Geometry of canonical genus 4 curves

We apply the machinery of Bridgeland stability conditions on derived categories of coherent sheaves to describe the geometry of classical moduli spaces associated with canonical genus 4 space curves via an effective control over its wall-crossing. This article provides the first description of a moduli space of Pandharipande– Thomas stable pairs that is used as an intermediate step toward the description of the associated Hilbert scheme, which in turn is the first example where the components of a classical moduli space were completely determined via wall-crossing. We give a full list of irreducible components of the space of stable pairs, along with a birational description of each component, and a partial list for the Hilbert scheme. There are several long standing open problems regarding classical sheaf theoretic moduli spaces, and the present work will shed light on further studies of such moduli spaces such as Hilbert schemes of curves and moduli of stable pairs that are very hard to tackle without the wall-crossing techniques.


INTRODUCTION
Despite their very natural definition as the parameterization spaces of subvarieties in projective space, Hilbert schemes are among the most badly behaved moduli spaces in algebraic geometry; for example, Murphy's Law holds for these spaces [29], or see [15] for pathologies of the Hilbert scheme of points.
For ℙ 2 as an ambient space, the Hilbert scheme of points is well-studied, but already in the case of ℙ 3 , few general results are known, and not many examples are fully understood.Some classical results are known about some Hilbert scheme of curves in ℙ 3 : For instance, see [25] and [7] for the Hilbert scheme of twisted cubics, and [1,9] and [6] for the Hilbert scheme of elliptic quartics.Both Hilbert schemes have two irreducible components.Also see [21] and [22] for the Hilbert schemes of Cohen-Macaulay curves, and [12] for the Hilbert schemes of points.The first examples of wall-crossing for Hilbert scheme of curves in ℙ 3 can be found in [27,30] for twisted cubics, and in [8] for elliptic quartics.
Our goal is to describe the geometry of the Hilbert scheme  6−3 (ℙ 3 ) containing canonical genus 4 curves, as well as the associated moduli space of PT stable pairs.
The space  6−3 (ℙ 3 ) parameterizes more objects than only the canonical genus 4 pure curves; considering the formula () =  + 1 − g for the Hilbert polynomial of genus g and degree  in ℙ 3 , some curves of higher genus together with some extra points can have the same Hilbert polynomial.
A PT stable pair was defined by Pandharipande and Thomas to be a pair ( , ), where  is a 1-dimensional pure sheaf on  and  ∶   →  a section with 0-dimensional cokernel (Definition 2.8).We will prove the following two main theorems describing the space of PT stable pairs and the Hilbert scheme associated to the canonical curves of genus 4: Theorem 1.1 (See Theorem 8.1).The moduli space of PT stable pairs  ℙ 3 →  , where ℎ( ) = (0, 0, 6, −15) consists of components birational to the following eight irreducible components (the first one is 24-dimensional, the last one is 36-dimensional, and the rest are all 28-dimensional).
(2) A ℙ 17 -bundle over (2, 4) ×  2 , which generically parameterizes the union of a line and a plane quintic intersecting the line together with a choice of two points on the quintic.(3) A ℙ 18 -bundle over  ×  1 , which generically parameterizes the union of a line in ℙ 3 together with a choice of a point on it, and a plane quintic intersecting the line, together with a choice of a point on it.(4) A ℙ 19 -bundle over  (2,4) ×  1 , which generically parameterizes the disjoint union of a line in ℙ 3 and a plane quintic together with a choice of a point on it.(5) A ℙ 19 -bundle over ( × (2,4)  ) × (ℙ 3 ) ∨ , which generically parameterizes the union of a line in ℙ 3 together with a choice of two points on it, and a plane quintic intersecting the line.
(6) A ℙ 20 -bundle over  × (ℙ 3 ) ∨ , which generically parameterizes the disjoint union of a line in ℙ 3  together with a choice of a point on it, and a plane quintic.(7) A ℙ 21 -bundle over  2 , which generically parameterizes the union of a plane quartic with a thickening of a line in the plane.(8) A ℙ 21 -bundle over  6 , which generically parameterizes a plane degree 6 curve together with a choice of six points on it.
The first four components are irreducible.
As a remark, we note that we expect ℋ 6 to be irreducible of dimension 48.Notice that a smooth nonhyperelliptic genus 4 curve  embeds into ℙ 3 as a (2,3)-complete intersection curve.The question is how to compactify this 24-dimensional space of such curves.As we have an embedding in the projective space, a classical answer would be considering the Hilbert scheme of such curves.But this has many irreducible components; for example, plane sextics with six floating points added yields a component of dimension 48.It would be hard to guess and list the rest of the irreducible components.
Therefore, instead of studying the Hilbert scheme directly, we consider Bridgeland stability conditions on D  (ℙ 3 ).Depending on a choice of a stability condition  ∈ Stab(ℙ 3 ), we get   (1, 0, −6, 15), the moduli space of -stable complexes  with ch() = ch(  ).Following a path in the space of stability conditions, we want to understand how   (1, 0, −6, 15) changes via wall-crossing.We studied the beginning of this path in [26].For example, the first chamber gives a ℙ 15 -bundle over ℙ 9 where ℙ 9 corresponds to a choice of a quadric, and ℙ 15 corresponds to a choice cubic modulo the ideal generated by the quadric.This is evidently a very efficient compactification of the space of canonical genus 4 curves, which are (2,3)-complete intersections.On the other hand in the large-volume limit, via rough control over wall-crossing, we recover the geometry of the Hilbert scheme, for which there are no prior results.

