Constant cycle and co-isotropic subvarieties in a Mukai system

Combining theorems of Voisin and Marian, Shen, Yin and Zhao, we compute the dimensions of the orbits under rational equivalence in the Mukai system of rank two and genus two. We produce several examples of algebraically coisotropic and constant cycle subvarieties.

1 3 [5]. In this context, Voisin introduced in [32] the notion of an algebraically coisotropic subvariety, which is an generalization of Lagrangian subvariety.
The goal of this note is to investigate the Chow group of zero cycles for the Mukai system of rank two and genus two. Specifically, we produce several examples of algebraically coisotropic subvarieties fibered into isotropic constant cycle subvarieties.
Let (S, H) be a polarized K3 surface of genus 2, that is a double covering ∶ S → ℙ 2 ramified over a sextic curve and H = * O(1) is primitive. We consider the moduli space M = M H (0, 2H, s) of H-Gieseker stable coherent sheaves on S with Mukai vector v = (0, 2H, s) where s ≡ 1 mod 2. This is an irreducible holomorphic symplectic variety, which is birational to the Hilbert scheme S [5] of five points on S. A point in M H (0, 2H, s) corresponds to a stable sheaf E on S such that E is pure of dimension one with support in the linear system |2H| and (E) = s . Taking the (Fitting) support defines a Lagrangian fibration known as the Mukai system of rank two and genus two [3,23]. It enjoys many beautiful features and is studied from various perspectives. For example, one can view it as a compactified relative Jacobian, as a generalisation of the Hitchin system [11] or as the birational model of S [5] admitting a Lagrangian fibration.
For any irreducible, holomorphic symplectic manifold X of dimension 2n, a brute force approach to finding constant cycle subvarieties (see Sect. 3.1 for the definition) is to consider the orbit under rational equivalence of a point x ∈ X . This is the countable union of algebraic subvarieties defined by Then dim O x is defined to be the supremum over the dimensions of the components of O x . In [32], Voisin defines an increasing filtration F 0 X ⊂ F 1 X ⊂ … ⊂ F n X = X on the points of X, where is again a countable union of algebraic subvarieties. Our examples are based on the combination of two theorems. The first one is due to Voisin. The second theorem applies in the case that X = M (v) is a smooth projective moduli of Bridgeland stable objects in D b (S) and is due to Marian, Shen, Yin and Zhao. It establishes a link between rational equivalence in X and in S, which in particular results in a connection between Voisin's filtration F • X and O'Grady's filtration S • CH 0 (S) (see Sect. 3.1 for the definition).  Or one can prove, that given E ∈ M such that Supp (E) = D , then E ∈ F g(D) M , where g(D) is the geometric genus of D. Here, the geometric genus of D is the genus of the normalization of D (resp. of D red ) and the sum over the genera of the normalizations of the irreducible components if D is reducible. This way, we find algebraically coisotropic subvarieties over singular curves. Precisely, for i = 0, … , 4 let and set M V i ∶=f −1 (V i ). Actually, V i is reducible due to reducible and non-reduced curves in the linear system |2H|. For every component we find the isotropic fibration and comment on the resulting constant cycle subvarieties. Most of them are rational. However, over the component of non-reduced curves Δ ⊂ V 2 , we find three-dimensional constant cycle subvarieties that are not rational (cf. Proposition 4.7).
Another series of examples comes from Brill-Noether theory. Let B • ⊂ B be the locus of smooth curves and C • → B • the restricted universal curve. For any i, we have an isomorphism

3
We consider the closures for odd i. Namely, As M H (0, 2H, −3) and M are isomorphic (Lemma 2.1), Z 1 can also be seen as subvarieties in M. In particular, they are algebraically coisotropic.

Outline
In Sect. 2, we collect general results on the Mukai system and describe the nature of its fibers. This requires an analysis of the singular curves in |2H|. In Sect.

