Bundles on Pn with vanishing lower cohomologies

We study bundles on projective spaces that have vanishing lower cohomologies using their short minimal free resolutions. We classify the Betti numbers of these bundles and show that there are only finitely many possibilities with a given first Chern class and bounded regularity. Accordingly, we classify the Hilbert functions of these bundles and provide an efficient way to represent and to generate them. Finally, we show that the Betti numbers of bundles with a fixed Hilbert function form a graded lattice. We then describe the stratification of the space of isomorphism classes of bundles with a fixed Hilbert function by Betti numbers.


Introduction
In this paper, we study bundles (i.e. locally free coherent sheaves) E on P n such that These include all bundles on P 2 in particular. Note that ( †) is an open condition on a family of bundles by the semicontinuity of cohomologies. A rank r bundle E on P n satisfying ( †) admits a resolution by direct sums of line bundles of the form A minimal such resolution is unique up to isomorphism, and the integers a = (a 1 , . . . , a l ) and b = (b 1 , . . . , b l+r ) are invariants of E called the Betti numbers.
The main results in this paper are the following.
In Section 1, we classify all Betti numbers of rank r bundles on P n satisfying ( †), generalizing results from Bohnhorst and Spindler [3] for the case r = n. Accordingly, we classify all possible Hilbert functions of such bundles, and introduce a compact way to represent and to generate them. We show that there are only finitely many possible Betti numbers of bundles satisfying ( †) with fixed first Chern class and bounded regularity, generalizing the observation of Dionisi and Maggesi [4] for r = n = 2. We then give examples to show that the semistability of such a bundle is not determined by its Betti numbers in general, in contrast to the case when r = n discussed in [3].
In Section 2, we define natural topologies on VB † P n (H) and VB † P n (a, b), the set of isomorphism classes of bundles on P n satisfying ( †) with Hilbert function H and with Betti numbers (a, b) respectively. The topologies are induced from the rational varieties of matrices whose ideals of maximal minors have maximal depth. We show that all Betti numbers of bundles in VB † P n (H) form a graded lattice under the partial order of canceling common terms. This lattice is downward closed and infinite in general, where the subposet of Betti numbers up to any given regularity is a finite graded sublattice. Finally, we describe the stratification of VB † P n (H) by various subspaces VB † P n (a, b). We show that the closed strata intersect along another closed stratum, and that they form a graded lattice dual to the lattice of Betti numbers. An open subset VB † P n (H) ss of VB † P n (H) is a subscheme of the coarse moduli space M(χ) of semistable torsion-free coherent sheaves with Hilbert polynomial χ, similarly for an open subset VB † P n (a, b) ss of VB † P n (a, b). The same description applies to the stratification of VB † P n (H) ss by VB † P n (a, b) ss on the level of topological spaces.
The study of vector bundles on algebraic varieties is central to algebraic geometry. In particular, the study of bundles on projective spaces already presents interesting challenges. We do not attempt to give a survey of the subject here. Instead, we provide some historical perspectives to motivate the investigations in this paper.
Maruyama [18] proved that the coarse moduli space of rank two semistable bundles on a smooth projective surface exists as a quasi-projective scheme. In the same paper, it was shown that the coarse moduli space M P 2 (2, c 1 , c 2 ) s of stable rank two bunldes on P 2 with given Chern classes is smooth and irreducible. Following this development, Barth [2] showed that M P 2 (2, c 1 , c 2 ) s is connected and rational for c 1 even, and Hulek [17] did the same for c 1 odd. Their arguments contained a gap which was pointed out and partially fixed in [21] and independently in [8]. The existence of the coarse moduli space of semistable torsion-free sheaves of arbitrary rank on a smooth projective variety was finally established by Maruyama [19], see [20] for another exposition.
Despite the progress in the theory of moduli, there are many basic questions about bundles on projective spaces that are unanswered, see [16] for a problem list. For example, Hartshorne's conjecture [15] states that a rank r bundle on P n is the direct sum of line bundles when r < 1 3 n. In particular, the conjecture predicts that rank two bundles on P n are split when n ≥ 7. On the other hand, the only known example (up to twists and finite pullbacks) of an indecomposable rank two bundle on P 4 and above is the Horrocks-Mumford bundle [11]. It is fair to say that bundles of small rank on P n remain mysterious. This paper is motivated by two main objectives in the study of bundles.