Strategy of the proof
There are two nontrivial extremes on a path we are taking from the empty space to the large volume limit: one is an efficient compactification of the (2,3)-complete intersection curves and 1 We cross the walls along the yellow path that is close enough to the hyperbola.
the other is the Hilbert scheme.The wall after the blow-up of the compactification that introduces a novel birational phenomenon is described in [26].The strategy to reach the Hilbert scheme uses the space of PT-stable pairs as an intermediate step (see Figure 1).For each wall ⟨, ⟩ on the right side of of the left branch of the hyperbola Im( ,, ) = 0 (which will be denoted by ℍ) as in Theorem 3.10, the newly stable objects are given by an exact triangle  →  → .For each ,  this gives a locus ℙ(Ext 1 (, )) inside the moduli space after crossing the wall.We stratify the space of pairs (, ) by dim(ℙ(Ext 1 (, ))); for each stratum, we describe a general element and decide whether it is in the closure of other strata.This will describe the components of the moduli space of stable pairs (see Theorem 8.1).To recover the Hilbert scheme from this moduli space, we analyze the DT/PT wall (i.e., the wall between the two adjacent chambers associated with the Hilbert scheme and the space of stable pairs).We investigate which components in the space of stable pairs survive in the Hilbert scheme, and what new components are created (see Theorem 9.3).

⟨𝐴, 𝐵⟩:
The wall where strictly semistable objects have Jordan-Hölder factors of type  and

Conventions
When there is no confusion, the subobject and the quotient of the defining short exact sequence of any wall will be denoted by  and , respectively.Notice that we cross the walls toward the large volume limit.Note that by large volume limit, we mean the region in Stab(ℙ 3 ) where  → ∞.When we say near hyperbola, we mean a sufficiently small tubular neighborhood of the locus of the hyperbola in the plane, so that there is no other wall intersects that region.Let C ⊂ ℙ 3 be a 1-dimensional subscheme.If C is not pure dimensional, one can consider its maximal 0-dimensional subsheaf  ⊂  C so that  C ∕ =   , for some curve  ⊂ C as the maximal subscheme of C with pure dimension 1.Now, if a point  ∈ Supp( ) ∩ , then it is called an embedded points, while if  ∈ Supp( ), but  ∉ , then  is called a floating points.

BRIDGELAND STABILITY CONDITIONS AND STABLE PAIRS
In this section, we review Bridgeland stability conditions on ℙ 3 , and the notion of PT stable pairs.

Bridgeland stability conditions on ℙ 𝟑
In this subsection, we briefly define stability conditions on the bounded derived category D  (ℙ 3 ) of coherent sheaves on ℙ 3 , following the construction in [5].Let Coh(ℙ 3 ) be the abelian category of coherent sheaves (as an initial heart of a bounded t-structure with usual notion of torsion pair) on ℙ 3 .Let  > 0,  ∈ ℝ be two real numbers.We define the twisted Chern characters ch  () =  − .ch(),where H denotes the hyperplane class.For  ∈ D  (ℙ The new heart of a bounded t-structure can be defined as Coh  (ℙ 3 ) ∶= ⟨  [1],   ⟩.Now we are ready to define the notion of tilt-stability: The central charge and the corresponding slope function for the new heart can be defined as We denote by Stab tilt (ℙ 3 ), the space of all tilt-stability conditions.It was conjectured in [5] for arbitrary threefolds, and proved in by Macrì in [17] for ℙ 3 that tilting again gives a Bridgeland stability condition as follows: Definition 2.2.We define a torsion pair similarly as for the tilting case:  , ∶= { ∈ Coh  (ℙ 3 ) ∶  , () > 0 for all  ↠ },  , ∶= { ∈ Coh  (ℙ 3 ) ∶  , () ⩽ 0 for all  ↪ }.

