves. We then apply Gaeta resolutions to the study of strange duality, proving the injectivity of the strange morphism in some cases. One of the key points in the proof is to show that relevant Quot schemes are finite and reduced, which we accomplish using Gaeta resolutions. A parallel statement over was proved in [BGJ16]. Another crucial point is to enumerate the length of the finite Quot scheme, which was settled in our previous paper [GL22] via the study of the moduli space of limit stable pairs [Lin18].
We now set up the study. Let be a smooth projective algebraic surface over an algebraically closed field of characteristic , with a strong full exceptional sequence of line bundles
Given a coherent sheaf on , we would like to find a resolution of of the form111To avoid clumsy notation, we drop the direct sum symbol from the exponents. If we want to denote tensor products, we will use .
| (1.1) |
which is called a Gaeta resolution. If a Gaeta resolution exists, the exponents are uniquely determined by the numerical class of .
One of our main technical results is a criterion to determine when a Gaeta resolution exists. We state here the criterion for a two-step blowup of , where we know there are strong full exceptional sequences of line bundles, see § 3.3. Let denote the set of blown-up points in the first step, with corresponding exceptional divisors for . The second set of blown-up points, which lie on these exceptional divisors, is denoted , and the corresponding exceptional divisors are for . Using the exceptional sequence § 3.3(c), we get the following result.
Proposition 1.1 (Proposition 4.4(b)).
On a two-step blowup of , a torsion-free sheaf has a Gaeta resolution if and only if
-
(i)
vanishes for ;
-
(ii)
vanishes for and the divisors , , and ; and
-
(iii)
and for all and .
Section 4 contains a similar statement for , which is known.
We will apply Gaeta resolutions towards moduli problems, using the connections to prioritary sheaves. We impose a mild technical condition on a two-step blowup of , which we call admissibility (Definition 2.7). We use to denote both the class of the fibers of the ruling and the pullback of this class to . We denote a numerical class in the Grothendieck group by
where is the rank, is the first Chern class (equivalent to the determinant line bundle), and is the Euler characteristic. We have
Proposition 1.2.
Let be an admissible blowup of and be a polarization such that . For a numerical class with fixed rank and fixed , suppose the first Chern class is sufficiently positive. Then a general -semistable sheaf of class admits Gaeta resolutions.
We mostly use Gieseker stability and use either “-semistable” or “semistable”. If we want to use slope stability, we will specify. For the precise meaning of being sufficiently positive in Proposition 1.2, see the conditions in Proposition 4.5(a) as well as Proposition 5.1(b.ii). There is another statement, Proposition 6.2, in which the rank and first Chern class are fixed, which asserts that if the discriminant is sufficiently large then general semistable sheaves admit Gaeta resolution up to a twist by a line bundle. In each case, we can immediately deduce that the moduli space is unirational, which is known [Bal87].
By imposing stronger conditions on the numerical class and the polarization , we prove a refinement of Proposition 1.2:
Theorem 1.3.
Let be an admissible blowup. Assume the class is of rank , admits Gaeta resolutions in which the exponents are strictly positive and satisfy (6.2, 6.3), that the polarization is general and satifies (6.4, 6.5), and that the discriminant of is sufficiently large in the sense of (6.9). Then the closed subset of S-equivalence classes of semistable sheaves whose Jordan-Hölder gradings do not admit Gaeta resolutions has codimension in .
Corollary 1.4.
Assuming the conditions from Proposition 1.2 and appropriate conditions from Proposition 5.1(b), a general sheaf in is torsion-free, is locally-free if , satisfies the cohomological vanishing conditions in Proposition 4.4 (b.i-iii), and is globally generated if . Assuming the stronger conditions from Theorem 1.3, the same properties hold away from a locus of codimension in (with locally free replaced by torsion-free).
For the proof of Theorem 1.3, we will need the following statement which is of general interest.
Theorem 1.5.
Suppose is a rational surface other than and is a morphism where a general fiber is isomorphic to . Let be a general ample divisor such that . Assume there is a slope stable vector bundle with rank , first Chern class , and second Chern class over . Then
We will apply Gaeta resolutions towards the study of the strange duality conjecture. Let be or an admissible blowup of . In the Grothendieck group , let
be two numerical classes. Notice that they are orthogonal: . Let denote the hyperplane class on or a polarization satisfying (6.4, 6.5) on . Let and denote the moduli spaces of -semistable sheaves, where is isomorphic to the Hilbert scheme of points. Let be the universal subscheme and its ideal sheaf. For a coherent sheaf of class , consider the determinant line bundle
on , where and are the projections from to the first and second factors, respectively. There is also a similar line bundle on . On , the line bundle has a canonical section, which induces the strange morphism
The strange duality conjecture says that is an isomorphism. For a more general setup, see § 7.1.
In this context, we prove the following result in support of the strange duality conjecture. In the statement, for a vector bundle on , we write
| (1.2) |
Theorem 1.6.
Let be or an admissible blowup of , and , , and as above. If is sufficiently positive, then:
-
(a)
The rank of the strange morphism is bounded below by , for a vector bundle with numerical class ;
-
(b)
The strange morphism is injective.
We sketch the proof. Let be a vector bundle of class that admits a general Gaeta resolution and consider quotients of of class . Then the Quot scheme has expected dimension . If it is finite and reduced, a simple argument shows that its length provides a lower bound for . According to [GL22], in this case its length is . On the other hand, Theorem 7.4 relates this top Chern class to , and the determinant line bundle has no higher cohomology when is sufficiently positive, which finishes the proof. Thus, the crucial point is to establish that the Quot scheme is finite and reduced, which we prove by considering the relative Quot scheme over the space of Gaeta resolutions and calculating the dimension of the relative Quot scheme. The following theorem summarizes the key results related to the Quot scheme.
Theorem 1.7.
Let , , , and be as in Theorem 1.6, and be a vector bundle of class that admits a general Gaeta resolution. If is sufficiently positive, then:
-
(a)
The Quot scheme parametrizing quotient sheaves of with numerical class is finite and reduced;
-
(b)
For every point of , is an ideal sheaf for general and the kernel is semistable;
-
(c)
The length of is .
In [BGJ16], parts (a) and (b) were proved over and calculations were made that informed Johnson’s expectation that the counting formula (c) should be true for del Pezzo surfaces [Joh18]. The formula (c) was proved in [GL22] by the authors of the current paper, for a general smooth regular projective surface, assuming that the Quot scheme is finite and reduced.
The positivity conditions on in these theorems, which are stronger than for Proposition 1.2, are summarized in the appendix. Theorem 1.7 requires (A.2, A.3), and Theorem 1.6 requires (A.4) as well.
We organize the paper as follows. In § 2, we review basic facts about divisors and line bundles on Hirzebruch surfaces and their blowups. In § 3, we review exceptional sequences in the derived category. In § 4, we obtain criteria for the existence of Gaeta resolutions, including Proposition 1.1, and classify the numerical classes of sheaves admitting Gaeta resolutions. In § 5, we prove some general properties of such sheaves and relate them to prioritary sheaves. In § 6, we discuss connections to semistable sheaves and prove Theorem 1.3. In § 7, we set up the strange morphism and prove Theorem 1.6. In § 8, we prove Theorem 1.7. Finally, the appendix contains a summary of the positivity conditions on required in the proofs of Theorems 1.6 and 1.7.
Acknowledgment. YL would like to thank Alina Marian and Dragos Oprea for helpful correspondences. TG would like to thank Lothar Göttsche for providing updates on his work with Anton Mellit. YL is supported by grants from the Fundamental Research Funds for the Central Universities and Applied Basic Research Programs of Science and Technology Commission Foundation of Shanghai Municipality.
2. Hirzebruch surfaces and blowups
We review some basic results on divisors and cohomology of line bundles on Hirzebruch surfaces and their blow-ups. Along the way, we introduce two technical assumptions (2.1) and (2.2). The first is not a restriction, while the second is a mild condition.
2.1. Divisors on blowups of Hirzebruch surfaces
Let denote the Hirzebruch surface with , which is a rational surface that is ruled over . Note that . Letting denote the class of a fiber and the 0-section with self-intersection , and generate the effective cone of , and , , . The canonical divisor is . The divisor classes and generate the nef cone. To simplify notation, let
The linear system induces the morphism to giving the ruling, while induces a morphism ; if , this is the other ruling of , while if , this contracts to a point and maps the fibers of the ruling to distinct lines through that point in . In particular, suppose are distinct points on , where we allow to be infinitely near to . Then separates unless are distinct points on the same fiber or corresponds to the tangent direction along the fiber at . Similarly, separates unless are distinct points on or and is the tangent direction along at .
