Moduli spaces of semistable sheaves on singular genus 1 curves
We find some equivalences of the derived category of coherent sheaves on a Gorenstein genus one curve that preserve the (semi)-stability of pure dimensional sheaves. Using them we establish new identifications between certain Simpson moduli spaces of semistable sheaves on the curve. For rank zero, the moduli spaces are symmetric powers of the curve whilst for a fixed positive rank there are only a finite number of non-isomorphic spaces. We prove similar results for the relative semistable moduli spaces on an arbitrary genus one fibration with no conditions either on the base or on the total space. For a cycle of projective lines, we show that the unique degree 0 stable sheaves are the line bundles having degree 0 on every irreducible component and the sheaves supported on one irreducible component. We also prove that the connected component of the moduli space that contains vector bundles of rank is isomorphic to the -th symmetric product of the rational curve with one node.
Key words and phrases:Geometric integral functors, Fourier-Mukai, Cohen-Macaulay, fully faithful, elliptic fibration, equivalence of categories, moduli, singular curves
2000 Mathematics Subject Classification:Primary: 18E30; Secondary: 14F05, 14J27, 14E30, 13D22, 14M05
Elliptic fibrations have been used in string theory, notably in connection with mirror symmetry on Calabi-Yau manifolds and D-branes. The study of relative moduli spaces of semistable sheaves on elliptic fibrations, aside from its mathematical importance, provides a geometric background to string theory. In the case of integral elliptic fibrations, a complete description is already known and among the papers considering the problem we can cite [BBHM98, Bri98, BrM02, HMP02]. A study of these relative spaces for a more general class of genus one fibrations (for instance, with non-irreducible fibers and even singular total spaces) turns out to be an interesting problem.
On the one hand, for sheaves of rank 1 a fairly complete study of a class of these moduli spaces (compactified relative Jacobians), including those associated to relatively minimal elliptic surfaces, was carried out by one of the authors in [LM05, LM05a] (see also [Ca00, Ca2]). On the other hand, nowadays it is well understood the efficient key idea of the “spectral cover construction” discovered for the first time by Friedman-Morgan-Witten in [FMW99] and widely used later by many authors. The method shows how useful is the theory of integral functors and Fourier-Mukai transforms in the problem. The study developed by two of the authors in [HLS07] and [HLS08] on relative integral functors for singular fibrations gives a new insight in this direction.
From the results in that paper one gets new information about moduli spaces of relative semistable sheaves of higher rank for a genus one fibration , that is, a projective Gorenstein morphism whose fibers are curves of arithmetic genus one and trivial dualizing sheaf but without further assumptions on or . The fiber of the relative moduli space over a point is just the absolute moduli space of semistable sheaves on , so that in order to start with the relative problem one has to know in advance the structure of the absolute moduli spaces for the possible degenerations of an elliptic curve. There are some cases where the structure of the singular fibers is known. For smooth elliptic surfaces over the complex numbers, the classification was given by Kodaira [kod] and for smooth elliptic threefolds over a base field of characteristic different from 2 and 3, they were classified by Miranda [Mir83]. In both cases, the possible singular fibers are plane curves of the same type, the so-called Kodaira fibers. Nevertheless, in a genus one fibration non-plane curves can appear as degenerated fibers. So that our genus one fibrations may have singular fibers other than the Kodaira fibers. The study of the moduli spaces of vector bundles on smooth elliptic curves dates back to Atiyah [At57] and Tu [Tu94a], who proved that for an elliptic curve there is an isomorphism , where , between the moduli space of semistable sheaves of rank and degree and the symmetric product of the curve. A very simple way to prove this isomorphism is by using Fourier-Mukai transforms (cf. [Pol03, HePl05]). This method has been generalized to irreducible elliptic curves (i.e., rational curves with a simple node or cusp) in [BBH08, Chapter 6]) obtaining that , where also in this case. In the case of singular curves, the moduli spaces of semistable torsion free sheaves were first constructed and studied by Seshadri [Ses82]; his construction can now be seen as a particular case of the general construction of the moduli spaces of semistable pure sheaves due to Simpson [Simp96a]. The properties of these moduli spaces and their degeneration properties have been studied by many authors (see, for instance, [Se00, NaSe97, NaSe99, Ca00, Ca2, LM05, LM05a]).