Now define a new heart, central charge, and slope, respectively, as follows:
Coh , (ℙ 3 ) ∶= ⟨ , [1],  , ⟩,  ,, ∶= −ch The pair  ,, = (Coh , (ℙ 3 ),  ,, ) (when exists) is called Bridgeland stability condition.Before going further, we have a formal definition of a wall and chamber: [19,Definition 5.26]; [18, p. 161].Let Λ be a finite rank lattice, and ,  ′ ∈ Λ * two nonparallel classes.We fix the parameter  in the definition of  ,, , and define a numerical wall in Bridgeland stability with respect to a class  to be a nontrivial proper subset of the stability space that is defined as An actual wall is the subset  ′ of a numerical wall consisting of those points for which there exists properly semistable objects with class  (we can give a similar definition for tilt-stability).
A chamber is defined as a connected component of the complement of the set of actual walls in ℝ >0 × ℝ.
There is also a well-behaved wall-chamber structure in Stab tilt (ℙ 3 ).The last part of the following theorem was proven for surfaces in [16]: ) defined by (, ) → (Coh  (),  , ) is continuous.Moreover, walls with respect to a class  in the image of this map are locally finite.In addition, the walls in the tilt-stability space are either nested semicircles or vertical lines.Remark 2.7.Note that the Jordan-Hölder factors of an object on a wall are stable along the entire wall.
For more details on Bridgeland stability conditions on ℙ 3 , we refer to [27].

Pandharipande-Thomas stable pairs
To make the floating/embedded points for the curves parameterizing the Hilbert scheme of curves more under control, Pandharipande and Thomas introduced the following notion: Definition 2.8 [23, p. 408].A stable pair ( , ) consists of a 1-dimensional pure sheaf  on  and a sections  ∶   →  with 0-dimensional cokernel.
We refer to the 0-dimensional cokernel as "points on the curve".

WALLS
According to Theorem 2.6, there is a wall and chamber structure in the stability manifold.In this section, we numerically describe the walls in Stab tilt (ℙ 3 ) with respect to ch(  ), where  is a canonical genus 4 curve, and give a geometric description of the walls.
Consider the hyperbola ℍ in the (, )-plane defined by Im( ,, ()) = 0.For such , , , semistable objects of Chern character  have phase 0.Moreover, these semistable objects have positive and negative phases with respect to the stability conditions on the left and right side of the hyperbola, respectively.Thus, on the left we work with Coh  (ℙ 3 ) and Coh , (ℙ 3 ), for tilt and Bridgeland stability, respectively, and on the right side of the hyperbola we work in Coh  (ℙ 3 )[−1] and Coh , (ℙ 3 )[−1].
Theorem 2.6 gives an order for the walls or semicircles in Stab tilt (ℙ 3 ).We refer to the semicircle with the smallest radius as the first wall, and so on.
First, we recall a number of results from [26]: To describe the walls in the space of Bridgeland stability conditions, we need the following result relating tilt-stability and Bridgeland stability: Lemma 3.4 [4,Lemma 8.9].Let  ∈ Coh , (ℙ 3 ) be a  ,, -semistable object, for all  ≫ 1 sufficiently big.Then it satisfies one of the following conditions.
is either 0 or 0-dimensional torsion sheaf, and Hom(  , ) = 0 for all points  ∈ ℙ 3 , then  is  ,, -stable for all  ≫ 1 sufficiently large.Remark 3.5.Note that (a) and (b) apply to semistable objects of Chern character  with respect to the stability conditions on the left and right of the hyperbola ℍ, respectively.
Proof.Let  be such an object with Chern character , that is,  is stable near the hyperbola, which means it is still stable on the hyperbola as we reach it from the right or from the left.For objects on the hyperbola ℍ, semistability does not change as  varies; in particular, we can let  → +∞ and apply Lemma 3.4 to  [1].Therefore,  1  () is a torsion sheaf  with 0-dimensional support, and  0  () is  , -semistable.Notice that  is semistable for Im  ,, () < 0; this implies Hom(  [−1], ) = 0 and so Hom(  [−1],  0  ()) = 0 for all  ∈ ℙ 3 , as   [−1] is semistable of phase 0. But (1, −1, 1∕2, ) is the Chern character of a tilt-semistable sheaf of the form   (−1) where  is a 0-dimensional subscheme of length −1∕6 −  by Lemma 3.2.But if −1∕6 −  ≠ 0, then we would have Hom(  [−1],  0  ()) ≠ 0 for  ∈ , which is a contradiction.Therefore,  = −1∕6 and thus  0  () ≅  ℙ 3 (−1).Now by Serre duality, we have This means that we must have  1  () = 0; otherwise,  would be a direct sum and hence unstable.Therefore,  ≅  0  () ≅  ℙ 3 (−1).□ Let ℍ be the left branch of the hyperbola Im( ,, ()) = 0. Corresponding to the walls in Stab tilt (ℙ 3 ), we list all (tilt and Bridgeland) strictly semistable objects on each wall defined by two given objects having the same slope that can be obtained from the extensions of the pairs of objects: Theorem 3.10.For the walls in Stab tilt (ℙ 3 ), the walls on the both sides of ℍ (and sufficiently close to it) are given by the following pairs: where   's and  ′  's are 0-dimensional subschemes of length ,   's are lines plus  extra embedded/floating points,  2 is a conic,  is a quadric in ℙ 3 ,  a curve supported on , and   ∶  ↪ ℙ 3 is the inclusion map.Every wall in Stab(ℙ 3 ) that cuts the hyperbola ( 2 −  2 = 12) is induced by a wall in Stab tilt (ℙ 3 ).

