On the ne Simpson moduli spaces of 1-dimensional sheaves supported on plane quartics

A parametrization of the ne Simpsonmoduli spaces of 1-dimensional sheaves supported on plane quartics is given: we describe the gluing of the Brill-Noether loci described by Drézet and Maican, provide a common parameter space for these loci, and show that the Simpson moduli spaceM = M4m±1(P2) is a blowdown of a blow-up of a projective bundle over a smooth moduli space of Kronecker modules. Two di erent proofs of this statement are given.


Introduction
Fix an algebraically closed eld , char = . Let V be a -dimensional vector space over and let P = PV be the corresponding projective plane. Let P(m) = dm + c be a linear polynomial in m with integer coe cients, d > . Let M = M dm+c = M dm+c (P ) be the Simpson moduli space (cf. [1]) of semi-stable sheaves on P with Hilbert polynomial dm + c. As shown in [2], M is a projective irreducible locally factorial variety of dimension d + . In general, moduli space M parameterizes the s-equivalence classes, i. e., there is a bijection between the closed points of M and the s-equivalence classes of semistable sheaves on P with Hilbert polynomial dm+c. This way di erent isomorphism classes of sheaves could be identi ed in the moduli space. However, if gcd(c, d) = , every semi-stable sheaf is stable, s-equivalence coincides with the notion of isomorphism, and M is a ne moduli space whose closed points are in bijection with the isomorphism classes of stable sheaves on P with Hilbert polynomial dm + c. In this case there is a universal family of stable sheaves parameterized by M such that every family of (dm + c)-sheaves is obtained (up to a twist) as a pull-back of this universal family. As demonstrated in [2,Proposition 3.6], M is smooth in this case.
In [3] and [4] it was proved that M dm+c ≅ M d ′ m+c ′ if and only if d = d ′ and c = ±c ′ mod d. Therefore, in order to understand, for xed d, the Simpson moduli spaces M dm+c it is enough to understand at most The main result of the paper The aim of this paper is to "glue together" the parameterizations of the strata from [6]  which provides a common parameter space for the strata from [6] and gives a simple way to deform the sheaves from M to the ones from M . This parametrization can be seen as a natural generalization of the parametrizations of the strata from [6]. The new parameter space is deduced from the closer understanding of the complement B ′ of M in B. The way we obtain it immediately provides overB ∶= Bl B ′ B a family of stable sheaves on P with Hilbert polynomial m − and thus a map fromB to M. Under this map the exceptional divisor D of the blow-upB → B is a P -bundle over the closed stratum M , which leads to the statement of Theorem 3.1 that M is a blow-down to M of the exceptional divisor D of the blow-upB. This result coincides with the statement of [7, Theorem 3.1], which appeared earlier. Our methods are, however, signi cantly di erent. Theorem 3.1 can be also discovered by just looking at the geometric data involved and seeing the corresponding statement, which provides a very geometric proof. The exceptional divisor D can be naturally seen as a projective bundle over the closed stratum M with bres being projective lines. The varietyB can be blown down along these bres.

Structure of the paper
In Section 1 we review the description of the strata of M from [6] and give a description of the degenerations to the closed stratum. In Section 2 we present a geometric description of the bres of the bundle B → N and construct local charts around the closed subvariety B ′ ∶= B ∖ M . As a side remark we provide here a simple computation of the Poincaré polynomial of M that follows directly from [8] and [6]. The geometric description of the bres of B → N allows one to describe the blow upB → B geometrically and to see Theorem 3.1 in Section 3 by just looking at the geometric data involved. In Section 4 we construct a common parameter space for the sheaves in M and rigorously prove Theorem 3.1.

Some notations and conventions
Dealing with homomorphisms between direct sums of line bundles and identifying them with matrices, we consider the matrices acting on elements from the right. In particular, a section of a direct sum of line bundles

M m− as a union of two strata
As shown in [2] M = M m− is a smooth projective variety of dimension . By [6] M is a disjoint union of two strata M and M such that M is a closed subvariety of M of codimension and M is its open complement.