Remark 2.1.
The blowup of at any point is isomorphic to . The blowup of at any point is isomorphic to the blowup of at a point not on . For , the blowup of at a point on is isomorphic to the blowup of at a point not on . Thus, when considering the surfaces that arise from blowing up or Hirzebruch surfaces, it suffices to consider blowups of for where the blown-up points are not on [GH78, p.519]. So, quite often, in the case , we impose the condition that
| the blowup avoids | (2.1) |
in the sense that none of the blown-up points is on .
Let be obtained from a sequence of blowups
where is the th blowup at a point . Assume that the indices can be partitioned into two sets
such that the within each of the sets and commute. In other words, can be obtained from by up to two blowups, each possibly at multiple points. By Remark 2.1, it suffices to consider the case and that the blowup avoids .
We define a partial ordering on the set by if is on the exceptional divisor of , and say that the height of is 0 if is minimal with respect to , while otherwise has height 1. We can consider points of height 0 as lying on , while if then is infinitely near to and can be viewed as a tangent direction at on . For simplicity, we choose the partition and such that
Thus, we think of as being obtained from by first blowing up a collection of points on and then blowing up a collection of points on the exceptional divisors of the first blowup.
Let denote the total transform in of the exceptional divisor of . We abuse notation by writing for the pullbacks of the divisors on . The Picard group of is generated by , with the following intersections:
Here, is the Kronecker delta function. The canonical divisor is
Let denote the strict transform of . If has height 1, then , while if has height 0, then . Note that .
The following classification of base loci of certain linear systems on will be useful. When describing the linear systems below, we write a divisor in parentheses, as in , to indicate that it is a fixed part of the linear system.
Lemma 2.2.
Suppose on is the pullback of an effective divisor on . Then if ,
Proof.
Tensoring the short exact sequence
by and taking cohomology, we get an exact sequence
As , we see that as is a connected (possibly reducible) curve and every section of this line bundle must be 0 on the component . ∎
Remark 2.3.
If is a divisor on and is a point of height 0, then the curves in the linear series on are in bijection with curves in on that contain . Similarly, if , then curves in on are in bijection with curves in on that contain and have tangent direction at . The curves in the linear system on are obtained as pullbacks of the corresponding curves on , with one copy of the appropriate exceptional divisors removed.
The linear systems on in the following examples will play an important role. We assume that and that the blowup avoids .
Example 2.4.
If , let denote the index such that . Then
Moreover, if is the tangent direction along the fiber at , then
The curve can be obtained by considering to be the first blown-up point, taking the strict transform of the unique fiber containing under , and then taking the pullback of that strict transform under the remaining blowups, which may be reducible if other lie on that fiber or are infinitely near to points on that fiber. In particular, if is the tangent direction along the fiber at , then lies on , and taking the strict transform with respect to yields the curve .
Example 2.5.
For , let denote the index such that . Then
where is basepoint-free. This follows from the fact that separates points on (including infinitely near points) as long as the points are not contained on , and by assumption the blown-up points are not on .
Example 2.6.
For and , the base locus of can be described as follows:
-
(a)
If , then is basepoint-free unless is the tangent direction along the fiber containing , in which case
-
(b)
If , let denote the index such that . Then
and is basepoint-free unless and lie on the same fiber , in which case
We explain the two parts. For (b), the linear system contains the union of and a curve in . As the latter is basepoint-free, the only possible basepoints are on . By the same argument with the roles of and reversed, we see that the linear system is basepoint-free unless and lie on the same fiber.
For (a), we note that if the linear system has a basepoint , which we may assume is a point of , then the linear system of curves in that contain and must have as an (infinitely near) basepoint, which, by a similar argument as the one for (b), implies that and lie on the same fiber and is the tangent direction along that fiber.
Some of the calculations in later sections are simplified if the linear systems for are basepoint-free. For this purpose, we will often assume that
| the blowup avoids fiber directions | (2.2) |
in the sense that are distinct from the point where the strict transform of the fiber containing meets the exceptional divisor of , for all .
We summarize the assumptions on in the following definition:
Definition 2.7.
The rational surface is an admissible blowup of if it is an at most two-step blowup and the following conditions hold:
-
•
if , then ;
-
•
if , then the blowup avoids and avoids fiber directions.
We emphasize that or can be empty. In particular, the definition includes . Then, by the above discussion and particularly Example 2.6, we have shown that if is admissible, then every divisor in the set
| (2.3) |
is basepoint-free. This leads to the following result.
Proposition 2.8.
Suppose is an admissible blowup of . Suppose is the line bundle associated to a positive integral linear combination of all divisors in the set
Then is very ample.
Proof.
The linear system associated to contains unions of divisors in the linear systems associated to the divisors in , so since these divisors are all basepoint-free, it suffices to show that the divisors in collectively separate points and tangents on in the following sense:
-
1)
For any distinct points , there is a divisor linearly equivalent to a divisor in such that and ;
-
2)
For any , there are divisors that contain , are each linearly equivalent to a divisor in , and whose images in are linearly independent.
For 1), can be used to separate points on the complement of since it is very ample there, can separate two points on , can separate two points on , and separates any two points on . Separating points on different exceptional curves or a point on the exceptional locus from a point on the complement is similarly easy.
For 2), as is very ample on the complement of , it suffices to consider the cases , where is , for , or for . In each case, it suffices to choose transversal to at and which is a union of with another divisor that does not contain . These divisors can be chosen to be general in the following linear subsystems:
This completes the proof. ∎
As very ample is equivalent to 1-very ample and -very ampleness is additive under tensor products [HTT05], we immediately see that if is the line bundle associated to a positive integral linear combination of all divisors in in which the weight of each divisor is , then is -very ample.
2.2. Cohomology of line bundles
First, we summarize how to calculate the cohomology groups of line bundles on the Hirzebruch surface , following [CH18]. By Hirzebruch-Riemann-Roch,
Since the effective cone of is generated by and ,
Then Serre duality implies that
It suffices to assume that , as other cases can then be obtained via Serre duality. In this case, as vanishes and the Euler characteristic is known, it suffices to calculate , which can be done as follows:
-
(a)
, for all and .
-
(b)
if and otherwise.
-
(c)
Let . If , then
while if , then the equality
allows to be determined by induction on .
In particular, we deduce the following:
Lemma 2.9.
Let . Then
-
(a)
if and only if or ;
-
(b)
if , if and , or if and .
In order to use these calculations on to obtain information about the cohomology of line bundles on a two-step blowup of , we use the following general result.
Lemma 2.10.
Let be a blowup of a smooth projective surface at distinct (possibly infinitely near) points. For a line bundle on , .
Proof.
According to [Har77, V. Proposition 3.4], and for . Then and for by the projection formula. The spectral sequence gives the result. ∎
Then, letting denote a two-step blowup of , we have:
Lemma 2.11.
Let be a line bundle on such that for some . Then
-
(a)
for all ;
-
(b)
;
-
(c)
If the base locus of does not contain , then .
Proof.
For (a), consider the short exact sequence
Since has no cohomology, we get the result. For (b) and (c), consider the short exact sequence
Since , we immediately obtain (b). If the base locus of does not contain , then is surjective, which gives the result since . ∎
3. Exceptional sequences
We review basic facts about Hom functors and exceptional sequences in the bounded derived category of a smooth projective variety over . In particular, we discuss how to replace a general complex with a complex built from a strong full exceptional sequence , which we call an -complex.
3.1. Hom functors
If and are bounded complexes of coherent sheaves, then is the complex of vector spaces defined by
The degree-0 cohomology of this complex is the vector space of chain maps modulo chain homotopy. This complex is especially useful when it computes the derived functor
as in the lemma below. The graded summands are denoted . In the case when and are sheaves and in degree 0,
consists of the usual groups for sheaves in each degree .
Similarly, is defined by
If either or is a complex of locally free sheaves, then represents .
We also recall a few facts about derived functors. The cohomology groups of , often denoted , are called hypercohomology, and hypercohomology of a sheaf is just sheaf cohomology. The derived functor has the property that . Moreover, and there is a spectral sequence
See [Huy06] for more details.
Lemma 3.1.
If and are any complexes composed of locally free sheaves such that all higher Exts between and vanish for all , then computes . In particular, is the space of chain maps modulo chain homotopy.
