Interpolation and moduli spaces of vector bundles on very general blowups of the projective plane

In this paper, we study certain moduli spaces of vector bundles on the blowup of the projective plane in at least 10 very general points. Moduli spaces of sheaves on general type surfaces may be nonreduced, reducible and even disconnected. In contrast, moduli spaces of sheaves on minimal rational surfaces and certain del Pezzo surfaces are irreducible and smooth along the locus of stable bundles. We find examples of moduli spaces of vector bundles on more general blowups of the projective plane that are disconnected and have components of different dimensions. In fact, assuming the SHGH Conjecture, we can find moduli spaces with arbitrarily many components of arbitrarily large dimension.


Introduction
In this paper, we study certain moduli spaces of vector bundles on the blowup of P 2 C at n ≥ 10 very general points.We describe their geometry very explicitly and find that they can have many connected components of different dimensions.In fact, assuming the Segre-Harbourne-Gimigliano-Hirschowitz (SHGH) conjecture (see Conjecture 2.2), we find that they can have arbitrarily many components of arbitrarily large dimension.This is in strong contrast to the behavior of moduli spaces on minimal rational surfaces and certain del Pezzo surfaces.To the best of our knowledge, our examples are the first time these phenomena have been observed on rational surfaces.
Throughout the paper, let X be the blowup of P 2 at n very general points p 1 , . . ., p n .Let H denote the pullback of the class of a line.Let E i denote the exceptional divisor lying over p i , and set E = n i=1 E i .The canonical divisor K = K X = −3H + E has self-intersection K 2 = 9 − n.The geometry of the surface X changes dramatically based on the sign of K 2 .For n ≤ 8, the divisor −K is ample and X is a del Pezzo surface.For n = 9, the divisor −K is still effective.On the other hand, for n ≥ 10, the divisor −K is not even effective.Our results will show that this dramatic change is also reflected in the behavior of moduli spaces of vector bundles on X.
We polarize the surface X by an ample divisor of the form A t = tH − E. When n ≥ 10, Nagata conjectures that A t is ample if t > √ n; see [Nag59].Set B = A √ n = √ nH − E to be the conjectural nef ray.Nagata's conjecture is only known when n is a perfect square, but the full Nagata conjecture is a consequence of the SHGH conjecture.We record the numerical invariants of a vector bundle by (r, c 1 , χ), where r is the rank, c 1 is the first Chern class, and χ is the Euler characteristic.We let M X,A t (r, c 1 , χ) denote the moduli space of A t -semistable sheaves with numerical invariants (r, c 1 , χ).
In this paper, we study the moduli spaces M X,A t (2, K, χ).We mostly focus on the case where χ ≥ 1 is positive, with special emphasis on the case where χ is the maximal Euler characteristic of an A t -stable bundle.For 10 ≤ n ≤ 17, this maximal Euler characteristic turns out to be χ = 2. On the one hand, the moduli spaces M X,A t (2, K, χ) are easiest to describe when χ is as large as possible; on the other hand, these spaces are also typically among the most pathological moduli spaces.We show that when n ≤ 9, the moduli spaces M X,A (2, K, χ ≥ 1) are empty for any ample divisor A (see Proposition 3.7).Similarly, if n ≥ 10, then M X,A t (2, K, χ ≥ 1) is empty for t > n 3 (see Proposition 3.8).Hence, we will be interested in these moduli spaces when n ≥ 10 and the polarization is close to the Nagata bound.
Our first main result shows that for any bundle V with invariants (2, K, χ ≥ 1), there is a unique effective divisor D such that V fits in an exact sequence of the form where Z is a zero-dimensional scheme (see Theorem 3.2).We will say that V is a bundle of type D. The moduli spaces M X,A t (2, K, χ) are therefore stratified by the type of a bundle, and different types frequently give rise to different components in moduli spaces.However, the possible types D of stable bundles are extremely special: if there is an A t -stable bundle of type D, then D must satisfy 2B • D < B • K and χ(D) ≥ 1 (see Proposition 3.9).Given an effective D with χ(D) ≥ 1 and 2B • D < B • K, there is a unique value t D where 2A t D • D = A t D • K.A bundle V of type D can only be semistable for polarizations A t with t ≤ t D .As t decreases past t D towards √ n, the moduli spaces M X,A t (2, K, χ) can gain points parameterizing bundles of type D.
When n = 16 or 25, we are able to give the following description of moduli spaces that is unconditional on the Nagata or SHGH conjectures.In these cases, there are only finitely many possible types D of a semistable bundle.Similar arguments could extend these descriptions to higher perfect squares n.
In particular, part (2) gives an explicit example of a reducible moduli space of vector bundles on a rational surface.
On the other hand, for n which is not a perfect square, there are typically infinitely many effective divisors D satisfying χ(D) ≥ 1 and 2B • D < B • K.If we assume the Nagata conjecture, these divisors can be classified by solving a series of Pell's equations; we do this explicitly in Section 4. If we assume the SHGH conjecture, then the types D which could contribute to the moduli spaces have good cohomological properties which makes it possible to completely describe the moduli spaces.For example, if 10 ≤ n ≤ 16, then D is the class of a reduced, irreducible, rigid curve on X (see Theorem 5.2).The next theorem then summarizes our results from Section 6 which describe the structure of the moduli spaces.
(2) Suppose 11 ≤ n ≤ 15.As t decreases past n 3 , M X,A t (2, K, 2) acquires a component isomorphic to P n−11 .For 11 ≤ n ≤ 12, this component persists without modification as t decreases to √ n.For 13 ≤ n ≤ 15, this component is blown up at n points as t decreases past n−2 3 and then persists without modification as t decreases to √ n.
(3) For every nontrivial, nonexceptional divisor D satisfying χ(D) ≥ 1 and 2B • D < B • K, M X,A t (2, K, 2) acquires a new component isomorphic to P −χ(2D−K)−1 as t decreases past t D .This component persists without modification as t decreases to √ n. (4) This is a complete description of the components of M X,A t (2, K, 2), and they are all disjoint.
We list the first several components of each of the moduli spaces when 10 ≤ n ≤ 13 in tables in Examples 6.8 and 6.9.A detailed study of the possible divisors D shows that each of the moduli spaces in Theorem 1.3 will have arbitrarily many components of arbitrarily large dimension if t is sufficiently close to √ n.This observation implies the following corollary that applies to moduli spaces with arbitrary Euler characteristic.
Corollary 1.4.Assume the SHGH conjecture, and let 10 ≤ n ≤ 12. Let χ ≤ 2 be an integer, and let k and r be positive integers.There exists an ϵ > 0 such that if When the polarization is fixed, moduli spaces of sheaves on surfaces behave well as χ tends to negative infinity.For example, by a theorem of O'Grady, see [O'G96], the moduli spaces are irreducible, reduced, and normal.However, for arbitrary χ, the moduli spaces can be poorly behaved.For example, moduli spaces of sheaves on general type or elliptic surfaces can be reducible, nonreduced, and even disconnected (see [CH18a,CHK22,FM88,Fri89,Kot89,Mes97,MS11,OVdV86] for some examples).
Let Y be a birationally ruled surface, and let F be the class of the fiber.Let A be a polarization such that (K Y + F) • A < 0. Walter proves that the moduli space M Y ,A (v) is then irreducible provided that it is nonempty; see [Wal98].In particular, all nonempty moduli spaces of sheaves on P 2 , Hirzebruch surfaces and X with n ≤ 6 are irreducible for every polarization; these moduli spaces have been studied in detail (see for example [LeP97,CH21,LZ19]).Similarly, for any rational surface Y , there exist polarizations A satisfying (K Y + F) • A < 0. For example, this is the case on X for A t with t ≫ 0. The nonempty moduli spaces are irreducible on X for such polarizations and have been studied in [Zha22].In contrast, our results show that the irreducibility may fail when Walter's condition is violated.
Our results are in part inspired by questions concerning the topology of moduli spaces.Göttsche, see [Göt90], computed the Betti numbers of the Hilbert schemes Y [n] of n-points on a smooth projective surface Y and observed that they stabilize as n tends to infinity.In fact, the Betti numbers monotonically increase as n increases.Coskun and Woolf, see [CW22], conjectured that the Betti numbers stabilize for moduli spaces of sheaves in general as χ tends to negative infinity and that the stable Betti numbers are independent of the rank and the polarization.They proved the conjecture for moduli spaces on rational surfaces when the polarization satisfies K Y • A < 0 and the moduli space does not contain any strictly semistable sheaves.Our examples show that in the absence of the assumption K Y • A < 0, the topology of the moduli spaces can be fairly complicated.In particular, even on rational surfaces, the Betti numbers of moduli spaces are not monotonically increasing as χ decreases.Examples of this phenomenon were previously known on certain elliptic and general type surfaces; see [CHK22,Kot89,OVdV86].