The Mukai system
Let (S, H) be a polarized K3 surface of genus 2 such that the linear system |H| contains a smooth irreducible curve, i.e. S is a double covering ∶ S → ℙ 2 ramified over a smooth sextic curve R ⊂ ℙ 2 and H = * O ℙ 2 (1) is primitive. We consider the moduli space M = M H (0, 2H, s) of H-Gieseker stable coherent sheaves on S with Mukai vector v = (0, 2H, s) where s ≡ 1 mod 2. This is an irreducible holomorphic symplectic variety of dimension 10, which is birational to S [5] . A point in M H (0, 2H, s) corresponds to a stable sheaf E on S such that E is pure of dimension one with support in the linear system |2H| and (E) = s . Taking the (Fitting) support defines a Lagrangian fibration known as the Mukai system of rank two and genus two [3,23]. As tensoring with O S (H) induces an isomorphism it is immediate that the isomorphism class of M H (0, 2H, s) depends only on s modulo 4.
The following lemma shows that actually the isomorphism class is the same for all odd s. If Pic (S) = ℤ ⋅ H one could also characterize M H (0, 2H, s) for odd s as the unique birational model of S [5] admitting a Lagrangian fibration.

The linear systems |H| and |2H|
The geometry of the Mukai system is closely related to the structure of the curves in the linear systems |H| and |2H|, which we want to analyze in this section. A curve in the linear system |H| (resp. |2H|) has geometric genus 2 (resp. 5). We use the Segre map m ∶ |H| × |H| → |2H| to define the subloci Then Σ ≅ Sym 2 |H| is four-dimensional and its generic member is reduced and has two smooth irreducible components in the linear system |H| meeting transversally in two points. The subset Δ ≅ |H| ≅ ℙ 2 is the locus of non-reduced curves. If (S) = 1 , then Σ is exactly the locus of non-integral curves.
Recall that ∶ S → |H| ≅ ℙ 2 is a double covering, which is ramified along a smooth sextic curve R ⊂ ℙ 2 . We have and so in particular We conclude that every curve in |H| (resp. in |2H|) is the pullback of a line (resp. a quadric Q) in ℙ 2 . In particular, every curve in |H| (resp. in |2H|) has singularities depending on the intersection behavior of the ramification sextic R with (resp. Q) and has at most two (resp. four) irreducible components. For example, let ⊂ ℙ 2 be a line and C∶= −1 ( ) ∈ |H| . Assume that C is reducible. Then C consists of two irreducible components C 1 and C 2 , each isomorphic to ℙ 1 with C 1 .C 2 = 3 . This is only possible if (S) ≥ 2 . If (S) = 1 , then all curves C ∈ |H| are irreducible.
For an ample line bundle L on S and 0 ≤ i ≤ L 2 2 + 1 , we can consider the closed subvariety which is called a (generalized) Severi variety. We have dim V(i, |L|) ≤ i and there are various results about non-emptiness, irreducibility or smoothness of V(i, |L|) in the literature, e.g. [9]. In our situation, one easily gets a description of V(i, |H|) using the geometry of the covering ∶ S → ℙ 2 .  Finally, we let for i = 0, … , 4 , which turn out to be irreducible for i ≠ 0.

Lemma 2.4
The varieties Λ i are non-empty of dimension i for i = 0, … , 4. Moreover, if i ≠ 0, a general curve in Λ i has exactly 5 − i nodes as its only singularities.
Proof We apply the analogous considerations as the for the linear system |H|. This is, in order to analyze the structure of V i , we study the double coverings D → Q of irreducible quadrics Q ⊂ ℙ 2 that arise as pullback of ∶ S → ℙ 2 . Note that any such quadric is isomorphic to ℙ 1 . Assume that D is reducible, then as above, D is necessarily rational and this is only possible for (S) ≥ 2 . If D is irreducible, its singularities can be read off the intersection of Q with the ramification sextic R, yielding the desired result. For example, assume D is integral with g(D) = 4 . Then, either Q ∩ R consists of 11 points all of which have multiplicity one except one point which has multiplicity two, or Q ∩ R consists of 10 points all of which have multiplicity one except one point which has multiplicity three. This produces either one node or one cusp in D. The first case provides a codimension one locus in the linear system |O ℙ 2 (2)| ≅ ℙ 5 of quadrics in ℙ 2 , the second case has codimension two. ◻ We sum up our discussion in the following proposition.