EXT COMPUTATION OF THE WALLS ON THE STABLE PAIRS SIDE
So far, we have described all the possible walls in Stab(ℙ 3 ) close to the left branch of the hyperbola  2 −  2 = 12.To study the wall-crossing and describe the chambers (on the right side of the hyperbola) in Section 6, we need to compute all the necessary Ext 1 -groups associated with the walls.When there is no confusion, we use the notation  and  for the subobject and the quotient of the defining short exact sequence of any wall, respectively.
In the statement of the following lemma, all tensor products are in D  (): Lemma 4.4.Let  ≠  be two different points in a plane , such that  ⊂ , and  ⊄  ′ , where lines ,  ′ are in .Then   ⊗   fits into the short exact sequence Proof.All the cases either proved in [26,Lemma 4.2] or exactly a similar argument there implies the result.□ Lemma 4.5.For the wall (given by (2, 4)) together with a choice of one point on the line.Thus, we have is the parameter space of one point in a plane, which gives the flag variety  1 .Therefore, we have Ext 1 (, ) = ℂ 5 .
Let  be the zero locus of .Now we compute Ext 1 (, ) as follows: ) .
First, assume that  ⊄ : There are two cases.
There are three cases.

CHAMBERS OF THE STABLE PAIRS SIDE
In this section, we describe the corresponding moduli spaces to the chambers close to the hyperbola from the right.First, we need the following lemma that gives a condition under which we can realize what components can survive all the way to the large volume limit: Lemma 6.1.Suppose that  0 of a general object in an irreducible component created by a wall is an ideal sheaf.Then this irreducible component survives all the way up to the moduli space of stable pairs.Proof.As stability is an open condition, we only need to show this for one object that is created after each wall.If an object  is destabilized by a subobject  ′ , there is an injection  ′ ↪  in Coh , (ℙ 3 ) that induces an injection  0 ( ′ ) ↪  0 ().By assumption,  0 of a general object created by the wall is an ideal sheaf .On the other hand,  0 of the destabilizing subobjects are all of the form   (−), by Theorem 3.10.But we have Hom(  (−), ) = 0, therefore the induced map on  0 is not injective, which is a contradiction.This implies the result.□ We also need the following lemma: Lemma 6.2.Suppose that  is a sheaf with ch 0 () = 1 and ch 1 () = 0, such that it fits into   (−1) ↪  ↠   (−), for   an ideal sheaf of a subscheme  transverse to the plane , and  a positive integer.Then  is an ideal sheaf of a curve.
Proof.First, we observe that  *  (  ) =  ∩∕ .To show that  is an ideal sheaf of a curve, we need to show that it is torsion-free.We know that any subsheaf of   (−) contains a subsheaf of the form   (−) for some  < .It is enough to show that such   (−) does not lift to a subsheaf of .Using the identification Ext 1 (  (−), Hence,   (−) does not lift to a subsheaf of .□ Proposition 6.3 [26,Proposition 3.14].The moduli space for the first chamber below the first wall, which is given by all the extensions of two objects  ℙ 3 (−2),   (−3), is empty.
Let  1 be the first nonempty moduli space for the next chamber, which appears after crossing the smallest wall ⟨ ℙ 3 (−2),   (−3)⟩.In the following proposition from [26], we see that this moduli space gives a compactification of objects corresponding to (2,3)-complete intersection curves in ℙ 3 : Proposition 6.4 [26,Proposition 3.15].The moduli space  1 is a ℙ 15 -bundle over ℙ 9 .More precisely, the complement of (2,3)-complete intersections in  1 are parameterized by the pairs (, ) where  =  ∪  ′ is a union of two planes and  is a conic in one of the two planes.The associated objects are nontorsion free sheaves  given by   (−4) ↪  ↠   2 (−1), for a conic  2 .The generic element with  1 of length 0, is given by an ideal sheaf of (2,3)-complete intersection curves.
Proof.Again the only remaining part that was not needed in [26], is the description of the generic element as an ideal sheaf: this is easily obtained by noticing that H 1 (  2 (−1)) = 0, and then taking the long exact sequence of the defining sequence   2 (−1) ↪  ↠   (−4) and applying Lemma 6.2 to  =  2 in the general case.□ Let  2 is the space parameterizing flags  2 ⊂  ⊂ ℙ 3 where  is a plane and  2 a 0-dimensional subscheme of length 2. The next wall crossing was considered in [26]: Theorem 6.8 [26,Corollary 4.4].The moduli space  3 for the next chamber consists of two irreducible components: one is 2 that is birational to  2 ; the other is a new component,  ′ 3 that is a ℙ 17 -bundle over (2, 4) ×  2 .The latter generically parameterizes the union of a line and a plane quintic together with a choice of two points on the quintic.For any general element  in the new component,  0 () is an ideal sheaf, and  1 () ≠ 0.
Proof.Again the only remaining part that was not needed in [26], is the description of the generic element as an ideal sheaf: this is easily obtained by noticing that  1 (  (−1)) = 0, and then taking the long exact sequence of the defining sequence ), and applying Lemma 6.2 to  =  in the general case.The claim  1 being nonzero is obtained in a similar way and noticing that  1 (  *  ∨  2 (−5)) ≠ 0. □ Let  4 be the moduli space for the next chamber.Let  be the universal line over  (2,4), and   is the space parameterizing flags   ⊂  ⊂ ℙ 3 where  is a plane and   a 0-dimensional subscheme of length .Proposition 6.9.The moduli space  4 has four irreducible components: 2 ,  ′ 3 ,  ′ 4 , and  ′′ 4 .The first two are birational to their counterparts in  3 .The component  ′ 4 is a ℙ 18 -bundle over  ×  1 , and it generically parameterizes the union of a line in ℙ 3 together with a choice of a point on it, and a plane quintic intersecting the line, together with a choice of a point on it.The component  ′′ 4 is a ℙ 19 -bundle over (2, 4) ×  1 , and it generically parameterizes disjoint unions of a line in ℙ 3 and a plane quintic together with a choice of a point on it.For both new components,  0 of the generic element is given by an ideal sheaf, and  1 ≠ 0 of length 2.