Proof.
We calculate as follows. First, we represent by . Then, since and have no higher Exts between them, has no higher cohomology, so by the above spectral sequence we can calculate simply as . Thus, the complex represents , which gives the result. ∎
3.2. Exceptional sequences
The material reviewed in this section can be found in [GK04].
Definition 3.2.
An object is exceptional if
An exceptional sequence is a sequence of exceptional objects such that
It is strong if in addition
It is full if generates as a triangulated category.
Let be a strong full exceptional sequence of locally free sheaves on . The full triangulated subcategories generated by individual yield a semi-orthogonal decomposition of . Thus, for each object in , there is a diagram of morphisms
| (3.1) |
in which each triangle is distinguished and is in . Each can be constructed as the left mutation of through , namely as the cone
We call the factors of with respect to the exceptional sequence. The diagram is functorial and in particular, the factors of are unique up to isomorphism.
Using the diagram, the factors of can be assembled to produce a complex isomorphic to . We define an -complex to be a bounded complex such that each is a direct sum of sheaves in the exceptional sequence.
Lemma 3.3.
is isomorphic to an -complex whose sheaves are the same as in the complex (but with different maps).
Proof.
We prove that each is isomorphic to an -complex by induction on . Assume is isomorphic to an -complex . By the previous lemma, the morphism can be represented by a chain map , whose mapping cone is an -complex whose sheaves are the same as and which represents . ∎
Example 3.4.
Suppose and is a sheaf in degree 0 that has a resolution . Then the diagram (3.1) can be realized as
The proof gives an inductive algorithm for assembling the factors of into an -complex isomorphic to . We say that an -complex is minimal if, among all -complexes isomorphic to , the total number of sheaves in the complex is as small as possible. If each is represented by the minimal -complex described above, then the complex obtained from this algorithm is minimal as well, and we call it the minimal -complex of . By Lemma 3.1, it is unique up to quasi-isomorphism.
Using -complexes to represent objects in the derived category is useful because the sheaves in an exceptional sequence have no higher Exts between them, so Lemma 3.1 implies that any morphism between two objects represented by -complexes can be realized as a chain map between the -complexes.
There is a direct way to identify the factors of by making use of the dual of the exceptional sequence. The (left) dual of a full exceptional sequence is a full exceptional sequence with the property that
| (3.2) |
The dual sequence always exists, can be constructed from by mutations, and is characterized up to isomorphism by (3.2).
Lemma 3.5.
Suppose is a strong full exceptional sequence and is its dual. Then the factors of an object satisfy
Proof.
Let be the minimal -complex of , whose sheaves are the same as . Then, as the higher Exts between and the sheaves in vanish, computes . Thus, by property (3.2),
and tensoring by gives the result. ∎
3.3. Main examples of strong full exceptional sequences
In later sections, we will focus on the following choices of strong full exceptional sequences of line bundles on , Hirzebruch surfaces, and, more generally, two-step blowups of Hirzebruch surfaces.
-
(a)
On , the exceptional sequence is strong and full (see [Huy06, Corollary 8.29, Exercise 8.32] for a more general result on ).
- (b)
-
(c)
On a two-step blowup of , consider the exceptional sequence
(3.3) This sequence is obtained from the sequence on in (b) by standard augmentations, so it is strong and full ([HP11] Theorem 5.8). The exceptional and strong properties can easily be checked by using Lemma 2.11 to reduce to calculations on . Note that this example specializes to (b) by allowing the set of blown-up points to be empty.
The dual sequences are as follows and can be verified by checking (3.2):
-
•
On , the dual exceptional sequence is
where is the tangent sheaf, which can be checked by Bott’s formula [Bot57].
- •
-
•
On the two-step blowup of , the dual exceptional sequence is
This can be seen by using the fact that , the short exact sequences , and Lemma 2.11(a) to reduce to the calculations on .
4. Gaeta resolutions
We study two-step resolutions of coherent sheaves by the exceptional sheaves in the previous section. We call such resolutions Gaeta resolutions. We provide a general criterion and a criterion specialized for rational surfaces that detect when a sheaf admits a Gaeta resolution. We classify numerical classes over two-step blowups of of sheaves admitting Gaeta resolutions, including the case of allowing a twist by a line bundle on . These results lay the foundation for our applications of Gaeta resolutions in later sections.
4.1. Definition of Gaeta resolutions
Let be a strong full exceptional sequence on a smooth projective variety over . We are particularly interested in minimal -complexes of the following form.
Definition 4.1.
For a coherent sheaf , a resolution of of the form
is a Gaeta resolution. If the minimal -complex of is of this form, then we say that admits a Gaeta resolution. The non-negative integers are called the exponents of the Gaeta resolution.
Clearly, the exponents determine the numerical class of . Conversely, the class of determines the exponents inductively using semi-orthogonality as
The dual sequence can be used to obtain a criterion for when a sheaf admits a Gaeta resolution.
Proposition 4.2 (General criterion).
Let be a coherent sheaf. Then admits a Gaeta resolution if and only if
vanish for all and except possibly for
Moreover, if admits a Gaeta resolution, then the exponents are
4.2. Gaeta resolutions on rational surfaces
In the context of the strong full exceptional sequences in § 3.3, we will focus on the following Gaeta resolutions.
Example 4.3.
For these examples, by applying the general criterion for having a Gaeta resolution (Proposition 4.2) with the explicit dual sequences in § 3.3, we deduce a more explicit criterion.
Proposition 4.4.
Proof.
For (a), Proposition 4.2 includes the first two conditions as well as the condition that vanishes for . Applying to the short exact sequence and using the vanishing of shows that the first two conditions already guarantee .
For (b), Proposition 4.2 includes (i) and (ii) as well as the condition that vanishes for and for all . The vanishing is guaranteed since is torsion-free, while applying to the short exact sequence and using shows that , hence is an equivalent condition. ∎
4.3. Chern characters and Gaeta resolutions
Recall that the exponents in the Gaeta resolutions are determined by the numerical class of . The results in this subsection classify numerical classes that arise as cokernels of Gaeta resolutions. First, we review some useful numerical invariants.
On a smooth projective surface over , if is a coherent sheaf of positive rank , first Chern class , and second Chern character , set
which are called the total slope and discriminant, respectively, of . A simple calculation shows that the discriminant of a line bundle is 0 and the discriminant of is unchanged when is tensored by a line bundle. Using these invariants, the second Chern class of can be written as
Set . Then by Riemann-Roch the Euler characteristic can be written as
and similarly, if are sheaves and , , are the rank, total slope, and discriminant of , then the Euler pairing is
| (4.3) |
On the two-step blowup of a Hirzebruch surface, writing , we compute
We write the numerical class of a sheaf as a triple in which is a non-negative integer, is an integral divisor class, and is an integer. We say that a numerical class admits Gaeta resolutions if there is a sheaf of rank , first Chern class , and Euler characteristic , such that admits a Gaeta resolution. For a sheaf of class and a line bundle , we denote the class of as , and we write , and for the second Chern class, total slope, and discriminant of , which depend only on .
Proposition 4.5.
On a two-step blowup of , consider the numerical class
of positive rank. Then
-
(a)
admits Gaeta resolutions (b) if and only if , , , and the following three integers are all :
-
(b)
Assume and . If the discriminant is sufficiently large, then there is a line bundle pulled back from such that admits Gaeta resolutions.
Proof.
For (a), assuming admits Gaeta resolutions, by comparing first Chern classes, the exponent of must be and the exponent of must be . The remaining exponents can be easily calculated. Conversely, the inequalities show that we can define the exponents in the same way, and a simple calculation shows that the numerical class of the cokernel must be .
For (b), consider the numerical class on . An elementary calculation shows that
Let . Since , the following lemma ensures that we can choose a line bundle on such that admits Gaeta resolutions
in which . Here, the inequality follows from the following lemma. Hence, as well by comparing ranks. Then a simple calculation shows that the cokernels of Gaeta resolutions
have numerical class . ∎
Lemma 4.6.
On , fix a rank and a first Chern class and consider the numerical class . Let be a positive real number. Then there are constants depending only on such that for all such that
there is a line bundle such that and the class admits Gaeta resolutions.
Proof.
We use a setup similar to [CH20, Lemma 4.5]. Consider the curve in the -plane, where is the (in general non-integral) line bundle
Set . By Riemann-Roch,
so is the hyperbola , or, as a function of , .