. Closed stratum
The closed stratum M is a closed subvariety of M of codimension given by the condition h (E) ≠ .
The sheaves from M possess a locally free resolution with linear independent linear forms z and z on P . M is a geometric quotient of the variety of injective matrices z q z q as above by the non-reductive group The points of M are the isomorphism classes of sheaves that are non-trivial extensions where C is a plane quartic given by the determinant of z q z q from ( ) and p ∈ C a point on it given as the common zero set of z and z .
This describes M as the universal plane quartic, the quotient map is given by M is smooth of dimension . Let M be the closed subvariety of M de ned by the condition that p is contained on a line L contained in C. Equivalently, a matrix from ( ) represents a point in M if and only if it lies in the orbit of a matrix of the form z z q . The dimension of M is .
where C ′ is a cubic and L is a line.
On the ne Simpson moduli spaces

49
Proof. Consider the isomorphism class of F with resolution This gives the commutative diagram with exact rows and columns.
The subvariety M of such sheaves is closed in M and locally closed in M. Its boundary coincides with M .

. . Twisted ideals of points on a quartic
Let M denote the open complement of M in M . In this case the maximal minors of the linear part of A are coprime, and the cokernel E A of ( ) is a part of the exact sequence where C is a planar quartic curve given by the determinant of A from ( ) and Z is the zero dimensional subscheme of length given by the maximal minors of the linear submatrix of A. Notice that in this case the subscheme Z does not lie on a line. The proof follows from the considerations below.

. . Degenerations along M
Fix a curve C ⊆ P of degree , C = Z(f ), f ∈ Γ (P , O P ( )). Let Z ⊆ C be a zero-dimensional scheme of length contained in a line L = Z(l), l ∈ Γ (P , O P ( )). Let F = I Z ( ) be the twisted ideal sheaf of Z in C so that there is an exact sequence Proof. Let us construct a locally free resolution of F. Let g ∈ Γ (P , O P ( )) such that O Z is given by the resolution one concludes that f = lh − wg for some w ∈ Γ (P , O P ( )) and h ∈ Γ (P , O P ( )). This gives the following commutative diagram with exact rows and columns.
In particular, if l and w are linear independent, which is true if and only if f is not divisible by l, this is a resolution of type ( ), hence F is a sheaf from M . If l and w are linear dependent, then without loss of generality we can assume that w = , which gives an extension and thus a destabilizing subsheaf O C ′ of F. This concludes the proof. Over M one gets one-dimensional bres: over an isomorphism class in M , which is uniquely de ned by a point p ∈ C on a curve of degree , the bre can be identi ed with the variety of lines through p that are not contained in C, i. e., with a projective line without up to points.  For a xed line L and a xed cubic curve C ′ one can compute Ext (O C ′ , O L (− )) ≅ . Therefore, using [10] one gets a projective bundle P over PV * × PS V * with bre P and a universal family of extensions on it parameterizing the extensions This provides a morphism P → M and describes the degenerations of sheaves from M to sheaves in M .

Description of B
B is a projective bundle associated to a vector bundle of rank over the moduli space N = N( ; , ) of stable ( × ) Kronecker modules, i. e., over the GIT-quotient of the space V s of stable ( × )-matrices of linear forms on P by Aut( O P (− )) × Aut( O P (− )).
The projection B → N is induced by For more details see [8,Proposition 7.7].

. The base N
The subvariety N ′ ⊆ N corresponding to the matrices whose minors have a common linear factor is isomorphic to P * = PV * , the space of lines in P , such that a line corresponds to the common linear factor of the minors of the corresponding Kronecker module ( z z z w w w ).
Théorème 4]). The exceptional divisor H ′ ⊆ H is a P -bundle over N ′ , whose bre over ⟨l⟩ ∈ P * is the Hilbert scheme L [ ] of points on L = Z(l). The class in N of a Kronecker module ( z z z w w w ) with coprime minors corresponds to the subscheme of non-collinear points in P de ned by the minors of the matrix.
The bre over such a point consists of the orbits of injective matrices If two matrices lie in the same orbit of the group action, then their determinants are equal up to a multiplication by a non-zero constant. Vice versa, if the determinants of two such matrices are equal, lies in the syzygy module of , which is generated by the rows of ( z z z w w w ) by Hilbert-Burch theorem. This implies that q − Q is a combination of the rows and thus the matrices lie on the same orbit.