To understand the objects in the new components more precisely, for any  ∈ Ext 1 (, ), we first want to understand its image in Hom( 0 (),  1 ()) to see for which elements we get a nonzero map.Let  be the zero locus of .We notice that we have the exact triangle Applying RHom( 0 () =   (−5), −), we have: We consider the stratification of  ×  1 , given by dim(Ext 1 (, )), and consider a general object in each stratum: (1)  ⊄ , the zero locus of  is not  ∩  and  1 ≠  ∩ : In this case, using Lemma 4.1, we have and Hom( 0 (),   ) = 0, and therefore any class in Ext 1 ( 0 (), ) induces zero map in Hom( 0 (),  1 ()).Moreover, we have and therefore the image is exactly given by quintics containing  ∩  and  1 .Now, from as the connecting map is zero,  1 () have length 2.Moreover, the top row is a short exact sequence 0 →   (−1) ↪  0 () ↠   (−5) → 0. The induced extension class corresponds to quintics containing the intersection point.Using Lemma 6.2, this means that for a generic object in Ext 1 (, ), the sheaf  0 () is ideal sheaf of the union of a line in ℙ 3 and a plane quintic intersecting in  ∩ .This together with the description of  1 () implies the result for  ′ 4 .
( As im() = im(), the composition • is zero.Therefore, general  ∈ Ext 1 (, ) induces the zero connecting map in the above diagram 6, that is, it gives the following short exact sequences and Therefore, the plane quintic is smooth, and using Lemma 6.2, for  ∈ Ext 1 (, ) general, we have  0 () =  ∪ 5 corresponds to the connected union of the quintic with  intersecting in a node, and so the curve  ∪  5 is Gorenstein.Thus, we can first apply Lemma 2.9 to realize  as a stable pair, and then apply [24, Proposition B.5] to identify the stable pair  with  0 () =  ∪ 5 , with a length 2 subschemes of  ∪  5 .But from the second short exact sequence above,  1 () is supported at , and therefore this length 2 subscheme is supported at .On the other hand, any such subscheme is in the closure of the locus of  [2] ( ∪  5 ) corresponding to one point on  and one point on  5 .This implies that every object in the ℙ 20 -bundle is in the closure of the ℙ 18 -bundle over  ×  1 .□ Let  5 , be the moduli space for the next chamber.Let  4 ⊂  be a plane quartic and  a thickening of a line  ⊂ .The other notations as defined right before Proposition 6.9: Proposition 6.10.The moduli space  5 has seven irreducible components: The first four are birational to their counterparts in  4 .The component  ′ 5 is a ℙ 19 -bundle over ( × (2,4)  ) × (ℙ 3 ) * , and it generically parameterizes the union of a line in ℙ 3 together with a choice of two points on it and a plane quintic intersecting the line.The component  ′′ 5 is a ℙ 20 -bundle over  × (ℙ 3 ) * , and it generically parameterizes the disjoint unions of a plane quintic and a line in ℙ 3 together with a choice of a point on it.The component  ′′′ 5 is a ℙ 21 -bundle over  2 , and it generically parameterizes the union of a plane quartic with a thickening of a line in the plane.For  ′ 5 and  ′′ 5 , any generic element has  0 an ideal sheaf, and  1 nonzero.In  ′′′ 5 , any generic element is of the form  ∪ 4 .
To understand the objects in the new components more precisely, for any  ∈ Ext 1 (, ), we want to understand its image in Hom(,  1 ()) to see for which elements we get a nonzero map.Let  ∪  ′ be the zero locus of .We notice we have the short exact sequence Now, there are three cases: (1)  ⊄  and zero locus of  does not contain  ∩ : In this case, we have Ext 1 (, )= H 0 ( ∩ ( 5)) = ℂ 20 (Lemma 4.6) and Hom(,  1 ()) = 0, and therefore any class in Ext 1 (, ) induces zero map in Hom(,  1 ()).Now from as the connecting map is zero,  1 () have length 2, and the top row is a short exact sequence 0 →   (−1) ↪  0 () ↠   (−5) → 0. The extension class Ext 1 (  (−5),   (−1)) = H 0 ( ∩ (5)) corresponds to quintics containing the intersection point.Using Lemma 6.2, this means for a generic choice in Ext 1 (, ), the sheaf  0 () is the ideal sheaf of the union of a line in ℙ 3 and a plane quintic intersecting the line.This together with the description of  1 () gives the result about  ′ 5 .
Surjectivity of  implies that a general class in Ext 1 (, ) induces a surjective connecting map  →  1 ().This means that  1 () = 0, for any general element  in the component.Therefore, we have ker(connecting map) = ker(  (−5) ↠  ∪ ′ ) =  ∪ ′ ∕ (−5), and thus any general element  in the component fits into Now, let  be the double line obtained by thickening , with tangent direction of infinitesimal thickening contained in the plane at ( ∩  4 ) ∪  ∪  ′ .From  ∪  4 ⊂  ∪ , we get On the other hand, considering the composition  =  ∪ 4 ↠  ∪ ′ ∕ (−5) ↪   (−5) gives the exact triangle But ( →   (−5)) ≅ ( ℙ 3 (−1) →   (1)) = : Let  [1] be the cone of the composition  →   (−5).From the octahedral axiom, we get an exact triangle . Then the original exact triangle we started with implies Therefore, our  also arises via a class in Ext 1 (, ).This means for a generic class in Ext 1 (, ), the sheaf  is the ideal sheaf of a plane quartic with a thickening of the line .This completes the result about  ′′′ 5 .□ Let  6 , be the moduli space for the next chamber.The other notations as defined right before Proposition 6.9: Proposition 6.11.The moduli space  6 has eight irreducible components: , and  ′ 6 .The first seven are birational to their counterparts in  5 .The component  ′ 6 is a ℙ 21 -bundle over  6 , and it generically parameterizes plane degree 6 curves together with a choice of six points on it.For any general element  in the new component,  0 () is an ideal sheaf   6 , where  6 is a plane sextic curve, and  1 ≠ 0.

