LetCr(P2) denote the group of biregular automorphism ofP2 with homogeneous coordinates (z0 :z1 :z2). Under the identificationC2 ={(1 :z1 :z2)}with affine coordinates (x, y) the elements of G uniquely extend to elements of Cr(P2) in the following way:
((x, y)7→(µ0x+µ1y+µ2, ν0x+ν1y+ν2))→
1 0 0
µ2 µ0 µ1 ν2 ν0 ν1
∈P GL(3) ⊂Cr(P2)
((x, y)7→(x, y+xm))→((z0 :z1 :z2)7→(z0m :z0m−1z1 :z0m−1z2+z1m)) Therefore one can consider G as a subgroup of Cr(P2). Artamkin shows in [A2]
the following:
Theorem 7.2.1 There is a well defined action of G ⊂Cr(P2) on the set of µ-stable locally free sheavesE withc1(E) = 0, c2(E) =n trivial atl∞. The bijection between µ-stable locally free sheaves and its Kronecker modules given by theorem 4.0.5 is G-equivariant.
We give a sketch of the proof: Since a ∈ A2 is a bijection of P2 preserving l∞, a∗(E) is well defined and locally free. Sinceais just a coordinate transformation a look at the monad of E, that definesA, B shows the A2-equivariance.
The non trivial part is the E-action. exm is not a bijection of P2, therefore we cannot consider the push forward of E under exm. But since it is a birational map, there is a projective surfaceX and a diagram
X
ξ . & η P2 → P2
exm
with regular maps ξ, η (See [Sh2],p.210). Thus one defines (exm)∗(E) :=
η∗(ξ∗(E)). ξ consists of the following 2m−1σ-processes: 1. blow up the point (0 : 0 : 1), denote the exceptional divisor byC1. 2. blow up the intersection point of C1 and the proper preimage of l∞ with exceptional divisor C2. 3. blow up the intersection point of C2 and the proper preimage of C1. . .m. blow up: blow up the intersection point of Cm−1 and the proper preimage of C1. Let ˜X be the
surface resulting from thism σ-processes and D the zero set divisor of the map X˜ →P2 resulting from (z0 :z1 : z2) 7→zm−10 z2+z1m. The following σ-processes are blow ups of D resp. its proper preimage and the exceptional divisor of the blow up before.
η consists of the blow downs of the preimages of the following divisors in the following order: l∞, C1, C2, . . . , C2m−2. Since the blow ups and blow downs com-mute one can perform after the first blow up, which makes P2 to the rational ruled surface F1 the second blow up followed by the first blow down, which leads toF2. In this way one can factor exm through Fm:
Fm
ξ˜ . & η˜ P2 → P2
exm
with regular maps ˜ξ,η˜ and (exm)∗(E) = ˜ξ∗(˜η∗(E)). Let ψi : Fi−1 → Fi be the birational map resulting from a blow up blow down process. Now Artamkin shows that there is an isomorphism between the set of stable bundles on Fi and Fi−1 respectively the stable bundles onF1 and P2 trivial on certain divisors, and a 1-to-1 correspondence of those stable bundles and stable nets, which are linear maps
V ⊗H0(OFi(C+ (i)f))→V∗
with n dimensional vector spaces V, V∗ and C, f being generators of the Pi-card group of Fi. Furthermore he shows that under the natural inclusion H0(OFi−1(C+ (i−1)f)),→H0(OFi(C+if)) for a stable bundleE onFi−1 with net (γ0, γ1, γ2) the restriction of the net of (ψi)∗(E) toH0(OFi−1(C+ (i−1)f)) and the net of E coincide and that the net of (ψi)∗(E) is uniquely determined by its restriction to H0(OFi−1(C+ (i−1)f)).
Because of these facts and the circumstance that the lines z0 = 0 and z1 = 0 are invariant under the resolution ofemx one getexm(γ0) = γ0 =1andexm(γ1) = γ1 = A. exm(γ2) is obtained in the following way: One has η∗(Sz2)*ξ∗(H0(OP2(1))), where Sz2 ⊂H0(OP2(1)) consist of the section with {z2 = 0} as zero set divisor.
But for a representatives2 ∈Sz2 one getsη∗(s) = ξ∗(s)+twheret∈H0(OFi(C+ (i)f)) has a zero set divisor of the form C+mf2. Furthermore for s1 ∈Sz1 the zero set divisor ofξ∗(s1) isC+mf1, wheref1andf2 are two different fibers ofFm, and ξ∗(E) is trivial on f1. Now Artamkin shows that under this circumstances one hasγC+mf2 =γC+mfm
1 =Am (see [A2] Prop. 5). Thereforeexm(γ2) =B+Am, which shows the equivariance of theE-action.