Let denote the lattice in the plane of points such that is integral, which is a shift of the standard integral lattice. We say that a point is minimal if
-
•
is on or above the upper branch of , and
-
•
and are both on or below .
The minimal points exactly correspond to the line bundles such that admits Gaeta resolutions, according to Proposition 4.5(a).
We need to find a minimal point such that . For this, consider the line , which intersects at the point
The tangent line to at this point has equation and lies above the tangent line. See Figure 1.
Let be the unique minimal point such that satisfies . which lies between and the shift . Then satisfies . Let denote the vertical distance from to , which satisfies . Then . If , then gives the result, so assume on the contrary that . Then let be a positive integer and consider the lattice point . We wish to find as small as possible such that lies above , as then the point will be minimal and will be close to 1. See Figure 2 for an example in which this is achieved with .
To find such that lies above , the rise in between and should exceed , namely
should be positive. As this quantity exceeds the approximation obtained by truncating the geometric series at the first two terms, and as , it suffices to take such that
which yields . Setting , we then have
and for this to be we need
Let , which satisfies . Then the inequality simplifies to
Replacing by , by its upper bound , , , and by their upper bounds, and solving for , we get a sufficient bound for :
for constants that depend only on . ∎
5. Properties of sheaves with general Gaeta resolutions
Given an exceptional sequence from § 3.3 with chosen as in Example 4.3 and a sequence of non-negative integers , consider the vector space
We let denote the open subset of injective maps
which is non-empty if and only if . In this case, we set
let denote the numerical class of these cokernels, which have rank , and call the following projectivization the space of Gaeta resolutions:
| (5.1) |
Then satisfies various cohomology vanishing conditions (Proposition 4.4). The purpose of this section is to prove additional properties of sheaves admitting general Gaeta resolutions, including the prioritary condition and a weak Brill-Noether result.
5.1. Basic properties
We begin by proving some basic properties of sheaves admitting a general Gaeta resolution.
Proposition 5.1.
-
(a)
On and , we have the following for general :
-
(i)
If , then is torsion supported on the determinant of (true even if is not general).
-
(ii)
If , then is torsion-free.
-
(iii)
If , then is locally free.
-
(iv)
If , then is globally generated.
-
(i)
-
(b)
On an admissible blowup of , using the notation in (b), the same cases are true if we assume, for each such that is nonempty:
The key to the proof of this proposition is a Bertini-type statement concerning the codimension on which a general map between vector bundles and drops rank. The case when is globally generated is well known.
Proposition 5.2.
On a smooth projective variety , consider maps , where , are fixed vector bundles of ranks such that . Let
denote the locus of pairs such that the rank of at is .
-
(a)
Suppose is globally generated. Then the codimension of in is .
-
(b)
Suppose there are non-trivial decompositions and such that each is globally generated except that . Set and . If , then . If , then the codimension of in is .
Proof.
Part (a) is a special case of [Ott95, Teorema 2.8] (see also [Hui16, Proposition 2.6]) and can be proved as follows. Consider the map of global sections
which is surjective since is globally generated. This map induces a map of projective bundles
Let denote the locus in the target of points such that drops rank, and let . As has codimension in each and is surjective, has codimension in .
We prove (b) by adapting this argument. In this case, is not surjective, so the codimension of may drop if the image of is not transversal to . At each point , fixing bases, is a matrix of the form
in which are general if is general. Since , this matrix drops rank if and only if the columns are linearly dependent. If , the columns of are always linearly dependent, while if , there are two cases to consider in which the columns are linearly dependent:
-
(i)
The columns of are linearly dependent. This occurs in codimension in the space of such block matrices.
-
(ii)
The columns of are linearly independent. Then the linear dependence involves a column of , hence is in the span of the remaining columns, which occurs in codimension . (For such , intersects the image of transversally.)
Thus, has codimension in .
∎
Proof of Proposition 5.1.
For (a), as each for and is globally generated, Proposition 5.2 (a) implies that for general the locus where drops rank is either empty or has codimension , which proves and . Then follows from the fact that cannot have zero-dimensional torsion as it has a two-step resolution by locally free sheaves. For (iv), there is a commutative diagram
with exact rows, hence is surjective if and only if the induced map
is surjective. This is a map between a bundle of rank and a bundle of rank
hence by dualizing the same Bertini-type statement, a general such map is surjective on all fibers.
For (b), set
The complication is that for each , the global sections of the line bundle
vanish on , where is the index such that and is any line bundle in except for . Still, we can adapt the proof of the Bertini-type statement as follows.
Consider
which is not surjective on each , and the induced map
Let denote the locus in the target where the linear maps drop rank and . At points in the open complement of the exceptional locus , is surjective, hence the codimension of in is . On each , we apply Proposition 5.2 with
and the restrictions of the remaining line bundles. Assuming and using the fact that the global sections of each line bundle summand of lift to , we deduce that the codimension of in is
On , a similar argument with
shows that has codimension
in . Altogether, since and have codimension 1 in , we deduce that the general fiber of over is empty or has codimension at least
in . As this codimension is in the case and in the case assuming for all , we get the result for (b.ii) and (b.iii).
For (b.iv), we dualize and use a similar argument. ∎
5.2. Prioritary sheaves
We relate sheaves which admit general Gaeta resolutions to prioritary sheaves, which will facilitate the study of stability in § 6. We begin by reviewing the prioritary condition.
Definition 5.3.
Let be a smooth surface and be a divisor on it. A coherent sheaf on is -prioritary if it is torsion-free and . If is a Hirzebruch surface or its blowup, -prioritary sheaves are simply called prioritary sheaves.
We will need the following lemma ([CH21, Lemma 3.1]) comparing prioritary conditions with respect to different divisors.
Lemma 5.4.
Let be a smooth surface, and be two divisors such that . Then -prioritary sheaves are -prioritary.
Now let denote an admissible blowup of and be a fixed class of positive rank admitting Gaeta resolutions.
Proposition 5.5.
For a divisor , consider the locus
If , then the locus is empty. If , then the locus is empty if and otherwise has codimension in .
Proof.
We begin by proving the second statement. Let and take . We need to study . We twist the Gaeta resolution of by and then apply , obtaining the exact sequence
A calculation using the Gaeta resolution for shows that vanishes as, after tensoring by , the line bundles in degree 0 have no global sections and the line bundles in degree have no :
and, for ,
Similarly, for , we get since and . Thus, the map in the exact sequence reduces to the map obtained by applying to the map
from the Gaeta resolution. If this map is surjective, then must to be injective. The locus in of such that is not surjective has codimension , giving the desired estimate. If , then and and are empty since is admissible, from which it follows that .
The first statement follows from a similar argument with by checking that , , for , and all vanish. ∎
The case (where ) is in [Ped21, Proposition 2.20].
Proposition 5.6.
Fix exponents such that , the condition in Proposition 5.1(b.ii) holds, and . Then the open family of Gaeta resolutions whose cokernels are torsion-free and -prioritary is a complete family of -prioritary sheaves.
By Lemma 5.4, the same statement holds with replaced by any divisor , where is defined in (2.3). The inequality is not needed for the case .
Proof.
5.3. Weak Brill-Noether result
Let be or an admissible blowup of . Consider Gaeta resolutions on of the types in Example 4.3.
For integers and , set and assume the remaining exponents satisfy
| (5.2) |
On , we further assume
| (5.3) |
Consider a general map
By Proposition 5.1, is injective and the cokernel is torsion-free of rank . Furthermore, it has vanishing higher cohomology, and
Then, if is a zero-dimensional subscheme of length , and we have the following weak Brill-Noether result.
Proposition 5.7.
Proof.
As the vanishing of the cohomology of is an open condition on families of and of , it suffices to prove the claim for a single choice of . We construct as the direct sum of maps , each of which is a general map of the form
where the exponents, which depend on , are non-negative and satisfy
On , we need the additional condition that the exponent of is at least as large as the sum of the exponents of for such that , which can be ensured due to (5.3). Then, by Proposition 5.1, the cokernel of is of the form , where, for each , is a line bundle with vanishing higher cohomology, is a 0-dimensional subscheme, vanishing on imposes independent conditions on , and is -dimensional. Choose distinct points inductively so that each avoids the base loci of the linear systems of curves in that vanish on . Then has no cohomology. ∎
5.4. No sections vanishing on curves
Proposition 5.8.
Suppose is a general Gaeta resolution and is the cokernel.
-
(a)
On or , has no sections vanishing on curves.