Organization of the paper
In Section 2, we recall the Nagata and SHGH conjectures and collect basic facts concerning very general blowups of P 2 .In Section 3, we define the type of a bundle V with character (2, K, χ ≥ 1) and show that it is unique.In Section 4, we study effective divisors D that satisfy χ(D) ≥ 1 and 2B • D < B • K and explain how to classify them.In Section 5, we study the cohomology of such D and associated divisors which are relevant to the calculation of the tangent space of the moduli space.In Section 6, we classify the components of the moduli spaces and prove our main theorems.Finally, in Section 7, we study the cases where n is a perfect square, where we can make our results independent of the SHGH conjecture.

Notation
Throughout the paper, we work over the field C of complex numbers.Let X be the blowup of P 2 at n very general points p 1 , . . ., p n .The Picard group of X is where H is the pullback of a line in P 2 and E 1 , . . ., E n are the exceptional divisors.We have For brevity, we let E i 1 ...i k := k j=1 E i j .For example, E 123 = E 1 + E 2 + E 3 .We write O = O X and K = K X for the trivial bundle and canonical bundle, respectively, and note that We compute K 2 = 9 − n.

Ample divisors
In this paper, we will study polarizations of X of the form A t = tH − E, where t is a real number.Since The famous conjecture of Nagata claims that the converse is true once n ≥ 10.
Nagata shows the conjecture is true when n is a perfect square.For other n, partial results towards the Nagata conjecture can be proved by exhibiting ample divisors A t with t as close to √ n as possible.For α ≥ √ n, we will call the statement that A α is nef the α-Nagata conjecture.

Linear series and the SHGH conjecture
Consider a divisor class D = dH − i m i E i on X with d ≥ 0. In general, it is a highly nontrivial problem to compute the dimension of the linear series |D| or, equivalently, the cohomology of the line bundle O(D).The Segre-Harbourne-Gimigigliano-Hirschowitz (SHGH) conjecture provides an algorithm to compute this dimension.We will call Har86,Gim87,Hir89]).The divisor D is special if and only if it contains a multiple (−1)-curve in its base locus.
If n ≤ 9, then the SHGH conjecture is true (see e.g.[CM11]), so the conjecture becomes most interesting for n ≥ 10.
Remark 2.3.The following consequences of the conjecture are frequently useful: (1) If D is a reduced curve on X, then D is nonspecial and χ(D) ≥ 1.
(2) If D is a reduced and irreducible curve on X with D 2 < 0, then D is a (−1)-curve.Indeed, by Riemann-Roch, a large multiple kD has χ(kD) < 0, but kD is effective.Therefore, kD is special, and the only possibility is that D is a (−1)-curve.
(3) Suppose D = dH − mE is a homogeneous divisor class.If n ≥ 10, then D is nonspecial.
(4) The SHGH conjecture implies the Nagata conjecture.For suppose that t > √ n and A t is not ample, so that by the Nakai-Moishezon criterion, there is an irreducible curve class C = dH − i m i E i with C • A t < 0. Then C is nonspecial, and if we permute the exceptional divisors, we get additional nonspecial classes.Summing over the symmetric group, we can obtain an effective homogeneous √ n and D 2 < 0. Then large multiples kD have χ(kD) < 0 and they are effective, contradicting that they are nonspecial by (3).
Since the full SHGH conjecture is quite challenging, it is useful to have results which make partial progress towards the SHGH conjecture.Here there are two main flavors of result: either one can bound the multiplicities m i (see e.g.[DJ07]

Moduli spaces of vector bundles
Let A be an ample divisor on X.Let V be a torsion-free sheaf on X with Chern character v.In this paper, it will be convenient to record v = (r, c 1 , χ) by the rank r, the first Chern class c 1 (V ), and the Euler characteristic χ(V ).The A-slope µ A (V ), the Hilbert polynomial P A,V (m), and the reduced Hilbert polynomial p A,V (m) are defined by for m ≫ 0. Gieseker, see [Gie77], and Maruyama, see [Mar78], constructed projective moduli spaces M X,A (v) parameterizing A-semistable sheaves.We refer the reader to [HL10] and [LeP97] for the properties of these moduli spaces.