CHAMBERS OF THE HILBERT SCHEME SIDE
In this section, we look at the loci created by walls ⟨  2 (−1),   (−5)⟩, ⟨  1 (−1),   1 ∕ (−5)⟩ and ⟨  (−1),   2 ∕ (−5)⟩.We denote by   , the parameterizing space of a plane, a line, and  floating points.Note that in the proofs below, as we use corresponding statement in Section 5 regarding ext-calculation (with precise reference in the statements below), we end up with expressions like Hom(   ∕ (−),    ), where   ,   are 0-dimensional subschemes.But Hom(   ∕ (−),    ) is the same as H 0 (   ∕ ()), for some 0-dimensional subscheme   , which parameterizes degree  plane curves together with some extra points whose length and intersection information come from the proof of corresponding lemmas in Section 5.As this is just H 0 (  (5)), it corresponds to a plane quintic in  containing  =  ∩ ; this induces a ℙ 19 -bundle over a locus that parameterizes a plane, a line and two floating points.Hence, the component has dimension 19 + (3 + 4 + 2 × 3) = 32.Similarly, the nongeneric cases give loci of dimensions 28, 29 in the closure of the generic case.
Similarly, for  ⊂ ,  ′ ⊄  and ,  ′ ⊂ , as in Lemma 5.1, we get ℂ 21 and ℂ 22 , respectively.The first one corresponds to a plane quartic and a double line in the plane together with one point on the line and one floating point outside the plane, which gives a locus of dimension 20 + (3 + 2 + 2 + 3) = 30.The second one corresponds to a plane quartic and a double line in the plane together with two points on the line, which gives a locus of dimension 21 + (3 + 2 + 2 + 2) = 30.
In both cases as we have plane curves, we have local complete intersections, and hence by using [6, Theorem 1.3], we can detach the point from the curve and deform it to two floating points outside the plane; hence both loci are in the closure of the generic case, that is, in ℋ 2 .□ Proposition 7.2.The locus ℋ 1 created by the wall ⟨  1 (−1),   1 ∕ (−5)⟩ is a ℙ 20 -bundle over  1 , of dimension 30 and generically parameterizes the disjoint union of a line in ℙ 3 and a plane quintic together with 1 floating point.
This corresponds to a plane quintic in  containing  =  ∩  and not containing  1 ; this induces a ℙ 19 -bundle over a locus which parameterizes a plane, a line a double point in the plane and another floating point.Hence, the component is contained in the closure of ℋ 2 , and has dimension 19 + (3 + 4 + 2 + 3) = 31.□ In the statement of the following proposition, we refer to the part (3) of Proposition 6.10, for the precise meaning of thickening.