-
(b)
On an admissible blowup of , assume
-
(i)
for all ,
-
(ii)
for all .
Then has no sections vanishing on curves.
-
(i)
Proof.
Let . It suffices to prove that for all minimal nonzero effective divisors . Twisting the Gaeta resolution by and taking cohomology, we get a long exact sequence
We need to show that the induced map is injective when is general.
(a) On , the only minimal effective divisor is the hyperplane class , and vanishes, so injectivity of is trivial. Similarly, on , the minimal effective divisors are and , and an easy calculation using § 2.2 shows that and both have vanishing .
(b) Now consider the sequences on the two-step blowup of . We argue by cases.
Case 1: is minimal nonzero effective not equal to any for or for . In this case, as above, we show that the domain of vanishes. Fix a curve in the linear equivalence class of . As is minimal effective, must be connected. As is ample on and is not contained in , there is a curve in the linear series that is connected, does not contain , and intersects . Then is a connected curve in the linear series , so by Lemma 5.9.
We use a similar argument to show the vanishing of for . Let be the index such that . By Lemma 2.2,
where is basepoint-free (Example 2.5) and contains connected curves that intersect , so the union of such a curve and is connected. Thus, for minimal effective not equal to any or , (and this vanishing holds for as well).
Case 2: for . Note that if is a line bundle with no cohomology whose restriction to is trivial, then taking cohomology of the short exact sequence
yields an isomorphism . Moreover, because and , we can view as the map obtained from
by restricting to and then taking the induced map on global sections. As the restriction to of each of these exceptional sheaves is trivial, is of the form
Thus, a necessary condition for to be injective is , or equivalently, as ,
To find additional sufficient conditions for to be injective, we observe that certain blocks of are injective when is general. Let denote the index such that . Then:
-
•
The block is injective for general if
because the linear series is basepoint-free. In fact, this inequality is necessary because is in the base locus of , , and for all such that and all , hence all other blocks involving these are 0.
-
•
For a similar reason, for each , is injective for general if
(though this inequality may not be necessary for to be injective).
As all blocks corresponding to are general if is general because the linear systems , , and for are all basepoint-free, these inequalities suffice to ensure that is injective when is general.
Case 3: for . Similar to the previous case, if is a line bundle on with no cohomology whose restriction to is trivial, then
while . Thus, can be viewed as the map obtained by restricting
| () | ||||
to and then taking the induced map on global sections. The restriction of each of the exceptional bundles in to is , except for and for , each of which has a two-dimensional space of global sections. Thus, the restriction of () to is
and hence is of the form
A necessary inequality for to be injective is , or equivalently,
As in the previous case, we obtain sufficient conditions for to be injective when is general by looking at various blocks.
-
•
The block is injective for general if
because the linear system is basepoint-free. As there are no nonzero maps , this condition is necessary for to be injective as all other blocks involving are 0.
-
•
For , the block is injective for general if
(but this condition may not be necessary).
The blocks involving are all general for general as the linear systems , , for are all basepoint-free and the curves in containing have no fixed tangent direction at , so the above inequalities are sufficient. ∎
Lemma 5.9.
Let be a nonzero effective divisor on a rational surface and denote its linear series. The following are equivalent:
-
(a)
contains a connected curve;
-
(b)
Every curve in is connected;
-
(c)
.
Proof.
Let be a curve in the linear equivalence class of and consider the corresponding short exact sequence
Taking cohomology and using , we see that if and only if is an isomorphism, which is true if and only if is one-dimensional, which holds exactly when is connected. (If is ample, also follows from Kodaira vanishing.) ∎
6. Gaeta resolutions and stability
For an admissible blowup, we study the connection between the existence of Gaeta resolutions and stability of a sheaf, which allows Gaeta resolutions to be applied in the study of moduli problems. First, we describe conditions ensuring that general stable sheaves in admit Gaeta resolutions (Proposition 1.2). Then, by imposing stronger conditions on and on the polarization , we show the locus of maps in the resolution space whose cokernels are unstable has codimension (Proposition 6.5), as well as the parallel statement in the moduli space that the locus of sheaves not admitting Gaeta resolutions has codimension (Theorem 1.3). In particular, the latter results imply that is non-empty and that general stable sheaves away from a locus of codimension satisfy various nice properties (Corollary 1.4).
We have discussed about prioritary conditions in the previous section. One of the motivations to consider the prioritary condition is the following statement, which is essentially in the proof of [Wal98, Theorem 1].
Lemma 6.1.
Over a smooth projective surface , if is a divisor and is a polarization such that , then any -semistable torsion-free sheaf is -prioritary.
Proof.
If is torsion-free and -semistable, then by Serre duality, by semistability and the fact that . ∎
As a warm-up, we prove Proposition 1.2. Recall that is an admissible blowup of , satisfies , and is a class of rank , Euler characteristic , and first Chern class satisfying the inequalities in Propositions 4.5(a) and 5.1(b.ii).
Proof of Proposition 1.2.
Proposition 6.2.
Let and be as in Proposition 1.2. Assume the class has fixed rank and first Chern class, and its discriminant is sufficiently large. Then there is a line bundle pulled back from such that for general semistable sheaves of class , admits a Gaeta resolution.
Since the space of Gaeta resolutions is rational, we immediately deduce the following special cases of [Bal87, Theorem 2.2].
Over Hirzebruch surfaces, similar results were proved in [CH20, Theorem 2.10, Proposition 4.4].
The rest of this section is dedicated to the proof of Theorem 1.3, which requires additional technical conditions on the exponents of the Gaeta resolutions and on the polarization. Before stating these conditions, we set some notation. For simplicity, we write the Gaeta resolutions of the form (b) on as
Let denote the locus of injective maps as in § 5. Let and denote the projections from to the first and second factors, respectively. Let denote the cokernel of the tautological map over :
| (6.1) |
Because it is the family of cokernels of injective maps, is flat over .
The technical conditions on the exponents are as follows. We assume
| (6.2) |
to ensure that the complement of in has codimension and that the cokernels are torsion-free away from a locus of codimension in (the proof is similar to the proof of Proposition 5.1, but we consider the image of the locus in Proposition 5.2 in ). We make the additional assumption
| (6.3) |
which ensures that the are -prioritary away from a locus of codimension in (Proposition 5.5), require all of the exponents to be strictly positive, and require the discriminant of the sheaves to be sufficiently large in the sense of (6.9).
To state the conditions on the ample divisor , which we assume is general, we write
for rational numbers 222As scaling does not affect stability, these weights could be taken as integers as well.. We assume that
| (6.4) |
namely that is a positive linear combination of all divisors in , where is defined in (2.3). Note that by Proposition 2.8, (6.4) implies that is ample. This condition implies that
as well as the condition that appears in Theorem 1.5. Moreover, we assume that no can be too close to in the sense that
| (6.5) |
Remark 6.4.
Our arguments can be extended to allow to include other divisors whose linear systems are basepoint-free with general member isomorphic to , for instance other divisors of the form for such that , which could expand the range of depending on the configuration of the blown-up points.
6.1. Locus of unstable sheaves in the space of Gaeta resolutions.
In this subsection we prove the following proposition.
Proposition 6.5.
As observed above, the complement of has codimension at least in , as does the locus of such that is not torsion-free. Moreover, by Proposition 5.5, the locus of such that is not -prioritary also has codimension in , so it suffices to show that the locus of unstable torsion-free sheaves that are -prioritary has codimension at least 2. We will do this by showing various Harder-Narasimhan strata have codimension .
For whose cokernel is -prioritary and unstable, let
be the Harder-Narasimhan filtration of with respect to Gieseker stability, let denote the grading and let , , and denote the rank, total slope, and discriminant of . Then and and for all .
Lemma 6.6.
The following subset of has codimension at least :
Proof.
All are basepoint-free, so Bertini’s Theorem implies that general divisors in the corresponding linear systems are nonsingular. Moreover, these general divisors are isomorphic to . Let be general in the linear system so that it avoids the singularities of . The restrictions of the cokernels in a neighborhood of , which are locally free on . As -prioritary implies -prioritary, is -prioritary, so the Kodaira-Spencer map
is surjective, according to [LP97, Corollary 15.4.4] or [CH20, Proposition 2.6]. The restrictions to provide a complete family of vector bundles. On the other hand, the inequality implies that over , . Thus, the subset has codimension at least 2, according to [LP97, Corollary 15.4.3]. ∎
The last step is to show that the locus of in satisfying the following two conditions has codimension :
-
(a)
is unstable and -prioritary;
-
(b)
For all , the inequality holds for all .