Types of bundles
Throughout this section, we let v be the Chern character v = (r, c 1 , χ) = (2, K, χ), where χ ≥ 1 is a positive integer.The first main result in the paper shows that the positivity assumption on χ allows us to neatly classify vector bundles of character v into various types.These will give rise to distinct components in moduli spaces.Definition 3.1.Let v = (2, K, χ) with χ ≥ 1, and let D ∈ Pic(X) be a (possibly trivial) effective divisor class on X satisfying χ(D) ≥ 1.A vector bundle V of character v has type D if it fits in an exact sequence of the form

Observe that χ(K(−D)) = χ(O(D)) and χ(K(−D)⊗I Z ) = χ(O(D))−l(Z)
, so the assumption on the length of Z is necessary to give χ(V ) = χ.The divisor D must also have χ(D) ≥ 1 in order for χ ≥ 1 to be possible.We first show that the type exists and is unique.Theorem 3.2.Let v = (2, K, χ) with χ ≥ 1.Any vector bundle V of character v is of type D for exactly one effective divisor class D.
Proof.First we show that a type exists.Since χ(V ) ≥ 1, at least one of h 0 (V ) or h 2 (V ) is nonzero.
Suppose h 2 (V ) 0. Then h 0 (V * ⊗ K) 0, so Hom(V , K) 0. Pick a nonzero map V → K, and let F ⊂ K be its image.Then F is of the form K(−D) ⊗ I Z for an effective divisor D and a zero-dimensional scheme Z ⊂ X.Consider the kernel Basic facts about homological dimension and the Auslander-Buchsbaum formula imply that G is locally free since it is the kernel of a surjective mapping from a vector bundle to a torsion-free sheaf on a smooth surface (see [HL10, Section 1.1, p. 4]).By Chern class considerations, we deduce G O(D), and V has type D.
If instead h 0 (V ) 0, we reduce to the previous case.Since H 0 (V ) Hom(O, V ), we pick a nonzero (hence injective) homomorphism O → V and consider its cokernel Let T be the torsion subsheaf of F, so we have an exact sequence The first Chern class of T is a positive Z-linear combination of any curves in the support of T , so it is a (possibly empty) effective divisor D. Then G is a rank 1 torsion-free sheaf with c 1 (G) = K − D, so it is of the form G = K(−D) ⊗ I Z for a zero-dimensional scheme Z. Then h 2 (G) 0, so h 2 (F) 0. Hence, h 2 (V ) 0, and we are reduced to the previous case.
For the uniqueness, suppose V has type D and type D ′ .Twisting the type D exact sequence by −D shows that V (−D) has a section.But twisting the type D ′ exact sequence by −D gives In particular, we have the following corollary.
Proof.The bundle V has type D for some D, and from the defining sequence, we see that V has the required cohomology.□ Remark 3.4.In the special case where χ = 1, all three cohomology groups H 0 (V ), H 1 (V ), and H 2 (V ) must be nonzero.For example, let us discuss what happens when n = 10 and χ = 1.The type O bundles fitting into sequences of the form is any component whose general member is a vector bundle, then every sheaf in that component must have nonvanishing cohomology in every degree.This exhibits a strong failure of the "weak Brill-Noether" property for these spaces, in stark contrast with known results for minimal rational surfaces and del Pezzo surfaces (see for example [CH18b,CH20,LZ19]).
The type of a bundle V can be determined cohomologically.
Corollary 3.5.Let V be a bundle of character v = (2, K, χ) with χ ≥ 1. Partially order Pic(X) by the relation Proof.Suppose V has type D. By definition, D is in the set.Let D ′ be any effective divisor with h 0 (V (−D ′ )) 0. Twisting the type D sequence by −D ′ gives This cohomological definition of type restricts the ways in which bundles of one type can specialize to another.
Corollary 3.6.Let v = (2, K, χ) with χ ≥ 1. Suppose V s /S is a flat family of vector bundles on X of character v, parameterized by an irreducible base S, and that V s has type D for a general s ∈ S.

Preliminary results on stability
The next result shows that once we are concerned with stability, the spaces M A (v) are not interesting until 10 or more points are blown up.
On the other hand, for n ≥ 10, we focus on polarizations of the form A t = tH − E.Here we find that the spaces 3 , we have A t • K < 0, and we proceed as in the previous proof.□ Next we investigate the stability of bundles of type D. The existence of a stable bundle of type D imposes strong restrictions on D. Proposition 3.9.Suppose V is a bundle of character v and type D, and assume there is a polarization A t 0 such that V is µ A t 0 -semistable.Then we must have and there is a unique polarization A t D such that O(D) and K(−D) have the same slope.It satisfies Proof.Suppose V is µ A t 0 -semistable and that it fits in an exact sequence We have t 0 ≤ n 3 by Proposition 3.8, and we must have Now for variable t, consider the relationship between 2A t • D and A t • K.Both quantities vary linearly in t.
For t = t 0 , we have Due to the proposition, we study curves D satisfying the inequality 2B • D < B • K in more detail in the next section.

Effective divisors D satisfying 2B • D < B • K
In this section, we classify the effective divisors D satisfying 2B • D < B • K and χ(D) ≥ 1, at least when n is small.For a complete answer to this question, we will need the Nagata conjecture to know that B is nef.However, with a little more care, we can avoid using the Nagata conjecture and instead classify divisors satisfying 2A t • D ≤ A t • K whenever A t is known to be an ample divisor.Notice that the inequality