Remark: It seems that this G-equivariance extends to the whole ˜M(n). To prove this, one has to ensure, that for torsion free sheaves on Fi trivial on the exceptional divisors of the blow ups and blow downs one has a monad uniquely determined by a datum (A, B, i, j) (for locally free sheaves those monads have been investigated in [Bu]) and that even for non stable nets the formula of [A2]
Prop. 5 is true.
Chapter 8 M ˜ reg nc (4)
In this chapter we will show, that there is no element in M˜regnc (4) that has a non reduced spectral scheme. In order to do that, we will consider the space of certain coherent systems, which is a blow up ofMGss(4) the set of G-semistable torsion free sheaves with (c1, c2) = (0,4), which includes under the identification of theorem 4.0.5 ˜Mregnc (4).
8.1 Coherent systems
We give a summary of coherent systems and coherent cosystems without proofs.
They were introduced by LePotier in [LP1].
Definition 8.1.1 A coherent system (Γ,F) on Pn of dimension d consists of a coherent sheaf F with dim(F) =d and a subspace Γ⊂H0(F).
A morphism of two coherent systems (Γ0,F0) → (Γ,F), is a map f : F0 → F such that f∗(Γ0)⊂Γ. (Γ0,F0) is called coherent subsystem of (Γ,F)if there is a morphism f : (Γ0,F0)→(Γ,F) of coherent systems that is injective.
Definition 8.1.2 Let F be a coherent sheaf onPn of codimension d and let PPn be the Hilbert polynomial of OPn andPF the Hilbert polynomial ofF. We define the reduced Hilbert polynomial of a coherent system (Γ,F) as follows:
p(Γ,F) := dim(Γ)PPn+PF
m(F) where m(F) is the multiplicity of F.
Definition 8.1.3 A coherent system (Γ,F) is called semistable if and only if
a) F is a torsion free sheaf on its support.
b) ∀(Γ0,F0)⊂(Γ,F) we have
p(Γ0,F0) ≤p(Γ,F).
(Γ,F) is called semistable concerning (Γ0,F0) if b) is satisfied for this fixed sys-tem.
As it is shown in [LP1] one can define families of coherent systems over algebraic varieties such that every fibre of such a family is a coherent system with a given Chern character ch. In particular one can define a functor Syst(ch) from the category of algebraic varieties to the category of sets for which there is a coarse moduli spaceSyst(ch). Due to [LP1, Th. 4.12.] Syst(ch) is a projective algebraic variety. It decomposes into the disjoint union ofSyst(ch, m), wherem=dim(Γ).
Dualizing the concept of coherent systems leads to so called coherent cosystems:
Definition 8.1.4 Acoherent cosystem (Λ,E)of dimensiondonPnconsists of a coherent sheafE of dimensiondand a subspaceΛ ⊂Extc(E,OPn)withc=n−d.
A morphism of coherent cosystems (Λ,E) → (Λ0,E0) of dimension d is a map f∗ : Extc(E0,OPn) → Extc(E,OPn) induced by a map f : E → E0 such that f∗(Λ0)⊂Λ, in particular if a map f induces a commutative diagram
Λ0 → Extc(E0,OPn)
↓ ↓
Λ → Extc(E,OPn) .
Once again we define a reduced Hilbert polynomial and a stability condition for them:
Definition 8.1.5 Let (Λ,E) be a coherent cosystem of dimension d on Pn. We set
p(Λ,E) := dim(Λ)PPn −PE
m(E) . We call the cosystem semistable if
a) E is torsion free on its support.
b) For any coherent cosystem (Λ0,E0) E0 of dimension d torsion free on its sup-port such that there is a surjective f :E → E0 that induces a morphism of coherent cosystems (Λ,E)→(Λ0,E0) one has
p(Λ0,E0) ≤p(Λ,E). We shall need the following results.
Lemma 8.1.6 (LP1, Lemma 5.8) A coherent cosystem(Λ,E)of dimension 1 on Pn is semistable if and only if the coherent system (Λ, Extn−1(E,OPn)) is.
Lemma 8.1.7 (LP1, Prop 4.4) Let (Γ,F)be a semistable coherent system of dimension d on Pn with Γ6= 0. Then the evaluation map
Γ⊗ OPn → F has a cokernel of dimension < d.
Lemma 8.1.8 (LP1, Prop 5.3) Let (Γ,E) be a semistable coherent cosystem of codimension d with Γ6= 0. Then the cokernel of
Γ⊗ OPn →Extd(E,OPn) is of codimension > d.
Furthermore in the special case of n= 2 one can reduce the semistability of the coherent system (Γ,F) to the G-semistability of F:
Lemma 8.1.9 (LP1, Lemma 6.6) Let(Γ,F) be a coherent system onP2 with dim(Γ) = rk(F). Then (Γ,F) is semistable if and only if F is G-semistable.