We will do this for each locally-closed stratum of this locus of fixed Harder-Narasimhan type, namely for an integer and polynomials , we let denote the locus of in such that (a) and (b) hold and the Harder-Narasimhan filtration of has length and the Hilbert polynomial of is . Our strategy for showing the codimension of in is is based on similar ideas for in [LP05, Chapter 15].
We begin with the following observation:
Lemma 6.7.
-
(i)
For , .
-
(ii)
Under the conditions (a) and (b) above, for all and .
Proof.
Part (i) follows from semistability.
For (ii), (b) implies that when , for all and . As is a positive linear combination of all divisors in , letting denote the minimum of the weights of and , can be written as plus a non-negative linear combination of divisors in . Thus, and this inequality is also true when since (6.4) implies . Thus, for all and . ∎
Let be the relative flag scheme of filtrations whose grading has Hilbert polynomial . Given and a point of the fiber over , there is an exact sequence
| (6.6) |
[LP97, Proposition 15.4.1] realizing the vertical tangent space as the group and the normal space of in at as the image of . Here the groups are defined with respect to the filtration of , and is the composite map
| (6.7) |
The Kodaira-Spencer map is surjective since parametrizes a complete family of prioritary sheaves. The idea is to show that is also surjective and that , which imply that has codimension in . For foundational material on the groups , see [DLP85].
There is a canonical exact sequence
showing that surjectivity of is guaranteed by the vanishing of . This group can be calculated using the spectral sequence
converging to . By Lemma 6.7 (b), vanishes when , hence , so is surjective.
To calculate , we use the spectral sequence
converging to . By Lemma 6.7, and the spectral sequence degenerates on the first page, yielding
Using (4.3), we thus calculate
| (6.8) |
where is the Hilbert polynomial of , as in § 4.3.
To finish the proof of the proposition, we will show that given the conditions for all and for all . As for each , we see that
which allows us to reduce to the case where the Harder-Narasimhan filtration has length 2. To see that the conditions (a) and (b) still hold, note that is a weighted average of the , hence we have and for all . As we know but no such inequality is guaranteed for the discriminant of , we need to apply the lemma below; if this does not hold, then a similar setup using and , which satisfies and , meets the conditions of the lemma. Thus, it suffices to prove the following:
Lemma 6.8.
Assume the condition (6.5) on . Suppose the class is the sum of classes and of positive rank with the property that for , that , that if , and that if . Then the condition
| (6.9) |
is sufficient to ensure that .
Proof.
Note that , , and . Using this, in the case , we write
As , it suffices to find an upper bound for . Setting and writing
this equals
The lemma below shows that given (6.5), an upper bound is
where since if then ample ensures that . This yields the bound
and, as , we can guarantee that the right side is by assuming satisfies (6.9).
The argument in the case is similar. ∎
Before stating and proving the lemma below, we introduce some useful notation. Set as above and consider the change of variables
The conditions can be written as , , , and the condition is . Moreover, thinking of as fixed,
is a quadratic function with maximum value
| (6.10) |
In particular, assuming and the constraints , , and , an upper bound is obtained by taking , as the constraints imply and this upper bound for yields a smaller value since .
Lemma 6.9.
Assume satisfies (6.5). Given the constraints , , , and , we have
Proof.
For , the result follows by bounding the first term on the left side by setting in (6.10), as well as the fact that as the maximum value of is .
Now suppose that . Then is also negative. By (6.10),
Using the inequality , where is the subset of indices for which is negative, as well as the Cauchy-Schwarz inequality to deduce , the two terms involving are bounded above by
| (6.11) |
Bounding each where by its maximum value , we see that is bounded above by
| (6.12) |
As for , the conditions on in (6.5) imply that the sum of (6.11) and (6.12) is bounded by , which completes the proof. ∎
By a similar argument, we can show the following statement whose proof will be sketched. The statement is similar to [LP97, Corollary 15.4.6], but the proof is more involved since the Picard group is not as simple as .
Lemma 6.10.
Suppose is an ample divisor satisfying (6.4) and (6.5). Consider a complete family of of semistable sheaves of a fixed class on , parametrized by a smooth algebraic variety . When the discriminant is large, say as in (6.9), the set of points such that is strictly semistable forms a closed subset of codimension .
Proof.
For such that is strictly semistable, consider one of its Jordan-Hölder filtrations and let be the corresponding sub-quotients and . The proof of Lemma 6.6 also shows that the set
has codimension .
We next consider such that (a) is strictly semistable and (b) , . Note that , for all . Thus, , which is calculated with respect to the fixed Jordan-Hölder filtration. Consider the relative flag scheme of filtrations of the same type as the Jordan-Hölder-filtration. We have
Moreover, the vanishing of implies . The codimension of set of satifying conditions (a) and (b) is bounded below by . ∎
Using this lemma, we can immediately strengthen Proposition 6.5 replacing “semistable” by “stable”.
6.2. Locus of semistable sheaves not admitting Gaeta resolutions
This subsection is devoted to the proof of Theorem 1.3.
First, the subset is indeed closed. As in the construction of the moduli space using geometric invariant theory [MFK94], let be the Quot scheme such that is a good quotient of the semistable locus with respect to the action by . According to Proposition 4.4 and upper semicontinuity, the subset of quotient sheaves that do not admit Gaeta resolutions is closed and invariant under the action of . Under the good quotient map, the image of this subset is closed and is exactly . 333If the Jordan-Hölder grading of a semistable sheaf admits a Gaeta resolution, the sheaf does as well.
Let
which acts on ([Ped21, § 4.3]). Let . There is an induced action of on . The universal cokernel (6.1) induces a map
| (6.13) |
which will be shown to be an isomorphism.
Since is an open subset in a vector space and its complement has codimension , is isomorphic to the character group and . The character groups can be explicitly described as follows:
and under this isomorphism,
| (6.14) |
Proposition 6.11.
The map in (6.13) is an isomorphism and it induces an isomorphism .
Proof.
We calculate using the isomorphism . In , for , let be the class of the dual bundle. Then the form a basis of . Let be a complex vector space of dimension for . We can calculate using the -equivariant short exact sequence (6.1):
The map takes the following matrix form:
Since the matrix is lower triangular with on the diagonal, is an isomorphism. For , if and only if , if and only if . ∎
Let denote the subset of cokernels which are semistable. According to Proposition 6.5, has codimension in . Let denote the subset of cokernels which are stable. The coarse moduli property provides a map
which factors through . According to Lemma 6.10, the restriction map induces isomorphisms and . The Donaldson map is an isomorphism according to Theorem 1.5, which will be proved at the end of the subsection. By the functoriality of the determinant line bundle construction, we have the following commutative diagram:
It is clear from the diagram that the restriction map is also injective. Since the polarization is general, is locally factorial ([Yos96, Corollary 3.4]). Therefore, has codimension in . We have proven Theorem 1.3.
Proof of Corollary 1.4.
We are left to provide
7. Strange duality
In this section, we review Le Potier’s strange duality conjecture for rational surfaces over and use our study of Gaeta resolutions to prove Theorem 1.6, which states that the strange morphism is injective in various cases on and on an admissible two-step blowup of . The argument is similar to what was shown on in [BGJ16]. We assume Theorem 1.7, which will be proved in § 8.
7.1. Strange morphism
Let be a smooth projective rational surface over and be an ample divisor. Let and denote two classes in the Grothendieck group . On , there is a pairing given by .
Let and be the moduli spaces of -semistable sheaves of class and respectively. For the moment, suppose there are no strictly semistable sheaves, namely , and there is a universal family over . Let be a sheaf of class , then we have a determinant line bundle on ,
Here, and are projections from to the first and second factors respectively. The isomorphism class of does not depend on but only on its class in , so we are justified in using the subscript . Two universal families may differ by a line bundle pulled back from , but if we assume , then they will provide isomorphic determinant line bundles on the moduli space. Thus, from now on, we assume , so that is independent of the choice of .
Even if there does not exist a universal family, we can still define by carrying out the construction on the Quot scheme coming from the GIT construction where there is a universal family, and then showing that it descends to . If there are strictly semistable sheaves of class , we need to further require for ; see [LP92, (2.9)]. These conditions will be satisfied in our setting. Similarly, we can construct a determinant line bundle
Orthogonal classes and are candidates for strange duality if the moduli spaces and are non-empty, and if the following conditions on pairs are satisfied:
-
(a)
and for all away from a codimension subset in , and
-
(b)
for some .