General restrictions on D
In this subsection, we prove several preliminary results which restrict the possibilities for a divisor D.
Proposition 4.1.Suppose D is an effective divisor and A t is an ample divisor with Expanding and rearranging, we get 4D • (D − K) < −K 2 , and so The required inequality follows from the Riemann-Roch formula We say that the multiplicities are balanced if |m i −m j | ≤ 1 for all i, j.
On the other hand, if D is not balanced, we construct a sequence D = D 0 , . . ., D k of divisors by iteratively increasing one of the smallest multiplicities by 1 and decreasing one of the largest multiplicities by 1, stopping when we arrive at a balanced divisor D k .We say that D is k steps away from having balanced multiplicities.The number k is independent of any choices made in this construction.Lemma 4.3.Suppose D 0 = dH − i m i E i is any divisor and that D 0 is k steps away from having balanced multiplicities.Let D k be the balanced divisor obtained from D 0 as in Definition 4.2.Then Proof.We claim that each step of "rebalancing" the multiplicities increases the Euler characteristic by at least 1.Without loss of generality, suppose that D 0 is not balanced, m 1 − m 2 ≥ 2, and Let χ max be the maximal Euler characteristic among effective divisors D satisfying this inequality.If we put ℓ = χ max − χ(D), then D is at most ℓ steps away from having balanced multiplicities.
Next we quantify how far away a balanced divisor is from having equal multiplicities.
Definition 4.5.Let D = dH − i m i E i be a balanced divisor, so that |m i − m j | ≤ 1 for all i, j.Then there are k copies of some multiplicity m + 1 and n − k copies of multiplicity m.We let ℓ = min{k, n − k}, and we say that D is ℓ steps away from having equal multiplicities.
Proposition 4.6.Suppose D is a balanced effective divisor with χ(D) ≥ 1 and A t is an ample divisor with Suppose that D is ℓ steps away from having equal multiplicities.Then In particular, if 10 ≤ n ≤ 12, then we get χ(D) = 1, ℓ = 0, and D has equal multiplicities.
Proof.Suppose there are 1 ≤ k ≤ n − 1 copies of multiplicity m + 1 and (n − k) copies of multiplicity m, so that the average multiplicity is m + k n .Let E ′ be the sum of the k exceptional divisors with multiplicity m + 1, and let E ′′ be the sum of the remaining n − k exceptional divisors, so that Observe that the sum of the multiplicities in G is zero, so G intersects any divisor with equal multiplicities in 0. In particular, as in the proof of Proposition 4.1, the Hodge index theorem still gives , so we must have the inequality Solving the inequality for ℓ and recalling that by definition ℓ ≤ n/2, we get the required upper bound on ℓ. □ In fact, the results in this section have the following partial converse, which will help us to find all the divisors D. Then This inequality is which factors as a difference of squares The first factor in this product is just 1 Now letting D be the divisor in the statement of the proposition, we form a balanced Q-divisor D ′ with the same average multiplicity as in the proof of Proposition 4.6.As in the proof of Proposition 4.6, we find that 1 8 Our upper bound on ℓ then implies that (2D ′ − K) 2 < 0, and by the argument in the first paragraph,

Ten through twelve points
When 10 ≤ n ≤ 12, it is now a matter of number theory to describe all the possible divisors D.
Theorem 4.8.For 10 ≤ n ≤ 12, we consider the sequence of convergents Here the odd convergents are less than √ n, and the even convergents are greater than √ n.
(1) For any positive odd integer k, we let d k = 1 2 (p k − 3) and m k = 1 2 (q k − 1), and we define a divisor Then there is a positive odd integer k such that D = D k . Proof.
(1) The continued fraction expansion of √ n is as follows: We write a = 6/(n − 9), so that in every case = 3 1 , and for odd k ≥ 3, we have the recurrence Thus for odd k, we have p k ≡ p k−2 (mod 2) and q k ≡ q k−2 (mod 2), so p k and q k are both odd.Therefore, D k is an integral divisor.
To show that D k is effective, we will show that For k = 1, the equality is trivial.From the fact that the continued fraction expansion of √ n has period dividing 2, we can manipulate continued fractions to see that Here the numerator and denominator are already coprime because the matrix 3a+1 9a+6 a 3a+1 has determinant 1, so for k odd, we have the recurrence Then we compute Here the second equality is a direct computation, and the third follows since a = 6/(n − 9).Thus p 2 k − nq 2 k = 9 − n holds in every case.
Next we show that 2B As t decreases to √ n, we get the required inequality.
Thus (x, y) = (2d + 3, 2m + 1) is a positive solution to the generalized Pell's equation √ n (which holds here since 10 ≤ n ≤ 12), every positive solution (x, y) to this equation is of the form (x, y) = (p k , q k ) (see [Sho67,Theorem 20,p. 204]).For k even, we have p 2 k − nq 2 k > 0. Therefore, there must be an odd integer k such that D = D k .□ When 10 ≤ n ≤ 12, we can now give a complete list of all the possible curves D. They are closely related to the odd convergents in the continued fraction expansion of √ n.
Remark 4.9.For n = 10, the convergents in the continued fraction expansion of

Thirteen through seventeen points
For the blowup at 13 to 17 points, divisors D must still be balanced, but they are no longer required to have equal multiplicities.This makes the classification more complicated, but it is still tractable in each case.It becomes necessary to solve several quadratic Diophantine equations, according to how unequal the multiplicities can be.We discuss the cases n = 13 and n = 16 in more detail; the remaining cases can be handled identically.
Example 4.10.Let n = 13.Here we classify the effective divisors D such that χ(D) ≥ 1 and 2A t • D ≤ A t • K for some ample divisor A t .We know that D must have χ(D) = 1 and that D is balanced.Furthermore, D is at most one step away from having equal multiplicities (see Proposition 4.6).Then D has the form and k is one of 0, 1, or 12.We must have Considering the three possible values for k separately gives us three possible Diophantine equations to solve.Conversely, note that χ(D) can be determined from (2D − K) 2 , so that each solution to the Diophantine equations will necessarily give a divisor D with χ(D) = 1.Proposition 4.7 furthermore shows that the solutions with d ≥ 0 and m ≥ 1 give divisors D satisfying 2B • D < B • K, and so 2A t • D ≤ A t • K for some ample A t if we are assuming the Nagata conjecture.
In more detail, we can expand the equation (2D − K) 2 = 9 − n to obtain Specializing to the values k = 0, 1, 12 gives the three associated equations: Quadratic Diophantine equations like this can be solved using Lagrange's method to transform them to generalized Pell's equations.We used the online Alpertron generic two integer variable equation solver (1) to find all the solutions in each case.
(1) https://www.alpertron.com.ar/QUAD.HTM Case k = 0.In this case, there are four fundamental solutions (d, m) = (−3, 0), (0, 0), (0, −1), and (−3, −1) (note that the third and fourth solutions can be obtained from the first two by Serre duality).Given a solution (d, m), new solutions can be obtained by applying the transformation (d, m) −→ (649d + 2340m + 2142, 180d + 649m + 594) or the inverse of this transformation.In this way, each fundamental solution gives rise to infinitely many solutions indexed by the integers.The solutions fit into the following chains: In each case, the geometrically relevant solutions are the ones obtained from the fundamental solutions by applying the recurrence an even number of times.Note that (0, 0) is actually a geometrically relevant solution, corresponding to the divisor E 13 .Up to permuting the exceptional divisors, the corresponding divisors D form two infinite families beginning with In each case, applying the transformation an odd number of times to the fundamental solution gives a solution of geometric significance, and we get two infinite families of divisors beginning with the divisors The method is the same as the method for n = 13: we find several quadratic Diophantine equations and determine all their solutions.However, these Diophantine equations turn out to only have finitely many solutions, so our answer is considerably more concrete.
We need to find all solutions for k = 0, 1, 15.
Note that the pairs (d, m) = (0, 0) and (−3, 0) are solutions for every k.The solution (−3, 0) is not geometrically relevant.On the other hand, (0, 0) corresponds to a sum of k exceptional divisors.This will satisfy 2B In what follows, we ignore these solutions and search for additional solutions.