(1) To classify certain invariants of bundles on P n . To expand on this point, the Hilbert polynomial is an important invariant of a bundle that is constant in a connected flat family, and thus indexes the connected components of the moduli space (of some subclasses of bundles, e.g. semistable). Note that the Hilbert polynomial can be computed from the Chern polynomial and vice versa. Thus the classification of Hilbert polynomials is equivalent to the classification of Chern classes. The Hilbert function eventually agrees with the Hilbert polynomial, and thus provides finer information. Furthermore, the Hilbert function can be computed from the Betti numbers of a free resolution of (the section module of) a bundle. Therefore the Betti numbers are even finer invariants of a bundle. Consequently, the classification of Betti numbers of bundles will lead to a classification of the Hilbert functions and Hilbert polynomials (equivalently Chern classes). In this paper, we take the first step by classifying the Betti numbers of bundles when their resolutions are short. It turned out that this condition implies that these bundles have rank greater than the dimension of the ambient projective space with the exceptions of direct sums of line bundles. (2) To provide examples of bundles with given invariants. In the best scenario, the moduli space or the space of isomorphism classes of bundles with given invariants is unirational, in which case the image of a random point in the projective space will give us a "random" bundle with given invariants. For example, Barth's parametrization of M P 2 (c 1 , c 2 ) s using nets of rank two quadrics [2] allows us to produce "random" rank two bundles on P 2 with given Chern classes. Here we can see the importance of using finer invariants. Since M P 2 (c 1 , c 2 ) s is irreducible, a general bundle produced using Barth's parametrization will be presented by a matrix of linear and quadratic polynomials by the main theorem in [4]. Therefore producing a bundle that is presented by a matrix of forms of other degrees, which is special in the moduli M P 2 (c 1 , c 2 ) s , is like looking for a needle in a haystack. On the other hand, if we stratify M P 2 (c 1 , c 2 ) s using the finer invarints of Betti numbers (a, b), then each piece M P 2 (a, b) s is still unirational and we can thus produce a "random" bundle that is presented by a matrix of forms of given degrees whenever possible.
The results in this paper are implemented in the Macaulay2 [12] package BundlesOnPn [23], which generates all Betti numbers of bundles satisfying ( †) up to bounded regularity as well as "random" bundles with given Betti numbers.
Acknowledgement. The author thanks his advisor David Eisenbud for support and for pointing out that the earlier versions of the results may be generalized.

Free resolutions of bundles
Throughout, we work over an algebraically closed field k. We fix R := k[x 0 , . . . , x n ] to denote the polynomial ring of P n . For a coherent sheaf F on P n , we write H i * (F ) for the R-module t∈Z H i (F (t)). We write VB † P n for the set of isomorphism classes of bundles on P n satisfying ( †).
We start with a standard observation on the relation between the vanishing of lower cohomologies of a coherent sheaf and the projective dimension of its section module.
denote the i-th local cohomology module supported at the homogeneous maximal ideal m of R. There is a four-term exact sequence By the vanishing criterion of local cohomology, we have depth M = inf{i | H i m (M) = 0}. Finally, the Auslander-Buchsbaum formula states that pdim M = n + 1 − depth M. The statement follows. Definition 1.2. Let E be a rank r bundle on P n satisfying ( †). By Proposition 1.1, the R-module H 0 * (E ) admits a unique (up to isomorphism) minimal graded free R-resolution We always arrange the numbers a 1 ≤ · · · ≤ a l and b 1 ≤ · · · ≤ b l+r in ascending order, and write a and b for brevity. We call (a, b) the Betti numbers of E .
Note that E is isomorphic to a direct sum of line bundles iff H 0 * (E ) is a free R-module iff l = 0 and the sequence a is empty.
The resolution ( * ) of graded R-modules sheafifies to a resolution of E by direct sums of line bundles. Conversely, a resolution (⋆) of E by direct sums of line bundles gives rise to a free resolution ( * ) of the R-module H 0 * (E ) under the functor H 0 * (−). With this understanding, we shall speak of these two resolutions of modules and sheaves interchangeably. In particular, the morphism ϕ is called minimal iff the corresponding map of R-modules φ is minimal, i.e. φ ⊗ R k = 0.
1.1. Betti numbers. In this subsection we classify the Betti numbers of bundles in VB † P n .
For a pair (a, b) of finite sequences of integers in ascending order, we write VB † P n (a, b) for the set of isomorphism classes of bundles with Betti numbers (a, b). For a sequence of integers d := (d 1 , . . . , d l ), we define Theorem 1.3. Let a = (a 1 , . . . , a l ) and b = (b 1 , . . . , b l+r ) be two sequences of integers in ascending order for some l ≥ 0 and r > 0. The set VB † P n (a, b) is nonempty iff a is empty or r ≥ n and a i > b n+i for i = 1, . . . , l.

(A)
In this case, the cokernel of ϕ represents the class of a bundle in VB † P n (a, b) for a general minimal map ϕ ∈ Hom(L (a), L (b)).
This generalizes the results of Bohnhorst and Spindler [3] for r = n. Likewise, we say a pair of ascending sequences of integers (a, b) is admissible if it satisfies the equivalent conditions of Theorem 1.3.
The fact that a bundle E satisfying ( †) that is not a direct sum of line bundles must have rank r ≥ n also follows from the Evans-Griffith splitting criterion [9,Theorem 2.4].
In order to prove Theorem 1.3, we need two lemmas regarding depth of minors of matrices.
Let S denote a noetherian ring and let φ : S p → S q be a map between two free S-modules. For any integer r, the ideal I r (φ) of (r × r)-minors of φ is defined as the image of the map ∧ r S p ⊗ S (∧ r S q ) * → S, which is induced by the map ∧ r φ : ∧ r S p → ∧ r S q .