Under these conditions, there is a line bundle
with a canonical section whose zero locus is given by
(see [LP05, Proposition 9]). The see-saw theorem implies that
see [LP05, Lemme 8]. Then, using the Künneth formula, the canonical section of induces a linear map
that is well-defined up to a non-zero scalar. Following Le Potier, we call this the strange morphism.
Conjecture 7.1 (Le Potier).
If is nonzero, then it is an isomorphism.
We focus on the case when is or , an admissible blowup of . For the former, we take to be the hyperplane class, while for the latter, we assume is general and satisfies (6.4, 6.5). Let
be the numerical class of an ideal sheaf of points, so that is the Hilbert scheme of points on . Clearly . Every numerical class orthogonal to has the form
We assume and are fixed, that , and that
The condition on ensures that general sheaves in are locally free, and it is no restriction in the study of strange duality as strange duality is known over all surfaces in the case when and both have rank one. Assumptions about the positivity of are required to apply many results in this paper, and we assume here that is sufficiently positive such that the positivity assumptions of every result we need are met. A precise statement of the positivity conditions we impose on can be found in the appendix.
In particular, since is sufficiently positive, admits Gaeta resolutions by Proposition 4.5(a), the discriminant condition (6.9) holds, and is non-empty by Proposition 6.5, so general sheaves in admit Gaeta resolutions by Proposition 1.2 or Theorem 1.3. In this situation, conditions (a) and (b) on p. 7.1 are satisfied. Let and be such that , which holds away from codimension . Under this condition for . Furthermore,
vanishes by semistability. Condition (b) also holds by Proposition 5.7. Thus, and are candidates for strange duality, and we will study the strange morphism
We begin with the determinant line bundle on the Hilbert scheme of points.
7.2. Determinant line bundles on the Hilbert scheme of points
We first review some general results about line bundles on the Hilbert scheme of points on surfaces. The Hilbert scheme of points, , is a resolution of singularities of the symmetric product . The resolution, which we denote by , is called the Hilbert-Chow morphism. Given a line bundle on , let be on the -fold product . There is an -action on such that is -equivariant. The line bundle descends onto , giving a line bundle . We denote its pullback to as
Via this construction, we can view as a subgroup of , by sending to . Fogarty [Fog73] showed that under this inclusion
Here, is the exceptional divisor of the Hilbert-Chow morphism, which parametrizes non-reduced subschemes. Furthermore, if is ample, then is nef, and the canonical divisor on is .
The determinant line bundle on induced by is
By the Kodaira vanishing theorem, if is ample on , then has no higher cohomology. Results of Beltrametti, Sommese, Catanese, and Göttsche [BS91, CG90] show that if is a line bundle on , then is nef if is ()-very ample and very ample if is -very ample. Thus, we deduce the following:
Lemma 7.2.
Suppose is sufficiently positive, for instance is the tensor product of an -very ample line bundle and (-1)-very ample line bundles. Then has vanishing higher cohomology.
Thus, the vanishing of the higher cohomology of follows from the assumption that is sufficiently positive. A precise statement of a sufficient condition on is (A.5) in the appendix.
7.3. Injectivity of the strange morphism
To prove Theorem 1.6, we will make use of a Quot scheme argument. As above, let and be orthogonal classes, where and are fixed, so sheaves of class are expected to be locally free, and we assume is sufficiently positive in the sense explained in the appendix. Consider the class
The first part of the argument is to show that if is a vector bundle of class that admits a general Gaeta resolution, then is finite and reduced, which will be proved in § 8. The strategy is to show that the relative Quot scheme over the space of Gaeta resolutions has relative dimension 0. This is known for by [BGJ16], so we write the argument for , an admissible blowup of , though it works for as well. Since admits Gaeta resolutions, an easy calculation using Proposition 4.5(a) shows that also admits Gaeta resolutions, with the same exponents and and with each larger by and smaller by .
The starting point is the following result.
Lemma 7.3.
There is a vector bundle of class such that has a Gaeta resolution and contains an isolated point.
Proof.
Choose a vector bundle of class that admits a Gaeta resolution and a general ideal sheaf of class such that , which is possible by Proposition 5.7. Since and is general, we can choose a quotient such that is surjective. Let denote the kernel of . Since is sufficiently positive, is large (for instance, (A.1) suffices). Then a general extension of by is locally free (see the proof of [BGJ16, Lemma 5.9]) and there are short exact sequences
which together give a long exact sequence whose dual
is a point of . As the tangent space at this point is , this is an isolated point of . Thus, all that remains is to show that admits a Gaeta resolution.
We do this in two steps using Proposition 4.4, by first showing that admits a Gaeta resolution. First, we see that for using the corresponding vanishings for and the fact that is surjective. Similarly, for all , as the induced map is still surjective. Finally, the vanishing of for and one of the divisors appearing in Proposition 4.4 (b.ii) follows from the same vanishing for and the vanishing of the higher cohomology of .
Second, since for , . Using the vanishings for , we obtain that for and that for all . For each divisor appearing in Proposition 4.4 (b.ii), for all , so and this vanishes for . Therefore, also admits a Gaeta resolution. ∎
The lemma establishes that the relative Quot scheme contains a point with vanishing relative tangent space, which can be deformed to give an open set with this property. The main technical task is to prove that the relative Quot scheme cannot have any other components of the same dimension, which is carried out in the next section for , and which was done for in [BGJ16]. The proof requires strong positivity assumptions on .
The second part of the argument is to count the points of and compare the result to . In previous work [GL22], we showed that if the Quot scheme is finite and reduced, then the number of points is
where is the tautological vector bundle defined as in (1.2), whose rank is and whose fiber at a point is . By the following theorem, this top Chern class calculates the Euler characteristic of the determinant line bundle:
Theorem 7.4.
Let be a smooth projective surface with . Then
Over Enriques surfaces, this statement was proved by Marian-Oprea-Pandharipande [MOP22, Proposition 3.2] . In general, it was conjectured by Johnson and shown to be equivalent to another conjecture ([Joh18, Conjecture 1.3, Theorem 4.1]). The second conjecture was proven by the works of Marian-Opera-Pandharipande [MOP19, MOP22] and Göttsche-Mellit [GM22] combined. See for example [GM22, Corollary 1.2].
Now, using the fact that sufficiently positive ensures that has vanishing higher cohomology, we can deduce the injectivity of the strange morphism.
Proof of Theorem 1.6.
Above, we checked the conditions for the strange morphism to be well-defined. Let be a vector bundle of class that admits a general Gaeta resolution. By Theorem 1.7, is finite and reduced and the points of are short exact sequences
for sheaves of class that are locally free and semistable. Since the Quot scheme is finite and reduced, the tangent space is 0, while if then follows from semistability, as otherwise the induced map would be zero, hence would factor through , yielding an equality as subsheaves of , identifying the two points of the Quot scheme. It follows that the hyperplanes in determined by the points map under to linearly independent lines . Thus, the rank of is at least
namely is injective. ∎
8. Finite Quot schemes
This section is devoted to the proof of Theorem 1.7. We will show that for an admissible blowup of over , under appropriate conditions, Quot schemes are indeed finite and reduced. This is an extension of the corresponding result in [BGJ16] for . We then apply our previous work [GL22] to enumerate the finite Quot scheme.
We first sketch the ideas. For a vector bundle admitting a Gaeta resolution, instead of directly studying ideal sheaf quotients , we replace by the dual of the Gaeta resolution of and by the canonical map . Namely, we consider commutative diagrams of the form (8.2), which we can view as a family over an open subset of the resolution space (8.3). We prove in § 8.2 that the main component of the family has the same dimension as , hence we deduce that an open subscheme of the main component is isomorphic to a relative Quot scheme over an open subset of . The fibers of the relative Quot scheme have dimension 0, and generically the relative Zariski tangent space has dimension . We conclude that when is general with appropriate numerical constraints, the Quot scheme is finite and reduced.
8.1. Finite Quot schemes
Over an admissible blowup of , as in the previous section, let be the class of an ideal sheaf of points for some fixed and be an orthogonal class for some fixed and satisfying the positivity conditions in the appendix. In particular, the need for the positivity condition (A.2) will be seen in the proof of Proposition 8.1. Then admits Gaeta resolutions, with exponents denoted as in (b). Since is sufficiently positive, all these exponents are large except for .