The paper is divided in two parts. In the first part, we consider an arbitrary Gorenstein genus one curve with trivial dualizing sheaf. The group of all integral functors that are exact autoequivalences of is still unknown and a criterion characterizing those Fourier- Mukai transforms that preserve semistability for a non-irreducible curve of arithmetic genus 1 seems to be a difficult problem. Here we find some equivalences of its derived category of coherent sheaves that preserve the (semi)-stability of pure dimensional sheaves. One is given by the ideal of the diagonal and the other is provided by twisting by an ample line bundle (see Theorem LABEL:t:preservation). Our proof follows the ideas in [Bri98, Pol03] where the result was proved for a smooth elliptic curve and in [BBH08] for an irreducible singular curve of arithmetic genus one. The results of this section allow to ensure that for rank zero, the moduli spaces are the symmetric powers of the curve whilst for a fixed positive rank there are only a finite number of non-isomorphic moduli spaces (see Corollary LABEL:c:isom). Unlike the case of a smooth curve, moduli spaces of semistable sheaves on a curve with many irreducible components are not normal (even in the case where the rank is 1). Its structure depends very strongly on the particular configuration of every single curve. The difficulty in determining the stability conditions for a sheaf in this case points out the relevance of the identifications of Corollary LABEL:c:isom. In fact, for a curve with two irreducible components endowed with a polarization of minimal degree, they reduce the study either to the case of rank 0 or degree 0. Coming back to the relative case, the section finishes with Corollary LABEL:c:isom2 which establishes new identifications between certain relative Simpson moduli spaces of (semi)stable sheaves for a genus one fibration.
In the second part, we focus our study in a curve of type and in the case of degree 0 which is particularly interesting as in this case semistability does not depend on the polarization. Proposition LABEL:p:grupoK computes the Grothendiek group of coherent sheaves for any reduced connected and projective curve whose irreducible components are isomorphic to . The discrete invariants corresponding to the Grothendieck group behave well with respect to Fourier-Mukai transforms and are important tools for the analysis of the moduli spaces. Although a description of all torsion-free sheaves on a cycle of projective lines is known, as we mentioned above, it is by no means a trivial problem to find out which of them are semistable. For instance, contrary to what happens for an elliptic curve, semistability is not guaranteed by the simplicity of the sheaf. For , that is, a rational curve with one node, this was done in [BuKr04] for the degree zero case and in [BuKr05] otherwise. Using the description of indecomposable torsion-free sheaves on given in [BBDG07] and the study of semistable torsion-free sheaves on and on tree-like curves of [LM05], Theorem LABEL:t:stables proves that a degree 0 stable sheaf is either a line bundle having degree 0 on every irreducible component of or for some irreducible component . Then Corollary LABEL:c:JHfactors gives the possible Jordan-Holder factors of any degree 0 semistable sheaf. In the integral case, if the sheaf is indecomposable all Jordan-Hölder factors are isomorphic to each other. This is no longer the case for cycles of projective lines. Proposition LABEL:p:graded computes the graded object of any indecomposable semistable sheaf of degree 0. The structure of the connected component of the moduli space that contains vector bundles of rank is given in Theorem LABEL:t:sym. Namely, it is isomorphic to the -th symmetric product of the rational curve with one node. Having studied the case of degree zero, the results of the first part of the paper allow to cover other cases (see Remark LABEL:r:othercases). In particular, the connected component of the moduli space that contains vector bundles of rank and degree , where is the degree of the polarization, is also isomorphic to .