Similarly, let ϕ : be a morphism of sheaves on P n A over a noetherian ring A. Set S := A[x 0 , . . . , x n ] and let φ : denote the corresponding morphism of graded free S-modules given by H 0 * (ϕ). For any integer r, we define I r (ϕ) = I r (φ) as an ideal in S. The depth of a proper ideal I in a noetherian ring S is defined to be the length of a maximal regular sequence in I. The depth of the unit ideal is by convention +∞. Recall that if S is Cohen-Macaulay, then depth I = codim I for every proper ideal I. Lemma 1.4. Let A be a finitely generated integral domain over k, and let S be a finitely generated A-algebra. Suppose φ : S q → S p is a morphism of free S-modules with p ≥ q. For a prime P of A, let φ P denote the morphism φ ⊗ A k(P ) of free modules over the fiber ring S ⊗ A k(P ). For any integer d, the set of primes P in A such that depth Proof. Note that I q (φ) = I q (φ * ). Let K • (φ * ) be the Eagon-Northcott complex associated to φ * as in [6]. Note that the formation of the Eagon-Northcott complex is compatible with taking fibers, i.e.
is exact after position p − q + 1 − d by the main theorem in [6]. The statement of the lemma follows from the general fact that the exactness locus of a family of complexes is open, see E.G.A IV 9.4.2 [13]. Lemma 1.5. Let S be a standard graded finitely generated k-algebra.
be a morphism of graded free S-modules with p ≥ q, and assume that φ is minimal, i.e. φ ⊗ S k = 0. Suppose that relative to some bases, the matrix of φ has a block of zeros of size u × v.
Proof. For the case of generic matrices over a field, this is a result of Giusti-Merle [10] . We fix, once for all, bases of the domain and target of φ, and let Z ⊂ {1, . . . , p} × {1, . . . , q} be the u×v rectangle where the matrix of φ has zero entries. Consider the polynomial ring A : , which is the coordinate ring of the affine space of (p × q)matrices with a zero block of size u ×v in position Z. Let ψ : A(−1) q → A p be the morphism given by the generic matrix (x ij ). Then codim The general case follows from Serre's result on the superheight of prime ideals in a regular local ring. The map φ corresponds to a morphism of k-algebras A → S, where x ij is sent to the entry of the matrix of φ relative to the fixed bases. In particular, note that I q (φ) = SI q (ψ). Let m and m ′ denote the homogeneous maximal ideals of A and S respectively. Since all entries of φ are in m ′ by assumption, we have an induced morphism on the localizations Let P be a prime above A m I q (ψ) of the least codimension. Since S m ′ P ⊂ m ′ , Serre's result on superheight on prime ideals in a regular local ring [22] implies that codim S m ′ P ≤ codim P . Now I q (ψ) and SI q (ψ) are homogeneous, and The following is a simple fact that allows us to translate between bundles and homogeneous matrices whose ideals of maximal minors have maximal depth. Proposition 1.6. Let a = (a 1 , . . . , a l ) and b = (b 1 , . . . , b l+r ) for some l > 0 and r ≥ 0. For a map ϕ ∈ Hom(L (a), L (b)), the cokernel of ϕ is a rank r bundle on P n iff depth I l (ϕ) ≥ n + 1. In this case, we have a resolution of E := coker ϕ by direct sums of line bundles Proof. The rank of coker ϕ is r iff I r (ϕ) is nonzero iff ϕ is injective at the generic point of P n iff ϕ is injective. The ideal I r (ϕ) cuts out points on P n where coker ϕ is not locally free of rank r. Thus coker ϕ is a rank r bundle iff I r (ϕ) is the unit ideal or is m-primary, where m is the homogeneous maximal ideal of R. In either case depth I r (ϕ) ≥ n + 1.
Proof of Theorem 1.3. If a is empty, then E := L (b) has Betti numbers (a, b). Suppose a is nonempty and (a, b) satisfies condition (A). Consider the minimal map ϕ : L (a) → L (b) given by the following matrix Since ϕ drops rank nowhere on P n , we conclude that E := coker ϕ is a rank r bundle with a resolution by direct sums of line bundles by Proposition 1.6. Since ϕ is minimal, it follows from Proposition 1.1 that E ∈ VB † P n (a, b). Conversely, suppose VB † P n (a, b) is nonempty and a is nonempty. Then there is a minimal map ϕ ∈ Hom(L (a), L (b)) where coker ϕ is a rank r bundle E . Since ϕ is minimal, it follows that I l (ϕ) ⊂ I 1 (ϕ) ⊂ m is a proper ideal. By Proposition 1.6, we have depth I l (ϕ) = n + 1. By the main theorem in [6], we have depth I l (ϕ) ≤ l + r − l + 1 = r + 1. It follows that we must have r ≥ n. Now suppose on the contrary that there is an index 1 ≤ i ≤ l where a i ≤ b n+i . Since ϕ is minimal, we see that the (n + i, i)-th entry in the matrix of ϕ must be zero. In fact, since a and b are in ascending order, we must have a block of zeros of size (l + r − n − i + 1) × i as the following By Lemma 1.5, we conclude that This is a contradiction to the fact that depth I l (ϕ) = n + 1. Now we prove the last statement. It is obvious when a is empty, so we assume a is nonempty. The set Hom(L (a), L (b)) has the structure of the closed points of an affine space A N . The subset of minimal maps is an affine subspace A M . There is a tautological morphism Φ : where the fiber Φ P for a closed point P of A M is given by the minimal map that P corresponds to. By Lemma 1.4, the set U of points in A M where depth I l (Φ P ) ≥ n + 1 is open. Since there is a morphism ϕ ∈ Hom(L (a), L (b)) whose cokernel is a bundle E ∈ VB † P n (a, b), by Proposition 1.6 the map ϕ corresponds to a closed point in U. It follows that U is open and dense in A M .