Twenty-five points
Only the solution (1, 1, 24) is relevant, and it corresponds to the divisors of the form H − E i for some i.In the rest of the paper, we will only need our classification when χ(D) = 2, but we briefly state the classification for χ(D) = 1 for completeness.Here we have to allow for the possibility that D does not have balanced multiplicities since Proposition 4.4 only guarantees that the multiplicities of D are at most one step away from being balanced.If D does not have balanced multiplicities, then the balanced divisor D 1 obtained from D will satisfy χ(D 1 ) ≥ 2 and 2B • D 1 < B • K.But then we must actually have χ(D 1 ) = 2, and D 1 must be of the form H − E i from the previous case.For the divisor D to have χ(D) = 1 and become equal to H − E i after a single rebalancing step, we must have that D is of the form The other possibility is that D does have balanced multiplicities.By Proposition 4.6, the multiplicities of D are at most four steps away from being equal.Using the methods from the previous classifications, we list all the divisors D satisfying χ(D) ≥ 1 and 2B • D < B • K in the following table, ordered so that t D is decreasing.For brevity, we only list divisor classes up to permutations of the exceptional divisors.

Cohomological properties of D and the SHGH conjecture
When we analyze stability, we will need to know various cohomological properties about the divisors D. To analyze these, we will assume the SHGH conjecture.Case 1: D 2 > 0. In this case, 2D is an effective divisor satisfying 2B • 2D < B • K, and This contradicts Proposition 4.1.Case 2: D 2 < 0. In this case, the SHGH conjecture implies D is a (−1)-curve, so D 2 = −1 and D • K = −1.We write  When analyzing the tangent space to the moduli space, we will need to understand the cohomology of O(2D), so we now compute this.(1) (1) By Theorem 5.2, we just need to make sure that D is not a (−1)-curve.If D is a (−1)-curve, so D 2 = −1 and D • K = −1, then as in the proof of Lemma 5.1, we have This implies Proof.Since 2B • D < B • K, we have A t • (2D − K) < 0 for t slightly greater than √ n, so 2D − K is not effective.There is no h 2 because the coefficient of H is positive.Thus χ(O(2D − K)) ≤ 0, and it remains to show that the inequality is strict unless D is trivial and n = 10.We use Riemann-Roch to write Since χ(D) = 1, we know that In order to have χ(O(2D − K)) = 0, we must then have both n = 10 and (2D − K) • (−K) = −1.
When n = 10, we have fully classified the possible divisors D using the odd convergents in the continued fraction expansion of √ 10.Suppose k ≥ 3 is odd and D = D k in the classification of Theorem 4.8.Then Since k ≥ 3, we have q k ≥ 37, so (2D − K) • (−K) is less than −1.□ Remark 5.5.Let 10 ≤ n ≤ 12, and let D k be the divisor of Theorem 4.8.To avoid the special case D = O, we may as well assume k ≥ 3.By a similar analysis, we can give a formula for χ(O(2D k − K)) which shows more explicitly how this quantity grows with k.As in the proof of the lemma, we have We estimate the final term from above as The error in this approximation is only Here in the final step we have used the well-known fact, see [Dav99, Section IV.6], that the convergents of the continued fraction expansion of a real number x satisfy together with the observation that q k+1 is considerably greater than 3 since k ≥ 3. Thus in fact The Euler characteristic χ(O(2D k − K)) has the same growth rate as the denominators of the continued fractions of √ n.

Stability, components, and the SHGH conjecture
Throughout this section, we assume the SHGH conjecture holds, and we prove our main theorems on the components of the moduli spaces M A t (v), where v = (r, c 1 , χ) = (2, K, 2) and 10 ≤ n ≤ 16.These spaces are particularly nice because this is the maximal possible value of the Euler characteristic χ.We will see that this causes the moduli spaces to be smooth and the irreducible components to be disjoint from one another.It also causes every semistable sheaf to be a vector bundle, so the moduli space is stratified by the type of a bundle.Lemma 6.1.Every µ A t -semistable sheaf V with rank 2 and c 1 = K has χ(V ) ≤ 2. In particular, every µ A t -semistable sheaf of character v = (2, K, 2) is a vector bundle.
Proof.Let V be a µ A t -semistable sheaf with rank 2 and c 1 = K, and suppose χ(V ) ≥ 2. Let W = V * * be the double dual.Then there is an exact sequence where T is torsion and (at most) zero-dimensional.Then W is a µ A t -semistable vector bundle with rank 2 and c 1 = K.Furthermore, χ(W ) ≥ χ(V ) ≥ 2, so W has a type D. By Propositions 3.9 and 4.1, we have χ(O(D)) ≤ 1 and therefore χ(W ) ≤ 2. But then χ(W ) = χ(V ) = 2, so χ(T ) = 0 and T = 0 and V = W is a vector bundle.□ For the rest of the section, let 10 ≤ n ≤ 16 and v = (r, c 1 , χ) = (2, K, 2).We let D be a (possibly trivial) effective divisor satisfying χ(D) = 1 and 2B • D < B • K.We denote by V any bundle of character v and type D given by a nonsplit extension