Proposition 2.5 The Severi varieties
We compute the degree of its components.

Lemma 2.6 We have
In particular, the discriminant divisor of f has degree 45.
Proof The easiest way, to see that deg[Σ] = 3 is a geometric argument. Choose 4 points x 1 , … , x 4 in general position and consider the line There is a unique (resp. no) curve C ∈ |H| passing through two (resp. three) points in general position. Hence, deg Σ = #( ∩ Σ) = 3 , corresponding to the three possible partitions of x 1 , … , x 4 into pairs of two points.

3
Alternatively, after a choice of coordinates Σ ≅ Sym 2 ℙ 2 is embedded into ℙ 5 via the map induced by One checks that the image is cut out by the equation, where the coordinates f i of ℙ 5 are ordered as in (2.4).
sing consists of 48 points. As a general pencil contains three curves in Σ which have two singular points and a generic integral singular curve has exactly one nodal singularity, we conclude deg Λ 4 = 42 . ◻

Remark 2.7
In [28, §5] the computation (2.5) serves as a demonstration for a formula of the degree of the discriminant locus of a Lagrangian fibration with 'good singular fibers'. An example of such a fibration is the Beauville-Mukai system over a primitive curve class and the discriminant divisor is irreducible of degree 6(n + 3) , where n is the dimension of the base of the fibration. However, in our example the fibers over Δ are not 'good singular fibers' and we find a different result.

Fibers of the Mukai morphism and structure of M
In this section, we collect some information on the fibers of the Mukai morphism.
Let us make this more precise for generic points: • In the first case, let x ∈ B⧵Σ correspond to a smooth curve D, then f −1 (x) ≅ Pic 3 (D).
• In the second case, let x ∈ Σ⧵Δ correspond to the union D = D 1 ∪ D 2 of two smooth curves meeting transversally in two points.  (2, 1) and (1,2). • In the third case, let x ∈ Δ correspond to a non-reduced curve with smooth underlying curve C ∈ |H| . Then f −1 (x) has two non-reduced irreducible components, which we denote as follows The first component M 0 2C consists of those sheaves, that are pushed forward from the reduced curve C. With its reduced structure it is isomorphic to the moduli space of stable vector bundles of rank two and degree one on C. The other component M 1 2C is the closure of those sheaves that can not be endowed with an O C -module structure. All these sheaves fit into a short exact sequence where i ∶ C ↪ S is the inclusion, and L ∈ Pic 1 (C) is the torsionfree part of E| C and x ∈ C is the support of the torsion part of E| C . This extension is intrinsically associated to E , for details see [15]. Following, [8, Propositions 3.7.19 & 3.7.23], the decomposition (2.7) remains valid if the underlying curve C is irreducible with nodal or cuspidal singularities. (In particular, in the case that (S) = 1 , it holds for every x ∈ Δ.) In the case of a K3 surface of higher Picard rank, the general picture remains the same. But due to reducible curves in the linear system |H| or B⧵Σ , the fibers could exceptionally have more irreducible components. For example, if x ∈ B⧵Σ corresponds to a reducible curve with two smooth components, then f −1 (x) still contains a dense open subset parameterizing line bundles. However, following de Cataldo et al. [8,Lemma 3.3.2] one finds, that the numerical restrictions imposed by the stability now allow partial degrees (5, −1), (4, 0), The decomposition (2.7) also exists globally over the locus of curves D = 2C with C ∈ |H| smooth, which we denote by Δ • ⊂ Δ . Here, we have where M 0 Δ • is a relative moduli space of stable vector bundles and M 1 Δ • the closure of its complement, [8, Proposition 3.7.23]. We set is reduced and irreducible if x ∈ B⧵Σ is reduced and has two irreducible components if x ∈ Σ⧵Δ has two irreducible components with multiplicities if x ∈ Δ.