Recall that the category of bundles on P n is a Krull-Schmidt category [1], i.e. every bundle E admits a decomposition E ∼ = E 0 ⊕ L , unique up to isomorphism, where L is the direct sum of line bundles and E 0 has no line bundle summands.
) is a minimal map whose cokernel is a bundle E , then we claim that ϕ ′ := π • ϕ is a minimal map in Hom(L (a), L (b)) whose cokernel is a bundle E ′ . To see this, observe that since a l ≤ b l+i for s < i ≤ r and ϕ is minimal, the last (r − s) rows of the matrix representing ϕ relative to any bases are zero.
In particular, we have I l (ϕ) = I l (π • ϕ). By Proposition 1.6, the cokernel of ϕ ′ is a bundle. It follows from the snake lemma that E ∼ = E ′ ⊕ L , where L is the kernel of the projection π. This shows that rank E 0 ≤ s. Observe that E 0 also satisfies ( †) and thus rank E 0 ≥ n by Theorem 1.3.

1.2.
Finiteness. In this subsection, we show that there are only finitely many possible Betti numbers of bundles in VB † P n with given rank, first Chern class and bounded regularity.
Recall that a coherent sheaf F on P n is said to be d- By the semicontinuity of cohomologies, being d-regular is an open condition for a family of coherent sheaves on P n . The notion of regularity also exists for graded R-modules. See [7] for an exposition.
Since the regularity depends only on the Betti numbers, we define reg(a, b) := max(b l+r , a l − 1) for any admissible pair (a, b). Proposition 1.8. There are only finitely many possible Betti numbers (a, b) of rank r bundles on P n satisfying ( †) with fixed first Chern class c 1 and regularity ≤ d.
the statement is evidently true for direct sums of line bundles. Thus we may consider the case l > 0. Since a i and b i are bounded above by d + 1, we only need to show that l is bounded above and b 1 is bounded below. Consider the following inequalities And similarly, This generalizes the observation of Dionisi-Maggesi [4] for the case n = r = 2.
1.3. Hilbert functions of bundles. In this subsection, we classify the Hilbert functions of bundles in VB † P n . We introduce an efficient way to represent and generate them. Recall that the Hilbert function of a bundle E on P n is the function H E (t) : Z → Z given by H E (t) = dim k H 0 (E (t)). For any function H : Z → Z, we define VB † P n (H) to be the subset of VB † P n consisting of isomorphism classes of bundles with Hilbert function H. Definition 1.9. The numerical difference of a function H : Z → Z is a function ∂H : Z → Z given by ∂H(t) := H(t) − H(t − 1). We inductively define ∂ i+1 H := ∂∂ i H.
Note that if H : Z → Z is a function such that H(t) = 0 for t ≪ 0, then H can be recovered by its i-th difference ∂ i H for any i ≥ 0. (1) ∂ n H(t) = 0 for t ≪ 0 and ∂ n H(t) = r for t ≫ 0, Proof. Let µ(d, t) denote the number of times an integer t occurs in the sequence d.
A routine computation shows that the leading coefficient of the Hilbert polynomial χ(E (t)) is r · t n /n!. Since the Hilbert function H eventually agrees with the Hilbert polynomial, we see that ∂ n H(t) = 0 for t ≪ 0 and ∂ n H(t) = r for t ≫ 0. Let (a, b) be the Betti numbers of E . If a is empty, then E is a direct sum of line bundles and ∂ n H is monotone nondecreasing and thus satisfies both conditions. We prove the case where a is non-empty. Consider the minimal free resolution 0 → L(a) → L(b) → H 0 * (E ) → 0. A simple calculation shows that ∂ n+1 H(R(−a), t) is the delta function at a. It follows from the minimal resolution that ∂ n+1 H(t) = µ(b, t) − µ(a, t). Suppose ∂ n H(t + 1) < ∂ n H(t) for some t, then ∂ n+1 H(t + 1) < 0 and thus µ(a, t + 1) > 0. Let j be the largest index where a j = t + 1. By Theorem 1.3, we have a j > b j+n and therefore (⇐=): Conversely, suppose H satisfies the conditions of the theorem. We define the ascending sequences of integers α and β by the property that for all t ∈ Z, By the first condition on H, the sequences α and β are finite. Furthermore, if α has length l then β has length l + r. The second condition on H implies that a i ≥ b i+n for all 1 ≤ i ≤ l. Since α and β share no common entries by construction, it follows that a i > b i+n for all 1 ≤ i ≤ l. By Theorem 1.3, there is a rank r bundle E on P n satisfying ( †) with Betti numbers (α, β). The Hilbert function of E is H by the reasoning of the previous direction.