. . Fibres over N ′
A bre over ⟨l⟩ ∈ N ′ can be seen as the join J(L * , PS V * ) ≅ P of L * ≅ PH (L, O L ( )) ≅ P and the space of plane cubic curves P(S V * ) ≅ P . To see this assume l = x , i. e., ⟨x ⟩ is considered as the class of Then the bre over is given by the orbits of matrices and can be identi ed with the projective space Abusing notations by renaming q ′ and q ′ into q and q respectively, rewrite the matrix ( ) as Its determinant equals This allows to reinterpret the bre as the projective space J(L * , P(S V * )) ∖ L * is a rank vector bundle over P(S V * ), whose bre over a cubic curve C ′ ∈ PS V * is identi ed with the isomorphism classes of the extensions ( ) from M with xed L and C ′ . This corresponds to the projective plane joining C ′ with L * inside the join J(L * , P(S V * )).

O. Iena
J(L * , P(S 3 V * )) \ L * is a rank 2 vector bundle over P(S 3 V * ), whose fibre over a cubic curve C ∈ PS 3 V * is identified with the isomorphism classes of the extensions (5) from M 01 with fixed L and C . This corresponds to the projective plane joining C with L * inside the join J(L * , P(S 3 V * )). The points of J(L * , P(S 3 V * )) \ L * parameterize the extensions (5) from M 01 with fixed L.

Description of B
B is a union of lines L * from each fibre over N (as explained above). It is isomorphic to the tautological Equivalently (cf. [6, p. 36]), B is isomorphic to the projective bundle associated to the tangent bundle T P * 2 . The fibre P 1 of B over, say, line L = Z(x 0 ) ⊆ P 2 can be identified with the space of classes of matrices (4) with zero determinant

Side remark: the Poincaré polynomial of M
Recall that the virtual Poincaré polynomial is a unique map P v (·)(t) from algebraic varieties to Z[t] that gives a ring homomorphism from the Grothendieck ring of varieties over k to Z[t] such that P v (X)(t) = P (X)(t) for smooth projective varieties X. In particular, this means that P v (X) = The points of J(L * , P(S V * )) ∖ L * parameterize the extensions ( ) from M with xed L.
. Description of B ′ B ′ is a union of lines L * from each bre over N ′ (as explained above). It is isomorphic to the tautological P -bundle over Equivalently (cf. [6, p. 36]), B ′ is isomorphic to the projective bundle associated to the tangent bundle T P * . The bre P of B ′ over, say, line L = Z(x ) ⊆ P can be identi ed with the space of classes of matrices ( ) with zero determinant .
Recall that the virtual Poincaré polynomial is a unique map P v (⋅)(t) from algebraic varieties to Z[t] that gives a ring homomorphism from the Grothendieck ring of varieties over to Z[t] such that P v (X)(t) = P(X)(t) for smooth projective varieties X. In particular, this means that P v (X) = P v (Y)+P v (X∖Y) if Y is a closed subvariety of X and P v (X × X ) = P v (X ) ⋅ P v (X ). The latter property implies that the virtual Poincaré polynomial of a locally trivial bration over X with bre Y equals to the product P v (X) ⋅ P v (Y).

Proposition 2.1. The Poincaré polynomial of M equals
Proof. Since M is a smooth projective variety, P(M) = P v (M). Since M is a closed subvariety in M and M is its open complement, since B ′ is a closed subvariety in B and its complement B ∖ B ′ is isomorphic to M , we On the ne Simpson moduli spaces Since B is a projective bundle over N with bre P , one gets P v (B) = P v (N) ⋅ P v (P ). Similarly, since B ′ is a P -bundle over N ′ ≅ P and the universal quartic M is a P -bundle over P , we obtain P v (B ′ ) = P v (P ) ⋅ P v (P ) and P v (M ) = P v (P ) ⋅ P v (P ). Therefore,

Lemma 2.4. N ′ is cut out in U by the equations a
Clearly these minors have a common linear factor if a, b, c, d vanish. On the other hand the condition c = d = is necessary to ensure the reducibility of these quadratic forms. If c = d = , the conditions a = b = are necessary for the minors to have a common factor.

O. Iena
Lemma 2.5. The restriction of B to U is a trivial P -bundle. Identifying P with the projective space

i. e., a point in P is identi ed with the class of the triple of quadratic forms
one can identify U × P , and hence B U , with the classes of matrices Assuming one of the coe cients of q , q , q equal to , we get local charts of the form U× and local sections of the quotient W s → B.
Proof. It is enough to notice that as in ( ) one can get rid of x in the expressions of q and q .