Stability
We next study the stability of V .Stability behaves slightly differently for the trivial type D = O and nontrivial types, where D is actually effective.We focus on the latter case first.
Proof.Suppose that √ n < t < t D and that V is not µ A t -stable.Then there is a saturated line subbundle and the assumption t < t D gives µ A t (O(D)) < µ A t (K(−D)).Then the composition L → V → K(−D) must be nonzero, for otherwise there would be an inclusion L → O(D).Thus L takes the form K(−D ′ ) for an effective divisor D ′ , and D ′ − D must be nontrivial (otherwise V is split) and effective.In particular, by summing a curve of class D ′ − D and one of class D, it follows that D ′ can be represented by a nonintegral curve.However, the inequality (2) If 13 ≤ n ≤ 16, then the same result is true except that there are n points in P Ext 1 (K, O) parameterizing bundles V which are only µ A t -stable if t satisfies n−2 3 = t E 1 < t < t O .These points are given by the images of the inclusions of 1-dimensional spaces Proof.We begin as in the previous proof.If there is a t with √ n < t < t O such that V is not µ A t -stable, then there is a saturated line subbundle of V of the form K(−D ′ ), where D ′ is a nontrivial effective divisor satisfying 2B • D ′ < B • K and χ(D ′ ) = 1.Also, V fits as an extension and this extension cannot be split since then V would have both type O and type D ′ .However, . By Corollary 5.3, the only way this is nonzero is if D ′ is one of the exceptional divisors E i .
(1) When n = 11 or 12, the exceptional divisors do not satisfy 2B • E i < B • K, so V is always stable.
(2) Suppose 13 ≤ n ≤ 16.Fix one of the exceptional divisors; without loss of generality say it is E 1 .We seek to describe the extension classes e ∈ Ext 1 (K, O) such that the corresponding bundle V admits a nonzero map to O(E 1 ).
From the defining sequence and the restriction sequence we get the following commuting diagram which has exact rows and columns: We can compute many of the terms in this diagram to get a diagram ) is nonzero if and only if the map is zero, which holds if and only if the composition is zero.This composition is the same as the composition and this is zero if and only if the image of the first map is contained in the kernel of the second.The image of the first map is the 1-dimensional subspace of Ext 1 (K, O) determined by the bundle V , and the kernel of ) is nonzero if and only if V is the bundle defined by an extension class in the 1-dimensional image of the canonical map Such a bundle is not µ A t -stable for any t with t ≤ t E 1 because the map V → O(E 1 ) would destabilize V .Additionally, we note that if V fits in the exact sequence then Hom(V , O(E j )) = 0 for i j since n ≥ 13.Therefore, the n points in P Ext 1 (K, O) corresponding to such bundles are distinct.□

Tangent space
Now that we have determined when nonsplit bundles V of type D are stable, we investigate the components of the moduli space given by bundles of the various types.The tangent space to the moduli space at V is given by Ext 1 (V , V ), so we compute this space now.Lemma 6.4.(Assume SHGH ) The spaces Ext i (V , V ) have dimensions given as follows: (1) If D is not one of the exceptional divisors E i , then we have (2) In particular, if n ≥ 11 and D = O, we get (3) If D = E i (which forces n ≥ 13), then we also have Proof.Let V be a nonsplit bundle of type D, given by a nonsplit extension as We first apply Hom(−, O(D)) to this sequence.We display the dimensions ext i (A, B) for the relevant pairs (A, B) of objects in the following table: Finally, we apply Hom(V , −) to the sequence and get the following table: Clearly hom(V , V ) ≥ 1, but Hom(V , V ) → Hom(V , K(−D)) is an injection and hence an isomorphism.The rest of the table is immediate.This proves part (1), and specializing to the situations of (2) and (3) completes the proof.□

Components of nontrivial type
In fact, each nontrivial type D other than the exceptional divisors E i gives rise to a disjoint component of the moduli space M A t (v) whenever t < t D , and this component is a projective space of extensions.Theorem 6.5.(Assume SHGH ) Assume that D is not trivial or one of the exceptional divisors E i .If √ n < t < t D , then the nonsplit bundles of type D sweep out an irreducible component of M A t (2, K, 2) which is isomorphic to the projective space This component is disjoint from all other components of the moduli space.
Proof.By Proposition 6.2, every nonsplit extension gives rise to an A t -stable bundle of type D. Then the universal extension over To complete the proof, it suffices to show that φ is an injection and an isomorphism on tangent spaces.
In the proof of Lemma 6.4, we showed that Hom coming from applying various Hom functors to the defining sequence of V .The map α fits into the sequence so α is surjective.The map β fits into the sequence and therefore β is surjective since we compute Ext 1 (O(D), V ) = H 1 (V (−D)) = 0 from the sequence where H 1 (K(−2D)) = H 1 (O(2D)) = 0 by Corollary 5.3.
We conclude that d φe is surjective, with Ce contained in its kernel.The tangent space to is surjective.These spaces have the same dimension by Lemma 6.4, so dφ [e] is an isomorphism.This completes the proof.□

The component of trivial type
When n = 11 or 12, bundles of trivial type again sweep out a component.
and it is disjoint from all other components.
Proof.(1) The proof is the same as that of Theorem 6.6.
(2) We know from Theorem 6.5 that components of the moduli space parameterizing bundles of types D other than O or E i are disjoint from any components which parameterize a bundle of type O or E i .Thus we can let M ⊂ M A t (2, K, 2) be the subscheme parameterizing any bundles of type O or E i .Lemma 6.4, the tangent space of M has dimension n − 11 at every point V ∈ M.
Let us write q 1 , . . ., q n for the n points in P Ext 1 (K, O) corresponding to the inclusions Hom(K, O E i (−1)) → Ext 1 (K, O).Then M is projective and contains the subvariety which parameterizes the µ A t -stable bundles of type O. Let Y i ⊂ M be the locus of stable bundles of type E i , so that each Y i is the bijective image of the projective space P Ext 1 (K(−E i ), O(E i )).Then M is the disjoint union of U and the Y i .
The most important step of the proof is to construct the blowdown map π : M → P Ext 1 (K, O), which we now describe.Every bundle V ∈ M is either of type O or of type E i , so fits in one of the sequences In each case, V has a unique section, so we can canonically consider an exact sequence of the form In either case, Hom(K, F) is a 1-dimensional space, and applying Hom(K, −) to the sequence gives us an inclusion 0 −→ Hom(K, F) −→ Ext 1 (K, O) since Hom(K, V ) = 0.This inclusion therefore determines a point in P Ext 1 (K, O), which we denote by π(V ).Carrying out this construction in families defines a morphism π : M → P Ext 1 (K, O).
If V has type O, it is clear that π(V ) is precisely (the linear space spanned by) the extension class defining V .Thus π acts on U by the natural inclusion U → P Ext 1 (K, O).
Suppose that V has type E i .Then we claim that π(V ) = q i .In fact, we have an isomorphism Hom(K, F) Hom(K, O E i (−1)), so we just need to see that up to this isomorphism, the inclusion Hom(K, F) → Ext 1 (K, O) is the same map as the canonical inclusion Hom(K, O E 1 (−1)) → Ext 1 (K, O).We have the following diagram of short exact sequences: Applying Hom(K, −), we get the commutative square which shows that the image of Hom(K, F) in Ext 1 (K, O) is the subspace corresponding to q i .Next we will show that M is smooth and irreducible.Since M has a component of dimension n − 11 and every tangent space of M has dimension n − 11, it suffices to show that M is connected.Consider any mapping of a smooth curve C → P Ext 1 (K, O) having a point p ∈ C mapping to q i .Then the map C \ {p} → M extends to a regular map C → M, and the point p maps to a point representing a bundle of some type E j .By the continuity of the map the only possibility is that j = i.The locus Y i is itself connected, so we conclude that M is connected.Therefore, M is smooth.
By the universal property of the blowup, the map π : M → P Ext 1 (K, O) factors through the blowup of P Ext 1 (K, O) at {q 1 , . . ., q n } as The first map is a bijection between smooth varieties, so it is an isomorphism by Zariski's main theorem.□