The above theorem suggests that we use the finitely many intermediate values of ∂ n H to encode the infinitely many values of the Hilbert function H. Definition 1.11. A finite sequence of integers B = B 1 , . . . , B m for some m ≥ 1 is called a bundle sequence of rank r if it satisfies the following If E is a rank r bundle in VB † P n (H) for some Hilbert function H, then we set s 0 := inf{t | ∂ n H(t) = 0}, s 1 := sup{t | ∂ n H(t) = r}.
The sequence ∂ n H(s 0 ), ∂ n H(s 0 + 1), . . . , ∂ n H(s 1 + 1) is a bundle sequence of rank r by Theorem 1.10, which we call the bundle sequence of H and of E .
By Theorem 1.10, there is a one-to-one correspondence between the set of Hilbert functions of rank r bundles in VB † P n up to shift and the set of bundles sequences of rank r. The ambiguity of shift disappears if we deal with normalized bundles. Definition 1.12. We say a rank r bundle on P n is normalized if −r < c 1 (E ) ≤ 0. Since c 1 (E (t)) = c 1 (E ) + r · t, it follows that every bundle can be normalized after twisting by the line bundle O(−⌈c 1 (E )/r⌉).
We define the degree of a bundle sequence B = B 1 , . . . , B m , denoted by deg B, to be the sum B 1 + · · · + B m . Proposition 1.13. If a normalized rank r bundle E ∈ VB † P n has bundle sequence B, then reg E ≥ ⌈deg B/r⌉ − 2.
Proof. Suppose E has Betti numbers (a, b) and Hilbert function H. We set c := max(a l , b l+r ) and s 1 := sup{t | ∂ n H(t) = r}. It follows from the short exact sequence Since E is normalized, we must have c ≥ ⌈deg B/r⌉ − 1. Finally, regularity E is c or c − 1 depending on whether b l+r ≥ a l − 1 or not. It follows from Proposition 1.13 and Proposition 1.14 that we can inductively generate, in the form of bundle sequences, all Hilbert functions of normalized bundles satisfying ( †) up to any bounded regularity. The generation is reduced to a partition problem with constraints. Here we use t j to denote the sequence of j copies of t.

Semistability.
In this subsection, we address the following question. Do the Betti numbers determine the semistability of a bundle in VB † P n ? If so, what is the criterion?
Here we use µ-semistability, where µ(F ) := c 1 (F )/ rank(F ) for any torsion-free coherent sheaf F on P n . The results are similar for Hilbert polynomial semistability as in [19].
For r < n, all rank r bundles E satisfying ( †) are direct sums of line bundles by Theorem 1.3, which are not semistable except for O(d) r . The main result in [3] states that if E satisfies ( †) and has rank r = n, then E is semistable iff its Betti numbers (a, b) satisfy The latter condition is obviously necessary.
The following example demonstrates that for r > n, the semistability of a bundle in VB † P n is not determined by its Betti numbers in general.
The main reason to discuss semistability is that we might hope for a coarse moduli structure on the set VB † P n (a, b). However, the above example illustrates the difficulty. In Section 2.2 we will define a topology on VB † P n (a, b), where the semistable bundles form an open subspace VB † P n (a, b) ss . The space VB † P n (a, b) ss supports the structure of a subscheme of M(χ), the coarse moduli space of semistable torsion-free sheaves with Hilbert polynomial χ, whose existence is established by Maruyama [19].

The Betti number stratification
The set VB † P n is the disjoint union of VB † P n (H) for all possible Hilbert functions H which are classified by Theorem 1.10. In this section we define a natural topology on VB † P n (H) and study how VB † P n (H) is stratified by bundles with different Betti numbers. In the following, we fix a Hilbert function H satisfying the conditions of Theorem 1.10.
2.1. The graded lattice of Betti numbers. In this subsection we show that all possible Betti numbers of bundles in VB † P n (H) form a graded lattice, such that those with bounded regularity form a finite sublattice. We remark that Betti(H) is infinite in general without restrictions on regularity. This is due to the fact that the Hilbert function H only bounds regularity from below (see Proposition 1.13) but not above, as the following example demonstrates.
Example 2.2. Let (a, b) ∈ Betti(H). For some arbitrarily large integer c, regarded as a singleton sequence, the pair (a, b) + c is admissible by Theorem 1.3. Note that any bundle with these Betti numbers has a line bundle summand by Corollary 1.7.
We now define a partial order on all pairs of increasing sequences of integers.
Definition 2.4. Let a, b, c be three finite sequences of integers in ascending order. The sum a + c is defined be the sequence obtained by appending c to a and sorting in ascending order. It is clear that this operation is associative.