Charts B(γ) and B(δ)
In order to get charts around rewrite ( ), similarly to what we already did with ( ) in 2.2.2, in the form Putting γ = or δ = , we get charts around B ′ , each isomorphic to U × × . Denote them by B(γ) and B(δ) respectively. Their coordinates are those of U together with δ respectively γ and the coe cients of q i , i = , , . The equations of B ′ are those of N ′ in U, i. e., a = b = c = d = , and the conditions imposed by vanishing of q , q , q . Remark 2.6. Notice that these equations generate the ideal given by the vanishing of the determinant of ( ).

Description of M
Consider the blow-upB = Bl B ′ B. Let D denote its exceptional divisor.

. A rather intuitive explanation
Before rigorously proving this, let us explain how to arrive to Theorem 3.1 and see it just by looking at the geometric data involved. What follows in not completely rigorous but provides, in our opinion, a nice geometric picture.
Blowing up B along B ′ substitutes B ′ by the projective normal bundle of B ′ . So a point of B ′ represented by a line L ∈ P * and a point p ∈ L, which is encoded by some ⟨w⟩ ∈ PH (L, O L ( )), is substituted by the projective space D (L,p) of the normal space T (L,p) B T (L,p) B ′ to B ′ at (L, p).
As B is a projective bundle over N, and B ′ is a P -bundle over N ′ , the normal space is a direct sum of the normal spaces along the base and along the bre. Therefore, D (L,p) is the join of the corresponding projective spaces: of P = L [ ] (normal projective space to N ′ in N at L ∈ N ′ ) and P = P(S V * ) (normal projective space to L * in J(L * , P(S V * )) at p ∈ L ≅ L * ; notice that the normal projective bundle of L * ⊆ J(L * , P(S V * )), i. e., P ⊆ P , is trivial).
The bre J(L * , P(S V * )) of B → N over L ∈ N ′ is substituted under the blow-up by the bre that consists of two components: the rst component is the blow-up of J(L * , P(S V * )) along L * , the second one is a projective bundle over L * with the bre P = J(L [ ] , P(S V * )), the components intersect along L * ×P(S V * ).

A rather intuitive explanation
Before rigorously proving this, let us explain how to arrive to Theorem 3.1 and see it just by looking at the geometric data involved. What follows in not completely rigorous but provides, in our opinion, a nice geometric picture.
Blowing up B along B substitutes B by the projective normal bundle of B . So a point of B represented by a line L ∈ P * 2 and a point p ∈ L, which is encoded by some w ∈ PH 0 (L, O L (1)), is substituted by the projective space D (L,p) of the normal space T (L,p) B/ T (L,p) B to B at (L, p).
As B is a projective bundle over N , and B is a P 1 -bundle over N , the normal space is a direct sum of the normal spaces along the base and along the fibre. Therefore, D (L,p) is the join of the corresponding projective spaces: of P 3 = L [3] (normal projective space to N in N at L ∈ N ) and P 9 = P(S 3 V * ) (normal projective space to L * in J(L * , P(S 3 V * )) at p ∈ L ∼ = L * ; notice that the normal projective bundle of L * ⊆ J(L * , P(S 3 V * )), i. e., P 1 ⊆ P 11 , is trivial). The fibre J(L * , P(S 3 V * )) of B → N over L ∈ N is substituted under the blow-up by the fibre that consists of two components: the first component is the blow-up of J(L * , P(S 3 V * )) along L * , the second one is a projective bundle over L * with the fibre P 13 = J(L [3] , P(S 3 V * )), the components intersect along The space L [3] is naturally identified with the projective space of cubic forms on L * whereas P(S 3 V * ) is clearly the space of cubic curves on P 2 .
We conclude that the join of L [3] and PS 3 V * can be identified with the projective space corresponding to the vector space The space L [ ] is naturally identi ed with the projective space of cubic forms on L * whereas P(S V * ) is clearly the space of cubic curves on P .
Assume We conclude that the join of L [ ] and PS V * can be identi ed with the projective space corresponding to the vector space i. e., the space of planar quartic curves through the point p = Z(x , w).
So the exceptional divisor of the blow-up Bl B ′ B is a projective bundle with bre over (L, p) being interpreted as the space of quartic curves through p. This way we obtain a map from the exceptional divisor to the universal quartic M . Its bre over a pair p ∈ C is identi ed with the space of lines L ∈ P * through p, i. e., with a projective line. Contracting the exceptional divisor along these lines one gets M. The contraction is possible by [15][16][17], which can be seen as follows.
The bre of D → M over a pair p ∈ C may be identi ed with the bre B ′ p over p ∈ P of the map B ′ → P given by the projection to the second factor (cf. ( )). Every two points (L, p) and (L ′ , p) of B ′ p ⊆ B ′ are substituted by the projective spaces J(L [ ] , P(S V * )) and J(L ′[ ] , P(S V * )) respectively, each of which is naturally identi ed with the space of quartics through p. Assume without loss of generality p = ⟨ , , ⟩.
The bre B ′ p in this case is identi ed with the space of lines in P through p, i. e., with the projective line in N ′ = P * = PV * consisting of classes of linear forms αx +βx , ⟨α, β⟩ ∈ P . The bre has a standard covering The elements of the bre corresponding to the points of B ′ p, are the equivalence classes of matrices In this way, we have chosen, so to say, the normal forms for the representatives of the points in the bre B ′ p . For β = α − , i. e., on the intersection of B ′ p, and B ′ p, , these matrices are equivalent. One computes that gA h = A for matrices From ( ) it follows that the restriction of the ideal sheaf of D to a bre of the morphisms D → M is O P ( ). By [15][16][17], this means that one can blow down D inB along the map D → M . This gives the blow down Bl B ′ B → M that contracts the exceptional divisor of Bl B ′ B along all lines B ′ p .