In this paper, scheme means separated scheme of finite type over an algebraically closed field of characteristic zero. By a Gorenstein or a Cohen-Macaulay morphism, we understand a flat morphism of schemes whose fibers are respectively Gorenstein or Cohen-Macaulay. For any scheme we denote by the derived category of complexes of -modules with quasi-coherent cohomology sheaves. This is the essential image of the derived category of quasi-coherent sheaves in the derived category of all -modules [BoNee93, Corollary 5.5]. Analogously , and denote the derived categories of complexes which are respectively bounded below, bounded above and bounded on both sides, and have quasi-coherent cohomology sheaves. The subscript will refer to the corresponding subcategories of complexes with coherent cohomology sheaves. By a point we always mean a closed point. As it is usual, if is a point, denotes the skyscraper sheaf of length 1 at , that is, the structure sheaf of as a closed subscheme of , while the stalk of at is denoted .
We thank I. Burban for pointing out a mistake in the first version of this paper and for showing us the example of a simple not WIT sheaf described in Remark LABEL:rem:simplenotwit. We also thank C.S. Seshadri for drawing to our attention Strickland’s result [Str82] which implies Lemma LABEL:lem:reduced, and U.N. Bhosle for pointing out some inaccuracies and mistakes. The second and the third author would like to thank respectively the Warwick Mathematics Institute and the Mathematical Institute of Oxford for hospitality and very stimulating atmosphere whilst this paper was written.
1. Fourier-Mukai transforms preserving stability
1.1. A non-trivial Fourier-Mukai transform on genus one curves
Let and be proper schemes. We denote the two projections of the direct product to and by and .
Let be an object in . The integral functor of kernel is the functor defined as
and it maps to .
In order to determine whether an integral functor maps bounded complexes to bounded complexes, the following notion was introduced in [HLS07].
Let be a morphism of schemes. An object in is said to be of finite homological dimension over if is bounded for any in .
The proof of the following lemma can also be found in [HLS08, Proposition 2.7].
Assume that is a projective scheme and let be an object in . The functor maps to if and only if has finite homological dimension over .
Let us suppose that is a projective Gorenstein curve with arithmetic genus such that its dualizing sheaf is trivial. This includes all the so-called Kodaira fibers, that is, all singular fibers of a smooth elliptic surface over the complex numbers (classified by Kodaira in [kod]) and of a smooth elliptic threefold over a base field of characteristic different from 2 and 3 (classified by Miranda in [Mir83]). In these two cases, all fibers are plane curves. Here, we do not need to assume that our curve is a plane curve. Notice also that an irreducible curve of arithmetic genus one has always trivial dualizing sheaf, but this is no longer true for reducible curves. Therefore in [HLS07] a genus one fibration is defined as a projective Gorenstein morphism whose fibers have arithmetic genus one and trivial dualizing sheaf.
Using the theory of spherical objects by Seidel and Thomas [SeTh01], we have the following
Let be a projective Gorenstein curve with arithmetic genus 1 and trivial dualizing sheaf. Let be the ideal sheaf of the diagonal immersion . One has:
The ideal sheaf is an object in of finite homological dimension over both factors.
The functor is an equivalence of categories.
The integral functor , where is the dual sheaf, is a shift of the quasi-inverse of with and .
(1) Denote by with the two projections. By the symmetry, to see that is of finite homological dimension over both factors it is enough to prove it over the first one. Using the exact sequence
it suffices to see that has finite homological dimension over the first factor. We have then to prove that for any bounded complex on , the complex is also a bounded complex and this follows from the projection formula for .
(2) Since is a projective Gorenstein curve of genus one and trivial dualizing sheaf, is a spherical object of . By [SeTh01] the twisted functor along the object , is an equivalence of categories. Since , the statement follows.
(3) By [HLS08, Proposition 2.9], the functor is the right adjoint to where is the dual in the derived category, then it is enough to prove that is isomorphic to , the ordinary dual. Indeed, one has to check that for . Let us consider the exact sequence
Taking local homomorphisms in , we get an exact sequence
which proves our claim because is Gorenstein. ∎
We shall use the following notation: for any integral functor , denotes the -th cohomology sheaf of , unless confusion can arise. Remember that a sheaf on is said to be WIT- if . In this case, we denote the unique non-zero cohomology sheaf by .
Note that in our particular situation , since is flat over the first factor and the fibers of are of dimension one, for any sheaf on one has unless .