We define (a, b) + c to be the pair (a + c, b + c). If (a ′ , b ′ ) = (a, b) + c for some c, then we say (a, b) is a generalization of (a ′ , b ′ ) and write (a, b) (a ′ , b ′ ).
A direct consequence of Theorem 1.3 is that admissibility is stable under generalization. then so is (a, b).
Proof. By induction, it suffices to prove the case where a ′ and b ′ have a common entry c at index p and q respectively, and that (a, b) is obtained from (a ′ , b ′ ) by removing a ′ p and b ′ q . We may assume that p and q are the largest indices where a ′ p = c and b ′ q = c respectively. For i < p, we have a i = a ′ i . But i+n < q and b i+n = b ′ i+n for i < p since q > n+p by Theorem 1.3. Therefore a i > b i+n for i < p. In this case, b l+n = b ′ l+n and a l > b l+n . For i > p, we have a i−1 = a ′ i > c. In this case, either i + n ≤ q, in which case b i+n−1 ≤ c < a i−1 ; or i + n > q, and b i+n−1 = b ′ i+n thus b i+n−1 < a i−1 . We conclude that (a, b) is also admissible.
Corollary 2.6. Every (a, b) in Betti(H) is of the form (α, β) + c for some c.
The main theorem of this subsection is the following.
Theorem 2.7. The set Betti(H) has the structure of a graded lattice given by the partial order and the grading Betti(H) = q Betti q (H).
For the clarity of the proof, we establish the existence of meet in several steps. Proof. The lemma is simple, but the notations may make it appear more complicated than it is. Notheless, we include a proof here for the sake of completeness.
For an ascending sequence d and an integer t, let p(d, t) denote the largest index i where d i = t. We may assume c < d, and write (a ′ , b ′ ) : If c is an integer sequence and d is an integer (considered as a singleton sequence) not appearing in c, such that both (a, b) + c and (a, b) + d are admissible, then so is (a, b) + c + d.
Proof. By Lemma 2.5 and Lemma 2.8, the pair (a, b) + c 1 + d is admissible. Applying Lemma 2.8 again with (a, b) + c 1 in place of (a, b), we see that (a, b) + c 1 + c 2 + d is admissible. By induction it follows that (a, b) + c + d is admissible.
Proof of Theorem 2.7. If (a ′ , b ′ ) ∈ Betti i (H) and (a, b) ∈ Betti j (H) such that (a ′ , b ′ ) = (a, b) + c for some c, then obviously i ≥ j. The cover relations in Betti(H) are given exactly by adding singleton sequences. It follows that (Betti(H), ) is a graded poset.
Clearly (a, b) + min(c, c ′ ) (a, b) + c and thus is admissible by Lemma 2.5. It follows that (a, b) + min(c, c ′ ) is the meet of (a, b) and (a ′ , b ′ ) in Betti(H).
We claim that (a, b) + max(c, c ′ ) is admissible, and thus it is the join of (a, b) and (a ′ , b ′ ) in Betti(H). To see this, we may replace (a, b) by (a, b) + min(c, c ′ ) and assume that c and c ′ have no common entries. By Lemma 2.9, we see that (a, b) + c + c ′ 1 is admissible. Applying Lemma 2.9 again with (a, b) + c ′ 1 in place of (a, b), we conclude that (a, b) + c ′ 1 + c + c ′ 2 is admissible. By induction, it follows that (a, b) + c ′ + c is admissible. Proof. If (a, b) (a ′ , b ′ ), then reg(a, b) ≤ (a ′ , b ′ ). If (a ′′ , b ′′ ) is the join of (a, b) and (a ′ , b ′ ) in Betti(H), then the regularity of (a ′′ , b ′′ ) is the maximum of those of (a, b) and (a ′ , b ′ ) by the construction in the proof of Theorem 2.7. It follows that Betti(H) ≤d is a graded lattice. The finiteness of Betti(H) ≤d follows from Proposition 1.8. Thus there is a maximum element of the form (α, β) + c for some sequence c. By Lemma 2.5, we see that Example 2.11. Let H be the Hilbert function of a normalized bundle on P 3 with bundle sequence (5, 4). With the same notation as in Example 1.15, the minimal element of Betti(H) is given by α = (0) and β = (−1 5 ). The maximum element of Betti(H) ≤2 is (α, β) + c, where c = (0, 1, 2). In particular, is a subsequence of (0, 1, 2) of length q} and Betti(H) ≤2 is isomorphic to the lattice of subsequences of (0, 1, 2).

The stratification.
In this subsection, we define a natural topology on VB † P n (H). We then describe the stratification of VB † P n (H) by locally closed subspaces VB † P n (a, b).  A(a, b) and A 0 (a, b) respectively. For A = A(a, b) and A 0 (a, b), the tautological morphism Φ : gives a tautological family of sheaves E := coker Φ over A, which pulls back to a family of bundles E (a, b) and E 0 (a, b) satisfying ( †) over M(a, b) and M 0 (a, b) respectively by Proposition 1.  Proof. Clearly if ϕ, ψ are in the same G(a, b)-orbit then coker ϕ ∼ = coker ψ. Conversely, let E := coker ϕ and E ′ := coker ψ. Then the isomorphism of the R-modules H 0 * (E ) ∼ = H 0 * (E ′ ) lifts to an isomorphism of free resolutions It follows that ϕ, ϕ ′ are in the same G(a, b)-orbit.