8.1.1. Morphisms vs. chain maps
Consider a vector bundle of class that admits a general Gaeta resolution. Then is locally free, so its dual has a resolution
| (8.1) |
where
We wish to study quotients of of class , which are expected to be ideal sheaves of points. Instead of considering maps , we replace them by chain maps of complexes using (8.1) and the short exact sequence . Namely, we consider commutative diagrams
| (8.2) |
where is surjective and has length . Letting denote the open subset of the space of Gaeta resolutions for which the cokernel is locally free, dualization yields an inclusion
| (8.3) |
and a universal sequence
| (8.4) |
where and denote the projections from to the factors.
8.1.2. Relative Quot schemes
On the other hand, we consider the relative Quot scheme . Letting
denote the subset where the relative Zariski tangent space has dimension , is open (by upper semicontinuity), non-empty (by Lemma 7.3), and smooth.
We claim that quotients in can only be ideal sheaves of points. A sheaf of class could take the following forms: is isomorphic to an ideal sheaf , contains dimension torsion, or contains dimension torsion. The second case cannot occur as it violates our assumption on the relative tangent space. In the third case, let denote the torsion subsheaf of , so that is torsion free. Then the quotient provides a nonzero morphism where is a non-trivial effective curve given by . But this cannot happen under the conditions in Proposition 5.8, which are guaranteed when is sufficiently positive, see the appendix.
8.1.3. Finite Quot schemes
We next define a regular map
over by associating to each quotient a diagram of the form (8.2). It is enough to define it on the level of functor of points. Furthermore, it is enough to consider morphisms from an affine scheme. Let be the spectrum of some -algebra. A morphism is equivalent to a family of quotients over . Here, and is the ideal sheaf of a subscheme . We denote the projection maps on by and . Applying the functor to the pull-back of the sequence (8.4) via , we have the following exact sequence
According to our choice of , the first term and the last term are zero. Therefore, the family can be completed to a commutative diagram
Then the square on the right provides a morphism . Clearly, the square uniquely determines the left-most vertical morphism. We have obtained an injective morphism whose image is contained in the unique component with the maximal dimension.
The complement of the image has dimension . Then , the complement of the image of , is non-empty and open in . For each sheaf parametrized by , has an isolated quotient. By the definition of , each fiber of over , which parametrizes all non-zero maps of the form , is entirely contained in the image of . In particular, the maps have to be surjective. Therefore, induces an isomorphism .
We have thus proved Theorem 1.7(a) for every sheaf parametrized by .
8.1.4. Genericity of kernels and cokernels
We have seen that Quot schemes for in the nonempty open set are finite and reduced and that the quotient sheaves are all ideal sheaves . Moreover, in the proof of Lemma 7.3, the isolated point of has the property that is general, hence we can shrink further if necessary to ensure all the ideal sheaves arising as quotients are general. Similarly, by Proposition 1.2, general sheaves in admit Gaeta resolutions, and since is nonempty, we may choose in the proof of Lemma 7.3 to be semistable, hence the isolated quotient has kernel , which is also semistable. As semistability is an open condition in families, and as the relative Quot scheme can be viewed as a family of sheaves with invariants , there is a non-empty open set in the relative Quot scheme of quotients for which the kernel is semistable. Shrinking if necessary, we may assume that all kernels of the finite Quot schemes are semistable. This proves Theorem 1.7(b).
8.1.5. Length of finite Quot schemes
According to [GL22, Proposition 1.2], when the Quot scheme is finite and reduced, it is isomorphic to the moduli space of limit stable pairs, which consist of torsion free sheaves of class together with nonzero morphisms . Then the virtual fundamental class of agrees with the fundamental class, and its degree is as stated in Theorem 1.7(c), by [GL22, Theorem 1.1]. We refer the reader to [Lin18] for foundational material on the moduli space of limit stable pairs.
8.2. Dimension count
We prove that by counting diagrams of the form (8.2) for fixed (nonzero) and (surjective). For fixed and , the locus of is in the exact sequence
This locus, if it is non-empty, is an affine space in that is isomorphic to . Since is fixed, we may assume it is nonzero only on the first -summand of . For the locus to be non-empty, we need to be in the image of the middle map of
Thus, for the purpose of a dimension count we may assume that the image of contains .
The idea is to control the dimension of . In particular, if , which is true for general as we show below, then the dimension of is . Adding to this the dimension of the parameter spaces for and , we get
| (8.5) |
Using the fact that , , , and , we see that (8.5) equals , which equals . Thus, the main technical difficulty is to control the dimension of in the case when is not general.
Proposition 8.1.
Let
and assume
Then the space of commutative diagrams has the expected dimension, namely . Furthermore, there is only one component with this dimension, while other components have strictly smaller dimension.
Proof.
Choose an open set containing and trivializations on of and for . The components of the map generate subspaces of and , which we identify with subspaces of using the trivializations on . For or , we calculate bounds on by considering the dimension of the image of
Picking a trivialization of on (shrinking if necessary), this image is the image of the multiplication map
defined by . As is basepoint-free, there is a section that is nowhere zero on the support of , hence multiplying by is injective, which implies that the image of has dimension . Similarly, as is basepoint-free when (Example 2.6), we can choose that is nowhere zero on the support of , so the image of has dimension .
Define the stratum
by the conditions that , for , and the sum has dimension for . By the above, the stratum is empty unless when . For in this stratum, the image of has dimension and the image of has dimension . Thus, and , which yields the estimate
For strata with , then also , so and the dimension of the space of commutative diagrams for in the union of such strata is equal to the expected dimension. Thus, it suffices to show that for strata with , the codimension of the stratum in is at least as large as .
When , we will compare this estimate for to the codimension of the stratum in . A map in this stratum can be obtained by first choosing a subspace of dimension , then choosing subspaces and of dimension and , and finally using to define the map . Comparing a dimension count based on this description to , we see that the stratum has codimension
The difference between the upper bound for and this codimension is
Since , we obtain
As , we may assume that the image of for is exactly and that the image of has dimension exactly , as otherwise we can improve the upper bound for by subtracting , leading to a new that is non-positive. Similarly, as and are each , is non-positive unless . Thus, all that remains is the case when for all such that and the image of each has dimension . In other words, up to multiplication by units, is stable under multiplication by rational functions in for and for .
This final case cannot happen. Since is in the image of , must contain an element that restricts to a unit at each point of the support of . But the linear systems , , for , and for collectively separate points and tangents on (compare with the proof of Proposition 2.8), hence multiplying by the functions generates all of , so the only way can be stable under multiplication by all is if . This cannot happen as .
We finish the proof by arguing that there is a unique component with the maximal dimension. As shown above, over a general point , the fiber of is an affine space in isomorphic to . The fiber of any component with maximal dimension must contain a non-empty open set in this affine space for dimension reasons, but since any two non-empty open sets in an affine space must intersect, there can only be one such component. ∎
Appendix: Positivity Conditions
We rephrase the conditions on the exponents of Gaeta resolutions in Proposition 4.5(a) to explain the connection with the positivity of the first Chern class. We then summarize the inequalities required for the proof of Theorems 1.6 and 1.7.
Given the numerical class
on an admissible blowup of , the Euler characteristics in Proposition 4.5(a) can be written more explicitly as
Thus, rephrasing Proposition 4.5(a) in the case when and for all , we can say that admits Gaeta resolutions if and only if are all and
| (A.1) |
Now, as in § 7, consider the orthogonal classes and
where and are fixed, and set . The assumption that is sufficiently positive in § 7 includes three conditions. Let
The first set of conditions used in the proof of Theorem 1.6 is:
| (A.2) | ||||
These conditions ensure that and admit Gaeta resolutions (Proposition 4.5(a)), general Gaeta resolutions for and are locally free (Proposition 5.1), general Gaeta resolutions for have no sections vanishing on curves (Proposition 5.8(b)), Weak Brill-Noether holds (Proposition 5.7), and (Proposition 8.1).
The inequalities (A.2) imply that is -very ample, as can be expressed as a tensor product of very ample line bundles by Proposition 2.8 and -very ampleness is additive under tensor products ([HTT05]).
The second condition is the discriminant condition (6.9), which in this context is
| (A.3) |
The last condition is that
| (A.4) |
which by Proposition 2.8 and Lemma 7.2 can be ensured by the following inequalities:
| (A.5) | ||||
The sufficiency of these conditions follows from computing and observing that it can be decomposed as the tensor product of an -very ample line bundle and -very ample line bundles. For not too large relative to and , (A.5) is implied by (A.2).
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