3 3 Orbits under rational equivalence
Our strategy to find algebraically coisotropic subvarieties is to single out points whose orbit under rational equivalence has a high dimension. In this section, we explain how this can be done combining results of Voisin [31,32], Shen-Yin-Zhao [29] and Marian-Zhao [22] (for the precise references, see below).

Preliminaries
We start by recalling some general definitions. Let (X, ) be an irreducible holomorphic symplectic manifold of dimension 2n. For a smooth subvariety Y ⊂ X , we let where the first arrow is given by . [18] if all its points are rationally equivalent in X. Note that this is the case, if Y contains a dense open subset U, such that all points in U are rationally equivalent in X. Mumford's theorem [24] implies where Y reg denotes the regular part of Y.
T ⟂ Z reg ⊂ T Z reg ) and the corresponding foliation is algebraically integrable. For a subvariety of codimension i, this is equivalent to the existence of a 2n − 2i-dimensional variety T and a rational surjective map ∶ Z ⤏ T such that where Z • ⊂ Z is the open subset, where is defined and smooth, or equivalently Actually, T and are unique up to birational equivalence. We call the associated isotropic fibration. (iii) For a point x ∈ X , its orbit under rational equivalence is which is a countable union of closed algebraic subvarieties [30,Lemma 10.7]. Its dimension is defined to be the supremum over the dimensions of its irreducible components. Following Voisin 1 [32, Def 0.2], we set for i = 0, … , n . This is again a countable union of closed algebraic subvarieties and defines a filtration on the points of X   In particular, for E ∈ M (v) we have Using that the union of all constant cycle curves in S is Zariski dense and Theorem 3.2 allow one to prove the following theorem. Next, one could ask how the filtration F i M (v) interferes with the second Chern classes. The answer is to consider O'Grady's filtration on CH 0 (S) . Let us recall some results about CH 0 (S).
In [4], Beauville and Voisin prove that any point lying on a rational curve in S determines the same class which has the property that the image of the intersection product Pic (S) ⊗ Pic (S) → CH 0 (S) is contained in ℤ ⋅ c S . In [27], (1) The filtration is compatible with addition, i.e. if ∈ S i CH 0 (S) and ∈ S j CH 0 (S) , then + ∈ S i+j CH 0 (S). and d ∈ ℤ depending on the degree of ch 2 (E) , which is fixed. After knowing the result for S [n] , the theorem translates into the statement c S ∈ CH 0 (S), The two orbits can be compared by means of the incidence variety which is a countable union of Zariski closed subsets in M × S [n] . There exists an irreducible component R 0 ⊂ R which projects generically finite and surjective to both factors, and hence yields a correspondence between the two orbits (see [29, §2.3]). However, in order to compare their dimensions, one needs to know that the components of maximal dimension in every orbit under rational equivalence are dense. This is known for the Hilbert scheme, whence the inclusion S SYZ i CH 0 (M) ⊂ S V i CH 0 (M) always holds. The reverse inclusion is true if M is birational to S [n] but in general not known. ◻

Orbits under rational equivalence in M
We turn back to our favorite example M = M H (0, 2H, −1) with the goal in mind, to give explicit constructions of constant cycle subvarieties in M. We remark that any moduli space of Gieseker-stable sheaves on a K3 surface S (with respect to a generic polarization) can be realized as a moduli space of Bridgeland-stable objects in D b (S) [7, Proposition 14.2]. In particular, the above results can be applied to M. The first step is to understand the orbits under rational equivalence in M and the filtration Recall that M is birational to S [5] and thus by Theorem 3.5, we know and for 0 ≤ i ≤ 5 . The following lemma is a straightforward computation using the Grothendieck-Riemann-Roch theorem.