Proposition 1.6 and Lemma 2.13 imply that the set VB † P n (a, b) supports the structure of the quotient topological space M 0 (a, b)/G(a, b). Similarly, we let VB † P n (a, b) denote the subset of VB † P n consisting of isomorphism classes of bundles E that admit a (not necessarily minimal) free resolution of the form Then Lemma 2.13 also implies that the set VB † P n (a, b) supports the structure of the quotient topological space M(a, b)/G(a, b). Clearly the inclusion of sets VB † P n (a, b) ⊆ VB † P n (a, b) is an inclusion of topological spaces.
. In particular, VB † P n (a, b) is a subspace of VB † P n (a ′ , b ′ ) . Proof. Let (a ′ , b ′ ) = (a, b) + c for some c. Consider an injective morphism ι : M(a, b) → M(a ′ , b ′ ) given by ϕ → ϕ ⊕ Id L (c) . It is not hard to see that the ideal of maximal minors does not change under this map, and thus ι is well-defined. Suppose ϕ, ψ are two morphisms in M(a, b) such that ϕ ⊕ Id L (c) and ψ ⊕ Id L (c) are in the same G(a ′ , b ′ )-orbit. It follows that coker ϕ ⊕ Id L (c) ∼ = coker ψ ⊕ Id L (c) . Since coker ϕ ∼ = coker ϕ ⊕ Id L (c) and coker ψ ∼ = coker ψ ⊕ Id L (c) , we conclude that coker ϕ ∼ = coker ψ. It follows from Lemma 2.13 that ϕ and ψ are in the same G(a, b)-orbit. This shows that the composition For each integer d, the set Betti(H) ≤d is a lattice by Corollary 2.10 and thus has a maximum element (a ′ , b ′ ). It follows from Lemma 2.14 that every d-regular bundle E in VB † P n (H) admits a (not necessarily minimal) free resolution of the form It follows from Lemma 2.14 and the construction above that if d < d ′ , then VB † P n (H) ≤d is a subspace of VB † P n (H) ≤d ′ . Finally, we define a topology on VB † P n (H) by is the maximum element of Betti(H) ≤d , it follows that VB † P n (H) ≤d is irreducible and unirational, and so is VB † P n (H).
The main result of this subsection is the following.
Theorem 2.17. The closed strata VB † P n (a, b) in VB † P n (H) form a graded lattice dual to Betti(H) under the partial order of inclusion. Furthermore, the intersection of two closed strata VB † P n (a, b) and VB † P n (a ′ , b ′ ) is again a closed stratum VB † P n (a ′′ , b ′′ ), where (a ′′ , b ′′ ) is the join of (a, b) and (a ′ , b ′ ) in the lattice Betti(H).
The theorem needs several standard lemmas on the behavior of resolutions in families with constant Hilbert functions. We include proofs here for the lack of appropriate references. Lemma 2.18. Let E ′ ∈ VB † P n (a ′ , b ′ ) and suppose (a, b) (a ′ , b ′ ). Then there is a family of bundles E on P n over a dense open set U ⊂ A 1 containing the origin 0 ∈ A 1 , such that E 0 ∼ = E ′ and E t ∈ VB † P n (a, b) for any closed point 0 = t ∈ U. Proof. Suppose (a ′ , b ′ ) = (a, b) + c. By Lemma 2.5, the pair (a, b) is admissible. Let ψ ∈ M 0 (a ′ , b ′ ) be a minimal presentation of E ′ , and let ϕ ∈ M 0 (a, b) be a minimal presentation of a bundle E . Set ϕ ′ = ϕ ⊕ Id L (c) and consider the morphism Φ : L (b ′ ) × A 1 → L (a ′ ) × A 1 whose fiber over a closed point t ∈ A 1 is given by Φ t := ψ+t·ϕ ′ . By Lemma 1.4, the morphism Φ t ∈ M(a ′ , b ′ ) for all closed points t in an open dense set U ⊂ A 1 containing 0. This shows that coker Φ t ∈ VB † P n (a, b) for t ∈ U. We show that in fact coker Φ t ∈ VB † P n (a, b) for all 0 = t ∈ U. Let t = 0 be any closed point of U. Since ψ is minimal and ϕ ′ induces an isomorphism on the common summand L (c) ∼ − → L (c), it follows that Φ t also splits off the common summand L (c) t − → L (c). Since ϕ does not split off any common summands other than those of L (c), neither does Φ t by Nakayama's lemma. It follows that the free resolution contains a minimal one of the form Proof. Let t ∈ T be a closed point. We may base change to Spec O T,t and reduce to the case where T is an affine local domain. Let m be the maximal ideal of T with residue field k and set R T := T [x, y, z] and R := k[x, y, z]. The module E := l∈Z H 0 (E (l)) is finitely generated over R T since E is a bundle. Since the fibers over T have the same Hilbert functions, it follows that E is flat over is a minimal system of generators, then by Nakayama's lemma over generalized local rings, it lifts to a system of generators Since E is flat over T , so is ker d T and thus (ker d T ) ⊗ T k ∼ = ker d. Applying this procedure again, we find a free resolution of E that specializes to a minimal free resolution of E ⊗ T k. It follows that F • ⊗ T k(T ) is a free resolution of the generic fiber which contains a minimal free resolution of the form We conclude that the general fibers E t have the Betti numbers (a, b) (a ′ , b ′ ).