Proposition 7.3. The locus ℋ ′
created by the wall ⟨  (−1),   2 ∕ (−5)⟩ is of dimension 28 and generically parameterizes the union of a plane quartic with a thickening of a line in the plane, which is the same as  ′′′ 5 (Proposition 6.10).
(2) Ext 1 (  2 ∕ (−5),   (−1)): in this case, as in the proof of Lemma 5.3, This corresponds to a plane quintic in  not containing  =  ∩ ; this induces a ℙ 20 -bundle over a locus that parameterizes a plane, a line and a point in the intersection and another point in the plane.Hence, this is contained in the closure of ℋ This corresponds to a plane quintic in  containing  =  ∩  and not containing  2 ; this induces a ℙ 19 -bundle over a locus that parameterizes a plane, a line, and two points in the plane.Hence, the component is contained in the closure of ℋ

THE SPACE OF STABLE PAIRS
In this section, we give a full description of the irreducible components of the moduli space of PT stable pairs by summarizing the results in the previous section on the description of the associated moduli spaces to the chambers.
F I G U R E 2 Walls in Bridgeland stability space (in the ,  plane with  = 0.8).Note that we are only interested in the region close enough to the hyperbola so that there is no other wall appears in the region.
Proof.This is obtained from Propositions 6.4, 6.7, 6.9, 6.10, and 6.11 and Lemma 6.1.We notice that all the loci appearing after each wall are new irreducible components as they have the maximal dimensions and cannot be considered as a subset of the previous components.□ Now, we summarize the birational description of the components of the intermediate moduli spaces: ).The image of this path (outside of the walls) describes the moduli spaces of semistable objects with the fixed Chern character (1, 0, −6, 15) for the corresponding chambers that are as follows: ( 0 ): the empty space, ( 1 ): only contains the main component.( 2 ): the blow up of  1 along the smooth locus (ℙ 3 ) * × (ℙ  5), ( 6) and (7)  Proof.This is obtained from Propositions 6.4, 6.7, 6.9, 6.10, and 6.11, and Lemma 6.1.The birational description of the wall between  2 and  3 is followed by [26].□ Remark 8.3.From Lemmas 4.2, 4.3, 4.5, 4.6, 4.7, the destabilizing loci for each of the moduli spaces corresponding to a chamber along the path on the right-hand side in Figure 2 are described as follows.
( 4 ): There is a destabilizing locus with two strata in  4 : one is  × (ℙ 3 ) * of dimension 8, and the other is a ℙ 1 -bundle over  2 of dimension 8.
( 5 ): There is a destabilizing locus with three strata in  5 : one is a 14-dimensional locus parameterizing six points on a conic, the second locus is a ℙ 1 -bundle over a 12-dimensional locus parameterizing five points on a line, and the third locus is a ℙ 2 -bundle over an 11-dimensional locus parameterizing six points on a line.