Lemma 3.6 Let i ∶ D ↪ S be a reduced curve and let F be a vector bundle on D.
(i) Assume that D is irreducible and let ∶D → D be its normalization. Then where m p = length ( * OD∕O D ) p . In particular, (ii) Assume that D = D 1 ∪ D 2 has two irreducible components. Then (3.4) ch 2 (i * F) = ch 2 (i 1 * F| D 1 ) + ch 2 (i 2 * F| D 2 ) − rk (F)(D 1 .D 2 )c S ∈ CH 0 (S), were i k ∶ D k ↪ S, k = 1, 2 are the inclusions of the components. In particular, ◻ Example 3.7 Using Lemma 3.6 we compute ch 2 (E) for some cases of stable sheaves E occuring in M: inclusion and L ∈ Pic 3 (D) . We find (ii) Let E ∈ M be the pushforward of a line bundle L on its support D = Supp E and assume that D = D 1 ∪ D 2 has two smooth and connected components. We write 2C for a smooth curve C ∈ |H| , i.e. Supp (E) = 2C and E = i * E 0 , where i ∶ C ↪ S is the inclusion and E 0 is a vector bundle of rank 2 and degree 1 on C. Then (iv) Let E ∈ M 1 2C ⧵M 0 2C for a smooth curve C ∈ |H| , i.e. Supp (E) = 2C but E is not pushed forward along the inclusion i ∶ C ↪ S . However, E fits into a short exact sequence for some L ∈ Pic 1 (C) and x ∈ C . Hence

Corollary 3.8 Let E ∈ M and let D = Supp (E). Assume either that D is reduced and E is locally free on D, or D is non-reduced and D red is a smooth curve. Then where g(D) is the geometric genus of D. ◻
As before, the geometric genus of D is the genus the normalization of D (resp. of D red ) and the sum over the genera of the normalizations of the irreducible components if D is reducible. Proof It suffices to consider a dense open subset of F, in order to decide whether F is a constant cycle subvariety. First assume that i ∶ D ↪ S is reduced. Then F contains a dense ch 2 (i * F) ∈ S g(D 1 )+g(D 2 ) CH 0 (S). If D = 2C is non-reduced, we apply the same argument to the explicit description (2.7) of the fiber F. It is exclusively here, that we use (S) = 1 . ◻ Remark 3. 10 We expect that a case by case analysis also proves Corollary 3.9 when (S) > 1 . In this case it is no longer true, that all curves C ∈ |H| are irreducible with at worst nodal or cuspidal singularities.   In particular, they are algebraically coisotropic.

Horizontal examples from Brill-Noether loci
where D ∈ B • and ⊂ D is an effective divisor of degree i. Hence in CH 0 (S) and we conclude ch 2 (E) ∈ S i CH 0 (S) , which in turn gives 2H, i − 4) . By Theorem 3.2, this implies dim Z i ≤ 5 + i , whereas the reverse inequality is known from Brill-Noether theory [1,IV Lemma 3.3]. ◻ Actually, Z 1 is a projective bundle over S. Precisely, let D ∈ |2H| and L ∈ Z 1 ∩ f −1 (D) , i.e. L ∈ Pic 1 (D) is effective and can uniquely be written as O D (x) for some x ∈ D . This way, Z 1 is isomorphic the universal curve C ⊂ |2H| × S , which is a ℙ 4 bundle with respect to the second projection. With the same arguments, we also prove that Z 3 is generically a ℙ 2 -bundle over S [3] , which parameterizes the line bundles O D ( ) over ∈ S [3] .
In the following, we will consider Z 1 as a subvariety of M via the isomorphism In particular, over a smooth curve D ∈ |2H| , we have

Vertical examples from singular curves
In this section, we give examples of algebraically coisotropic subvarieties, that arise as preimages of subvarieties in B. In Corollary 3.9, we already treated the case of a fiber over a point D ∈ B . Namely, f −1 (D) is a constant cycle Lagrangian, if and only if D is a constant cycle curve. We set In particular, they are algebraically coisotropic.
Proof We saw in Proposition 2.5 that dim V i = i and in Corollary 3 In the following section, we find the isotropic fibrations for M V i .