. Here all closures are taken within VB † P n (H). Proof. (1) =⇒ (2): Suppose (a ′ , b ′ ) = (a, b) + c. Let ϕ ∈ M 0 (a, b) and ψ ∈ M 0 (a ′ , b ′ ). Consider the line Φ : For an open set U ⊂ A 1 containing 0, the image Φ(t) is contained in M(a ′ , b ′ ). By Lemma 2.18, the image of Φ(t) in the quotient VB † P n (a ′ , b ′ ) lies in VB † P n (a, b) for t = 0. It follows that the image of ψ in VB † P n (a ′ , b ′ ) is contained in the closure of VB † P n (a, b) inside the space VB † P n (a ′ , b ′ ) . Since ψ represents an arbitrary point of VB † P n (a ′ , b ′ ), we conclude that VB † P n (a ′ , b ′ ) is contained in the closure of VB † P n (a, b) in VB † P n (a ′ , b ′ ) , and therefore the same is true inside VB † P n (H).  reg(a, b), reg(a ′ , b ′ )). Let (a ′′ , b ′′ ) denote the maximum element of Betti(H) ≤d . Let π : M(a ′′ , b ′′ ) → VB † P n (H) ≤d be the quotient map and set V to be the preimage of VB † P n (a, b) under π, endowed with the structure of a (reduced) subvariety of M(a ′′ , b ′′ ). Let E be the pullback of the tautological family of bundles E (a ′′ , b ′′ ) on M(a ′′ , b ′′ ) to V . Since VB † P n (a, b) is dense in VB † P n (a, b), it follows that the fiber E v over a general point v ∈ V has Betti numbers (a, b). If p is a point in VB † P n (a ′ , b ′ ) that is in the closure of VB † P n (a, b), and q is a point in π −1 (p), then q ∈ V and E q has Betti numbers (a ′ , b ′ ). Finally, an application of Lemma 2.19 to the family E gives (a, b) (a ′ , b ′ ). Proof of Theorem 2.17. The first statement follows directly from Lemma 2.20. For the same reason, it is clear that VB † P n (a ′′ , b ′′ ) is in the intersection of VB † P n (a, b) and VB † P n (a ′ , b ′ ). Let p be a closed point in the intersection of VB † P n (a, b) and VB † P n (a ′ , b ′ ). We assume p ∈ VB † P n (c, d) for some (c, d) ∈ Betti(H) since M(H) is the disjoint union of these subspaces. By Lemma 2.20, it follows that (a, b) (c, d) and (a ′ , b ′ ) (c, d). Since (a ′′ , b ′′ ) (c, d) by the definition of join, another application of Lemma 2.20 shows that p ∈ VB † P n (a ′′ , b ′′ ).
Last but not least, we discuss the semistable case where the description of the stratification holds within the coarse moduli space.
By [20,Theorem 4.2], semistablity is open for a family of torsion-free sheaves. Furthermore, the set of semistable torsion sheaves with a given Hilbert polynomial χ is bounded in the sense of Maruyama, and thus have bounded regularity by [20,Theorem 3.11]. Let VB † P n (H) ss and VB † P n (a, b) ss denote the subset of isomorphism classes of semistable bundles in VB † P n (H) and VB † P n (a, b) respectively. It follows that VB † P n (H) ss and all VB † P n (a, b) ss are contained in VB † P n (H) ≤d for some large enough integer d. Since VB † P n (a, b) ss is open in VB † P n (a, b) and VB † P n (H) ss is open in VB † P n (H) by the similar reasoning as in Proposition 2.15, it follows that the stratification of VB † P n (H) ss by VB † P n (a, b) ss has the same description as given in Theorem 2.17.
Let M(χ) denote the coarse moduli space of semistable sheaves on P n with Hilbert polynomial χ. We show that the spaces VB † P n (H) ss and VB † P n (a, b) ss are subschemes of M(χ). Let M 0 (a, b) ss denote the open subscheme of M 0 (a, b) over which the fibers of the tautological family of bundles E 0 (a, b) are semistable. By the property of the coarse moduli space, there is a map p 0 : M 0 (a, b) ss → M(χ) inducing the family of semistable bundles. By Lemma 2.13, the isomorphism classes of the fibers are exactly given by the G(a, b)-orbits. Therefore VB † P n (a, b) ss is a subscheme of M(χ) with the image subscheme of p. Similarly, the space VB † P n (H) ≤d is also a subscheme of M(χ). Since VB † P n (H) ss is an open subspace of VB † P n (H) ≤d for some d ≫ 0, the same is true for VB † P n (H) ss .