THE HILBERT SCHEME
In this section, after describing the DT/PT wall and crossing it, we describe the Hilbert scheme  6−3 (ℙ 3 ).We note that using results in [14], we can interpret the DT/PT wall-crossing formula as a wall-crossing in the Bridgeland stability space (similar statements for different types of stability conditions can be found in [2, Section 6] and [28,Section 4.3]).
Proof.This is obtained from Castelnuovo inequality.□ Note that the constant term of the Hilbert polynomial for a curve is 1 −   + , where  is the number of floating or embedded points.Thus, as  ⩾ 0, for a Cohen-Macaulay curve  of Hilbert polynomial 6 − 3 we must have 1 −   () ⩽ 1 −   +  = −3, which means 4 ⩽   ().Combining this with Proposition 9.1 implies that for a curve of Hilbert polynomial 6 − 3, we have 4 ⩽   ⩽ 10, and it has   − 4 floating or embedded points.
Description of the hyperbola as an actual wall.Recall that the hyperbola is given by Im( ,, ) = 0, and so to describe the hyperbola as an actual wall, we need to find some objects  such that Im( ,, ch()) = 0. Lemma 9.2.As an actual wall, the hyperbola ℍ is given by ⟨   ,   [−1]⟩ for 1 ⩽  ⩽ 6, where   is a Cohen-Macaulay curve of degree 6 and genus 4 + , and   is a torsion sheaf of length .
Proof.For objects with Im( ,, ch()) = 0, semistability does not change as  varies; in particular, we can let  → +∞ and apply Lemma 3.4 to  [1].Therefore,  0  () is  , -semistable, and  1  () is a torsion sheaf   of length .Notice that we have ch( 0  ()) = (1, 0, −6, 15 + ), which is the Chern class of the ideal sheaf of a genus 4+i sextic curve   .By Proposition 9.1, the length cannot be more than 6.Notice that   is a Cohen-Macaulay curve: Otherwise, there would be a 0-dimensional subsheaf  of    , which induces an injection [−1] ↪    .But    is stable; so this is a contradiction.□ Finally, we give a description of the Hilbert scheme and prove Theorem 1.2: Theorem 9.3.Theorem 1.2 holds.
Proof.Theorem 8.1 describes the eight components of the space of stable pairs.We notice that crossing the DT/PT wall (described in Lemma 9.2) from the right side to the left side changes the heart from Coh , (ℙ 3 ) to Coh , (ℙ 3 )[−1].The component ℋ  (which is birational to the main component) and the component ℋ ′  (which is birational to  ′′′ 5 ) survive after crossing this wall: we just need to check this for one object in each; let  be either an ideal sheaf of a (2,3)-complete intersections or an ideal sheaf of the union of a plane quartic with a thickening of a line in the plane (see Propositions 6.4 and 6.10).Let   be a Cohen-Macaulay curve of degree 6 and genus 4 + , where 1 ⩽  ⩽ 6.As neither a (2,3)-complete intersection curve nor the union of a plane quartic with a thickening of a line in the plane can be contained in a   , we have Hom(   , ) = 0. Therefore,  cannot get destabilized at this wall.
We notice that apart from these two components of genus 4 Cohen-Macaulay curves, other six components in the space of stable pairs are destabilized once we cross the DT/PT wall: each object  in those components has  1 () ≠ 0 (see Propositions 6.8, 6.9, 6.10, and 6.11), and the short exact sequence  0 () →  →  1 ()[−1] destabilizes any object in those components.
Note that the underlying Cohen-Macaulay curves that are parameterized by the generic elements of the components remain the same in both the Hilbert scheme and the space of stable pairs (as the support of  , for a given stable pair ( , )).Therefore, the new components in the Hilbert scheme generically parameterize ideal sheaves of curves with underlying Cohen-Macaulay curves either the disjoint union of a line in ℙ 3 and a plane quintic, or the union of a line in ℙ 3 and a plane quintic intersecting the line, or a plane sextic.They have 1,2, or 6 floating/embedded points, respectively (see Propositions 7.2, 7.1, 7.3, and 6.11).The dimension of ℋ 1 is obtained from Proposition 7.2.
As no object in the components ℋ  and ℋ ′  gets destabilized after crossing the DT/PT wall, the dimensions and irreducibility of them are clear from the stable pairs side.For ℋ 1 , there is only one point involved and the underlying curve is a disjoint union of a line and a plane quintic that is a locally complete intersection curve of codimension 2; hence [6, Theorem 1.3] implies that we can always detach the point from the underlying curve in this case.Therefore, ℋ 1 is irreducible.
As for ℋ 2 , Corollary 7.4 implies the irreducibility.□ Notice that our wall-crossing picture confirms the classical result below by Hartshorne for canonical genus 4 curves, as the only nonplane curves that occur as underlying curves of independent components have arithmetic genera 4,5,6: Theorem 9.4 [11,Theorem 3.3].Let  be a locally Cohen-Macaulay curve in ℙ 3 , of degree  ⩾ 3, which is not contained in a plane.Then g() ⩽ ( − 2)( − 3) 2 .

A C K N O W L E D G E M E N T S
First and foremost, I would like to thank Arend Bayer for suggesting the problem, generosity of time, invaluable comments on preliminary versions, and helpful discussions.Special thanks to Benjamin Schmidt for the great suggestion to work on the stable pairs side, and for helpful discussions.Special thanks to Qingyuan Jiang for discussions on the irreducibility of projectivization loci.Special thanks to Dominic Joyce for the last-minute comments.This work benefited from useful discussions with Antony Maciocia.I am grateful for comments by Aaron Bertram, Tom Bridgeland, Ivan Cheltsov, Dawei Chen, Daniel Huybrechts, Marcos Jardim, Dominic Joyce, Emanuele Macrì, Scott Nollet, Balázs Szendröi, Richard Thomas, Yukinobu Toda, and Bingyu Xia.I am also grateful for careful reading, and several helpful comments and suggestions by the anonymous referee.with GeoGebra, and some computations were checked using Macaulay2 [10].We acknowledge that Loughborough University has covered the gold open access costs.

J O U R N A L I N F O R M AT I O N
The Proceedings of the London Mathematical Society is wholly owned and managed by the London Mathematical Society, a not-for-profit Charity registered with the UK Charity Commission.All surplus income from its publishing programme is used to support mathematicians and mathematics research in the form of research grants, conference grants, prizes, initiatives for early career researchers and the promotion of mathematics.
The author was supported by the University of Edinburgh Scholarships (PCDS and the School of Mathematics Scholarships), ERC Starting Grant WallXBirGeom, Number: 337039, and ERC Consolidator Grant WallCrossAG, Number: 819864.This material is partially based upon work supported by the NSF under Grant Number: DMS-1440140 while the author was in residence at the MSRI in Berkeley, California, during the Spring 2019 semester.The author was also supported by the EPSRC Grant EP/T015896/1, during final edits while in residence at Loughborough University.The revisions for journal publication were done while the author was supported by UKRI Grant Number: EP/X032779/1 (in lieu of an MSCA Grant).The pictures were generated 3 The pair (Coh  (ℙ 3 ),  tilt , ) is a weak stability condition also known as tilt-stability condition.
We note that Propositions 7.2, 7.1, and 7.3 imply that all the potential loci are contained in the closure of ℋ 2 .□ in Theorem 8.1, respectively.