Summary of results
We briefly summarize our description of the moduli spaces M A t (2, K, 2) for 10 ≤ n ≤ 15, assuming that the SHGH conjecture holds.In the next section, we will unconditionally describe the moduli space when n = 16. (

Smaller Euler characteristic
We have focused entirely on the moduli spaces M A t (2, K, 2) so far in this section, but our analysis makes it easy to prove a qualitative statement about the components of M A t (2, K, χ) for any χ ≤ 2. For concreteness, we will restrict our attention to 10 ≤ n ≤ 12, although with a detailed analysis of the divisors D, it would be easy to extend the statement to 10 ≤ n ≤ 15.Theorem 6.10.(Assume SHGH ) Suppose 10 ≤ n ≤ 12 and χ is an integer with χ ≤ 2. Fix positive integers k and r.There exists an ϵ > 0 such that if √ n < t < √ n + ϵ, then the moduli space M A t (2, K, χ) has at least k irreducible components of dimension at least r.
Proof.As t approaches √ n, the moduli space M A t (2, K, 2) gets components corresponding to the divisors D 3 , D 5 , D 7 , . ... We know that the dimensions of these components grow like the denominators in the continued fraction expansion of √ n by Remark 5.5.Then if t is sufficiently close to √ n, we can arrange that M A t (2, K, 2) has at least k irreducible components M 1 , . . ., M k of dimension at least r.Thus the theorem is true for χ = 2.If χ < 2, we can additionally arrange that the difference in the dimensions between any two of these components is as large as we want; for concreteness, let us say that any two of these components differ in dimension by more than 2 − χ.These components are projective spaces, and in particular they are smooth.
Recall that if V is a torsion-free sheaf and p ∈ X is a point where V is locally free, then an elementary modification of V at p is a sheaf V ′ fitting in a sequence If V is µ A t -stable, then so is V ′ .Then the locus in M A t (2, K, 1) parameterizing the elementary modifications of sheaves in M 1 is irreducible, so lies in an irreducible component M ′ 1 of M A t (2, K, 1).By the analysis in [CH18a, Section 3.3], the dimension of the component M ′ 1 satisfies Similarly, if we instead perform 2 − χ general elementary modifications to the bundles in M 1 , then the resulting bundles will lie in an irreducible component M (2−χ) 1 of M A t (2, K, χ) whose dimension satisfies If we carry out this process for each of the components M 1 , . . ., M k , we obtain a list of components

M
(2−χ) 1 , . . ., M (2−χ) k of M A t (2, K, χ).Our assumption on the dimensions of M 1 , . . ., M k implies that these components each have distinct dimensions, and they all have dimension at least r.□ Remark 6.11.In contrast, if the polarization A t is fixed but χ becomes arbitrarily negative, then the moduli spaces M A t (2, K, χ) become irreducible by O'Grady's theorem; see [O'G96].Thus it is necessary to choose the polarization A t after fixing the Euler characteristic χ in the previous theorem.

Moduli spaces for sixteen or twenty-five points
When n = 16, the results of the previous section can all be proven independently of the SHGH conjecture.In this section, we indicate the modifications that need to be made to the arguments to remove this dependency.We then also discuss the moduli space M A t (2, K, 4) when n = 25; by similar arguments, these spaces can also be completely described, independently of the SHGH conjecture.We begin with the following theorem, which summarizes our results in case n = 16.(1) For any t with t E 1 < t < t O , the moduli space M A t (2, K, 2) is isomorphic to P 5 (2) For any t with 4 < t < t E 1 , the moduli space M A t (2, K, 2) is isomorphic to the blowup of P 5 at sixteen points.Under the identification P 5 P Ext 1 (K, O), these sixteen points correspond to the images of the inclusions Hom(K, O E i (−1)) → Ext 1 (K, O).
Proof.We essentially have to repeat the sequence of arguments in Section 6, making modifications whenever the SHGH conjecture was used.The conjecture was primarily used when appealing to Section 5 to determine cohomological properties of possible divisors D which could lead to destabilizing objects.However, when n = 16, we have the complete unconditional classification of divisors D satisfying χ(D) ≥ 1 and 2B•D < B•K provided by Theorem 4.13: the only possible D are O and the E i .For these divisors, the statements in Section 5 become trivial, so this will be fairly straightforward.
In the proof of Proposition 6.2, if a bundle of type E i is destabilized, then it is destabilized by a line bundle K(−D ′ ) such that D ′ − E i is nontrivial effective, 2A t • D ′ < A t • K, and χ(D ′ ) ≥ 1.By Theorem 4.13, there are no such possible D ′ .
Similar modifications can be made to the first paragraph of the proof of Proposition 6.3, and the rest of the proof of that proposition does not refer to SHGH.
The only portion of Lemma 6.4 that is relevant is part (3), which clearly holds without SHGH.Theorem 6.5 only discusses components corresponding to nontrivial, nonexceptional divisors D satisfying χ(D) ≥ 1 and 2B • D < B • K; as there are no such divisors, the moduli space does not have any additional components.
The proof of Theorem 6.7 makes use of the previous results from Section 6, but does not make any additional use of SHGH.□ Next we consider the case n = 25 and the moduli space M A t (2, K, 4).Note that the maximal Euler characteristic of an effective divisor D satisfying 2B • D < B • K is χ(D) = 2.The argument in Lemma 6.1 then shows that the maximal Euler characteristic of an µ A t -semistable rank 2 bundle V with c 1 (V ) = K is χ(V ) = 4 and that any A t -semistable sheaf of character (2, K, 4) is a vector bundle.Theorem 7.2.Let n = 25.For any t with 5 < t < 27 5 , the moduli space M A t (2, K, 4) is isomorphic to a disjoint union of 25 copies of P 8 .The copies can be naturally identified with the spaces P Ext 1 (K(−D), O(D)), where D is one of the divisors H − E i .
Proof.Recall that by Theorem 4.15, any effective divisor D satisfying χ(D) ≥ 2 and 2B • D < B • K is of the form D = H − E i for some i.If a bundle V of type D = H − E i is not A t -stable, then it is destabilized by a line bundle L = K(−D ′ ).Here D ′ must be an effective divisor such that D ′ − D is nontrivial effective, 2A t • D ′ < A t • K, and χ(D ′ ) ≥ 2, as in the proof of Proposition 6.2.There are no such divisors D ′ .
For these divisors D, Lemma 6.4 clearly holds without SHGH.The proof of Theorem 6.5 goes through without further modification to complete the result.□ Example 1.2.Suppose the Nagata conjecture holds.If n = 10, then the effective divisors D satisfying χ(D) ≥ 1 and 2B • D < B • K form an infinite list O, 57H − 18E, 2220H − 702E, 84357H − 26676E, . . . .These divisors can be read off from the continued fraction expansion of √ 10; see Theorem 4.8.