Isotropic fibrations
In  . This is what M Λ i ⤏ T i encodes. Consider the universal curve over |2H| and let C i → Λ i be its restriction to Λ i ⊂ |2H| . By construction, the generic fiber of C i is singular and so must be the total space C i . Hence, the normalization Here, ℤ∕2ℤ = ⟨ ⟩ acts on the right hand side viã ⋅ (L 1 , L 2 ) = ( * L 2 , * L 1 ).
(By a slight abuse of notation, also denotes the map identifying the isomorphic components C 1 Σ⧵Δ and C 2 Σ⧵Δ ). The quotient of (4.3) by is what we are looking for, when restricted to the component, where the correct degree has been fixed. More precisely, we set and take Σ to be the above map, whose fibers are isomorphic to m . ◻ To Then, by (2.8) and Lemma 3.6 the class , where L∶=E| C ∕T ∈ Pic 1 (C) and x∶= Supp (T) ∈ C with T being the torsion subsheaf of E| C and i ∶ C ↪ S being the inclusion. Consequently, we define We want to compute the fibers of ( 1 Δ ) 2C . First, we forget the twist with −1 C . Then we can factor ( 1 Δ ) 2C as follows

3
The first arrow is defined outside the intersection M 0 2C ∩ M 1 2C and its fibers are a torsor under Ext 1 C (i * L, i * (L(x) ⊗ −1 C )) ≅ ℂ 2 , cf. [15,Corollary 3.5]. Thus the fibers of ( 1 Δ ) 2C are an 2 -bundle over the fibers of the second arrow, which we factor as follows Here, the first map is étale of degree 16 and the fibers of the second map are isomorphic to C. To see this, let M ∈ Pic 3 (C) and consider p 2 ∶ −1 (M) → C . As L(x) ≅ M for fixed x ∈ C determines L ∈ Pic 2 (C) , this projection is an isomorphism and the claim follows.

More examples
We construct some more examples of algebraically coisotropic subvarieties.

Horizontal constant cycle Lagrangians
To start with, we produce a constant cycle Lagrangian that dominates B. For example, any section of M → B would work. Unfortunately, f does not admit a section [2]. Below, we produce a multisection of degree of 2 10 . Recall that where B • = B⧵(Λ 4 ∪ Σ) is the locus of smooth curves, and there is an exact sequence [12, (9.2.11.5)] Next, we claim that The first isomorphism follows from the diagram with exact rows and columns where we use, that the restriction of the universal curve over Λ 4 and Σ , respectively is irreducible. The second isomorphism in (4.5) holds, because C ⊂ B × S is a ℙ 4 -bundle over S with O p 2 (1) = p * 1 O B (1) . After this identification, any L ∈ Pic (S) with n = 2H.L is mapped under the first morphism in (4.4) to the section If Pic (S) = ℤ ⋅ H , for example, one gets sections for n ≡ 0 mod 4. There is always a section of Pic 2 C • ∕B • that does not come from S.

Lemma 4.9
There is a section such that a curve D ∈ B • maps to the unique line bundle g 1 2 (D) ∈ Pic 2 (D) with h 0 (g 1 2 (D)) = 2. In particular, and g 1 2 is not of the form s L for L ∈ Pic (S).
Proof This is a consequence of the same phenomenon occurring for the universal family of smooth quadrics in ℙ 2 . We identify B = |O ℙ 2 (2)| and we will see that the lemma holds true The composition of the one map with the inverse of the other is the rational involution on M that comes from the natural involution [5] on S [5] . Now, we use the squaring map to construct a constant cycle Lagrangian from g 1 2 (B • ) . Specifically, we set

Lemma 4.11
The subvariety L 1 2 is a constant cycle Lagrangian in M, which is generically finite of degree 2 10 over B.