The main result
Now let us properly prove Theorem 3.1 by presenting here the main result of this paper.

. Exceptional divisor D and quartic curves
Notice that the subvariety W ′ in W s parameterizing B ′ is given by the condition det A = .  2) The bre D (L,p) of D → B ′ over (L, p) ∈ P * × P , p ∈ L, is isomorphic via the mapB → PS V * to the linear subspace in PS d V * of curves through p.
3) The morphism D → M is a P -bundle over M , its bre over a point of M given by a pair p ∈ C can be identi ed with the bre of B ′ → P over p.
Proof. 1) Let [A] ∈ B ′ with A as in ( ) and let a , a , a be the rows of A. Let B be a tangent vector at A, which can be identi ed with a morphism of type ( ). Let b , b , b be its rows. Then, since det A = , Then f A,B is a non-zero quartic form if B is normal to W ′ . One computes and thus f A,B vanishes at p, which is the common zero point of x and w.
2) Since the map D (L,p) → PS V * is injective, it is enough to notice that, for a xed A ∈ W ′ , every quartic form through p can be obtained by varying B. This gives a bijection and thus an isomorphism from D (L,p) to the space of quartics through p.

. Local charts
Let us describeB over B(δ) (cf. 2.5). Around points of D lying over [A] ∈ B(δ) there are charts. For a xed coordinate t of B(δ) di erent from α, β, γ, denote the corresponding chart of Bl B ′ ∩B(δ) B(δ) byB(t). Theñ B(t) can be identi ed with the variety of triples (A, t, B), such that the coe cient of B corresponding to t equals and A + t ⋅ B belongs to B(δ). The blow-up map B(t) → B(t) is given under this identi cation by sending a triple (A, t, B) to A + t ⋅ B.

. Family of ( m − )-sheaves onB
Notice that the cokernel of ( ) is isomorphic to the cokernel of

Lemma 4.2. For t ≠ consider the matrix
x y +x z x y +x z x y +x z q q q ty ty ty tz tz tz which concludes the proof.
Evaluating ( ) at t = gives

Lemma 4.3. The isomorphism class of the cokernel F of
is a sheaf from M with resolution ifx h − wg ≠ for g =x p + x p + x p , h =x q + x q + x q .
Proof. Consider the isomorphism class of F with resolution ( ). Then, using the Koszul resolution of O P , one concludes that the kernel of the composition of two surjective morphisms which concludes the proof.
For A + tB with A and B as in ( ) we obtain the morphism given by the matrix  Proof. By the universal property of blow-ups, there exists a unique morphismB φ → Bl M M over M, which maps D to the exceptional divisor E of Bl M M and is an isomorphism outside of D. This morphism must be surjective as its image is irreducible and contains an open set. Its bres must be connected by the Zariski's main theorem. Restricted to D we get a surjective morphism D → E of P -bundles over M . Over every point of M we have a surjective morphism P → P with connected bres. The only connected subvarieties of P are the subvarieties consisting of one point and P itself. The latter can not be a bre, since this would contradict the surjectivity. This implies that the map D → E is a bijection. Therefore, φ is a bijective morphism and thus an isomorphism.
This concludes the proof of Theorem 3.1.