Theorem 4. 13 .
Let n = 16.The only effective divisors D with χ(D) ≥ 1 and 2B • D < B • K are O and the exceptional divisors E i .

Theorem 4. 15 .
Let n = 25.The only effective divisors D with χ(D) ≥ 2 and 2B • D < B • K are the divisors H − E i .
we find that D • K = 0. Thus D has arithmetic genus 1, and D • H ≥ 3. Using the above decomposition of B shows seen to hold as long as 9 ≤ n ≤ 144.□ Theorem 5.2.(Assume SHGH ) Suppose 10 ≤ n ≤ 16.If D is an effective divisor with 2B • D < B • K, then D is a reduced, irreducible curve with h 0 (O(D)) = 1 and h 1 (O(D)) = h 2 (O(D)) = 0. Proof.If D is not reduced and irreducible, then we can write it as D = D ′ + D ′′ + D ′′′ , where D ′ and D ′′ are reduced and irreducible curves and D ′′′ is effective (possibly empty).Since 2B • D < B • K, at least one of 4B • D ′ or 4B • D ′′ is less than B • K, contradicting Lemma 5.1.Therefore, D is a reduced and irreducible curve.By the SHGH conjecture, χ(D) ≥ 1, so Proposition 4.1 gives χ(D) = 1.The cohomology of D then follows from the SHGH conjecture.□
but since 10 ≤ n ≤ 16, this inequality implies H • D < 1.The only such (−1)-curves are the exceptional divisors.□ Finally, we also compute the cohomology of O(2D − K) because bundles of type D are parameterized by an extension class in Ext 1 (K(−D), O(D)) H 1 (O(2D − K)).Proposition 5.4.(Assume SHGH ) Let 10 ≤ n ≤ 16, and suppose D is an effective divisor with 2B • D < B • K.The line bundle O(2D − K) has no h 0 or h 2 , and its h 1 is nonzero unless D is trivial and n = 10.
Here the first column of values ext i (K(−D), O(D)) are given by h i (O(2D − K)), which were computed in Proposition 5.4.The third column is clear, and the mapHom(O(D), O(D)) → Ext 1 (K(−D), O(D)) is injective because V is nonsplit.The values of ext i (V , O(D)) follow.Let e be 0 if D is not exceptional, and let e be 1 if D is exceptional.Then by Corollary 5.3, the line bundle O(2D) has h 1 (O(2D)) = e and h 2 (O(2D)) = 0.When we apply Hom(−, K(−D)) to the sequence, we get the following cohomology since in the third column, we have ext (V , K(−D)) = C.It follows that if V fits as an extension of K(−D) by O(D) in two different ways, then the corresponding classes in Ext 1 (K(−D), O(D)) are multiples of each other.Therefore, φ is injective.Let V be given by the extension class e ∈ Ext 1 (K(−D), O(D)), and let [e] denote its image in P Ext 1 (K(−D), O(D)).We let φ : Ext 1 (K(−D), O(D)) \ {0} −→ M A t (2, K, 2) be the composition of the quotient map Ext 1 (K(−D), O(D)) \ {0} → P Ext 1 (K(−D), O(D)) and φ.Then the derivative d φe factors as the composition of the natural maps Ext 1 (K(−D), O(D)) is naturally identified with Ext 1 (K(−D), O(D))/Ce, and d φe factors through dφ [e] to show that dφ [e] : Ext 1 (K(−D), O(D))/Ce −→ Ext 1 (V , V )
13 , 27801195H − 7710665E 1 − 7710664E 2,...,13 , . . .The solution (0, 0) corresponds to the divisor E 2,...,13 , which does not satisfy 2B • D < B • K. Let n = 13.If D is an effective divisor with χ(D) ≥ 1 such that 2A t • D < A t • K holds for some ample divisor A t , then D comes from one of the six infinite families (I)-(VI) discussed above.Conversely, if the Nagata conjecture holds for n = 13, then the divisors in these six families are precisely the effective divisors D with χ(D) = 1 such that 2B • D < B • K.Example 4.12.Let n = 16.Since the Nagata conjecture is true for 16 points, our results here are sharper, and we can completely classify effective divisors D such that χ(D) ≥ 1 and 2B Theorem 6.6.(AssumeSHGH ) Suppose n = 11 or 12.For any t with√ n < t < t O , the nonsplit bundles of type O sweep out a component of M A t (2, K, 2) isomorphic to P Ext 1 (K, O) P n−11 .On the other hand, bundles of type E i complicate the picture for 13 ≤ n ≤ 16.Nevertheless, we completely identify the moduli space in this case., the component of M A t (2, K, 2) containing stable bundles of type O consists of all the stable bundles of type O together with the nonsplit bundles of each type E i .This component is isomorphic to the blowup of the projective space P Ext 1 (K, O) at the n points determined by the canonical inclusions of 1-dimensional spaces 1) If t > t O = n 3 , then the moduli space M A t (2, K, 2) is empty.(2) If 11 ≤ n ≤ 15, a new component parameterizing bundles of type O arises when t decreases past t O .This component is isomorphic to P n−11 , and the component persists as t decreases to √ n.If 13 ≤ n ≤ 15, then as t decreases past t E 1 = n−2 3 , this component is blown up at n points, with the exceptional divisors parameterizing bundles of type E i .(3) For each nontrivial, nonexceptional effective divisor D satisfying χ(D) ≥ 1 and 2B • D < B • K, a new component of M A All components of the moduli space are disjoint from each other.We list, in order of decreasing t D , all of the wall-crossings where t D − √ 13 > 10 −13 .The "type" indicates the infinite family that the divisor comes from in Theorem 4.11.
t (2, K, 2) parameterizing bundles of type D arises when t decreases past t D .This component is isomorphic to P −χ(2D−K)−1 , and it persists and is unmodified as t decreases to √ n.Example 6.9.Since we also have the classification of divisors D for n = 13, we can similarly list all the wallcrossings in this case.The main additional complication is that D does not have to have equal multiplicities.For such a D, when t decreases past t D , many components will simultaneously arise by permuting the multiplicities of D.