Remark 4.12
Another example of a horizontal constant cycle Lagrangian is constructed more generally for any Lagrangian fibration by Lin [21]. The idea of Example 4.13 is to find a constant cycle surface in T Δ . Then the preimage under i Δ is a constant cycle Lagrangian in M contained in M i Δ . This idea is taken further in Example 4.14. Here, we find a surface in T Δ , that consists of line bundles whose first Chern class is a multiple of the Beauville-Voisin class, when pushed forward to S. By Theorem 3.5, the preimage of this surface is also a constant cycle Lagrangian in M.

Starting from
For simplicity, we assume from now on that Pic (S) = ℤ ⋅ H . Then every curve in |H| is integral and Pic 1 C |H| ∕|H| is representable by a smooth, quasi-projective scheme.
Example 4. 13 We construct a constant cycle subvariety of Pic 1 C |H| ∕|H| applying the same trick as for the construction of L By construction, these are constant cycle Lagrangians in M.
Example 4.14 The idea of this example is to consider the preimage of two-dimesional subvarieties in T Δ that are not constant cycle subvarieties themselves, but consist of line bundles whose first Chern class is the Beauville-Voisin class when pushed forward to S.
To begin with, we have an embedding Then, for example a vector bundle E ∈ M 0 Δ lies over Θ(C |H| ) if and only if its determinant line bundle is effective (of degree one). Therefore, In particular, ( i Δ ) −1 (Θ(C |H| )) is algebraically coisotropic. The isotropic fibration is given by the composition of the projection C → S with i Δ .
Refining this example yields constant cycle Lagrangians in M i Δ as follows. For example, let C cc ⊂ S be a constant cycle curve and set [5] We can also produce easily examples of algebraically coisotropic subvarieties in S [5] . As M and S [5] are birational, we have [13,Example 16.1.11] and algebraically coisotropic varieties that are not contained in the exceptional locus of a birational map can be transferred from S [5] to M and vice versa. Then E i ⊂ S [5] is closed subvariety of codimension 5 − i [6]. For example, E∶=E 4 is the exceptional divisor of the Hilbert-Chow morphism s ∶ S [5] → S (5) . The irreducible components E n i of E i are indexed by ordered tuples of positive natural numbers n = (n 1 ≥ n 2 ≥ … ≥ n i ) such that ∑ i k=1 n k = 5 . In particular, E 4 and E 1 are irreducible, whereas E 3 and E 2 consists of two irreducible components. To sum up By definition of E i and Theorem 3.2, we have

Example 4.17
This example is taken from Knutsen et al. [19]. As in (2.2), we consider the locus V(j, |H|) ⊂ |H| of curves C with g(C) ≤ j for j = 0, 1, 2 . Specifically, V(2, |H|) is everything, V(1, |H|) ⊂ |H| is irreducible and one-dimensional and the generic curve in V(1, |H|) has exactly one node, V(0, |H|) is the discrete set of rational curves. We let C j → V(2 − j, |H|) be the respective restriction of the universal curve C |H| → |H| . For 2 − j ≤ i ≤ 4 , consider the diagram in which the lower horizontal map turns out to be generically injective [19,Theorem 6.4]. Hence, f j i is generically finite. We define This is a subvariety of codimension 5 − i , which is irreducible for j ≠ 2 . We have the following table of inclusions: A generic point ∈ W j i corresponds to a subscheme in S that contains exactly 7 − i − j points, which lie on a curve C ∈ V(2 − j, |H|) and the other i + j − 2 points can move freely outside C. Hence, [ ] ∈ CH 0 (S) is contained in the (2 − j) + (i + j − 2)-th step of O'Grady's filtration. In other words, and W j i is algebraically coisotropic. g 3 ∶ ℙ 2 × S [3] ⤏ S [5] and g 1 ∶ Sym 2 (ℙ 2 ) × S ⤏ S [5] P i ∶= im (g i ) ⊂ S [5] for i = 1, 3. [5] .