We now adapt to our case some well-known properties about WIT sheaves.
The following results hold:
There exists a Mukai spectral sequence
Let be a WIT- sheaf on . Then is a WIT- sheaf on and
For every sheaf on , the sheaf is WIT- while the sheaf is WIT-.
There exists a short exact sequence
(1) and (2) follow from [BBH08, Eq. 2.35 and Prop. 2.34]. (3) is a direct consequence of (1) and (4) is the exact sequence of lower terms of the Mukai spectral sequence. ∎
1.2. Preservation of the absolute stability for some equivalences
1.2.1. Pure sheaves and Simpson stability
A notion of stability and semistability for pure sheaves on a projective scheme with respect to an ample divisor was given by Simpson in [Simp96a]. He also proved the existence of the corresponding coarse moduli spaces.
Let be a projective scheme of dimension over an algebraically closed field of characteristic zero and fix a polarization, that is, an ample divisor on . For any coherent sheaf on , denote .
The Hilbert polynomial of with respect to is defined to be the unique polynomial given by
This polynomial has the form
where and are integer numbers and its degree is equal to the dimension of the support of .
A coherent sheaf is pure of dimension if the support of has dimension and the support of any nonzero subsheaf has dimension as well.
When is integral, pure sheaves of dimension are precisely torsion-free sheaves. We can then adopt the following definition.
A coherent sheaf on is torsion-free if it is pure of dimension and it is a torsion sheaf if the dimension of its support is
When is a projective curve with a fixed polarization , the Hilbert polynomial of a coherent sheaf on is then
a polynomial with integer coefficients and at most of degree one. It is constant precisely for torsion sheaves.
The (Simpson) slope of is defined as
This is a rational number if and it is equal to infinity for torsion sheaves. It allows us to define (Simpson) -stability and -semistability for pure sheaves as usual.
A sheaf on is (Simpson) -stable (resp. -semistable) with respect to , if it is pure and for every proper subsheaf one has (resp. ).
With these definitions any torsion sheaf on is -semistable and it is -stable if and only if it has no proper subsheaves, that is, it is isomorphic to , the structure sheaf of a point . As a particular case of Simpson’s work [Simp96a], we have the following existence result. Fixing a polynomial , and a polarization on , if the class of -semistable sheaves on , with respect to , with Hilbert polynomial equal to is non-empty, then it has a coarse moduli space which is a projective scheme over . Rational points of correspond to -equivalence classes of -semistable sheaves with Hilbert polynomial .
When is an integral curve and is a coherent sheaf on it, one has classical notions of rank of , as the rank at the generic point of , and degree of , as . The Riemann-Roch theorem gives us what is the relation between the coefficients of the Hilbert polynomial and the usual rank and degree of , namely
where is the degree of defined in terms of the polarization . In this case, the Simpson notions of -stability and -semistability are equivalent to the usual ones for torsion-free sheaves. Thus, for integral curves -semistability does not depend on the polarization. This is no longer true for non-integral curves (see [LM05] for more details).
1.2.2. Invariants of the transforms and the WIT condition
In the rest of this section we assume that is a projective Gorenstein curve of arithmetic genus one with trivial dualizing sheaf and that is a fixed polarization of degree on .
Since the curve may be a singular curve, we will work with the Hilbert polynomial of a sheaf instead of its Chern characters that might not be defined. The following proposition computes the Hilbert polynomial of the transform of by the equivalences and of the previous subsection and by and . Remember that for a bounded complex , the Euler characteristic is defined to be the alternate sum
and the Hilbert polynomial is by definition .
Let be a sheaf on with Hilbert polynomial . Then
The Hilbert polynomial of the complex (resp. ) is equal to (resp. ).
The Hilbert polynomial of the sheaf (resp. ) is equal to (resp. ).
(1) Denote and consider the exact sequence
In the derived category , this induces an exact triangle
for any sheaf on . Since the Euler characteristic is additive for exact triangles in the derived category, the Hilbert polynomial of the complex is equal to
If is the projection of onto a point and are the natural projections, the base-change formula for the diagram