MINIMAL RESOLUTION OF GENERAL STABLE RANK-2 VECTOR BUNDLES ON ℙ^{2}

CARLA DIONISI AND MARCO MAGGESI

Sunto. – Abbiamo studiato gli elementi generici degli spazi di moduli 𝔐_{ℙ^{2}}(2, c_{1}, c_{2}) dei fibrati vettoriali stabili di rango 2 su ℙ^{2} e le loro risoluzioni libere minimali. Ne segue una dimostrazione piuttosto semplice dell’irriducibilità di 𝔐_{ℙ^{2}}(2, c_{1}, c_{2}).

Abstract. – We study general elements of moduli spaces 𝔐_{ℙ^{2}}(2, c_{1}, c_{2}) of rank-2 stable holomorphic vector bundles on ℙ^{2} and their minimal free resolutions. Incidentally, a quite easy proof of the irreducibility of 𝔐_{ℙ^{2}}(2, c_{1}, c_{2}) is shown.

1. Introduction

We investigate stable rank-2 vector bundles on the complex projective plane ℙ^{2} by means of their minimal free resolutions. Bohnhorst and Spindler in their paper [BS92] develop interesting techniques for the study of minimal free resolution of rank-n stable vector bundles on ℙ^{n} of homological dimension 1. In this work we derive a number of consequences for rank-2 vector bundles on ℙ^{2}.

As Bohnhorst and Spindler observe, Betti numbers define a stratification of the moduli space 𝔐_{ℙ^{2}}(2, c_{1}, c_{2}) of stable holomorphic vector bundles on ℙ^{2} by constructible subsets. We estimate the codimension of such strata and characterize Betti numbers of the general element of the moduli space. As a corollary, we get a simple proof of the irreducibility of 𝔐_{ℙ^{2}}(2, c_{1}, c_{2}). Other proofs of the irreducibility of moduli spaces of stable vector bundles can be found in [Bar77a, Ell83, Hul79, HL93, Pot79, Mar78]. The irreducibility of 𝔐_{ℙ^{2}}(2, c_{1}, c_{2}) was proved for c_{1} even in [Bar77a] and c_{1} odd indipendently in [Pot79] and [Hul79].

We would like to thank G. Ottaviani for his invaluable guidance and V. Ancona for many useful discussions.

2. Admissible pairs and resolutions

Let ℰ be a rank-2 vector bundle on ℙ^{2}. By Horrocks’ theorem [Hor64], ℰ has homological dimension at most 1, i.e., it has a free resolution of the form

(1)  0 ⟶ ⨁_{i=1}^{k} 𝒪_{ℙ^{2}}(−a_{i}) ⟶^{Φ} ⨁_{j=1}^{k+2} 𝒪_{ℙ^{2}}(−b_{j}) ⟶ ℰ ⟶ 0.

We do not assume that such a resolution is minimal. In what follows we suppose that ℰ has homological dimension 1 (so that k > 0); we are not interested in vector bundles of homological dimension 0, i.e., splitting vector bundles, since they are not stable.

We suppose that the two sequences a_{i} and b_{i} are indexed in nondecreasing order

(2)  \begin{aligned}a_{1} & ≤ a_{2} ≤ ⋯ ≤ a_{k}, \\ b_{1} & ≤ b_{2} ≤ ⋯ ≤ b_{k+2}; \end{aligned}

call (a, b) = ((a_{1}, …, a_{k}), (b_{1}, …, b_{k+2})) the associated pair to the resolution (1). If this resolution is minimal, we call (a, b) the pair associated to the bundle ℰ. Notice that the associated pair and Betti numbers encode exactly the same information; in particular max(a_{k} − 1, b_{k+2}) is the Castelnuovo-Mumford regularity of ℰ.

Chern classes c_{1}, c_{2} of ℰ are expressed in terms of the a_{i}, b_{j} by the formulas

(3)  c_{1} = ∑_{i=1}^{k} a_{i} − ∑_{i=1}^{k+2} b_{i},

(4)  2c_{2} − c_{1}^{2} = ∑_{i=1}^{k} a_{i}^{2} − ∑_{i=1}^{k+2} b_{i}^{2}.

Definition 2.1. A pair (a, b) is said to be admissible if

(5)  a_{i} > b_{i+2}  for all i = 1, …, k

For the sake of brevity, we say that the resolution (1) is admissible if its associated pair (a, b) is so.

More generally, we can consider the associated pair (a, b) to any vector bundle of homological dimension 1 on ℙ^{n} with n ≥ 2. In that case, we say that (a, b) is admissible if a_{i} > b_{n+i} for i = 1, …, k, as in [BS92].

Let us restate the main results of Bohnhorst and Spindler on admissible pairs (proposition 2.3 and theorem 2.7 in [BS92]) in our settings.

Theorem 2.2. Let ℰ be the rank-2 vector bundle on ℙ^{2} of resolution (1). Then

(1) resolution (1) is minimal if and only if it is admissible and every constant entry of the matrix Φ is zero;

(2) if resolution (1) is admissible then ℰ is stable (resp. semistable) if and only if b_{1} > −μ (resp. b_{1} ≥ −μ) where μ = c_{1}/2 is the slope of ℰ.

We denote by ℑ the set of all admissible pairs (a, b) associated to rank-2 vector bundles on ℙ^{2} with Chern classes c_{1}, c_{2} which satisfy the condition

b_{1} > −μ = \frac{1}{2}(∑ a_{i} − ∑ b_{j}).

Theorem 2.2 shows that the set ℑ contains the set of all possible associated pairs to a stable vector bundle in 𝔐_{ℙ^{2}}(2, c_{1}, c_{2}) and coincides exactly with it. Then

(6)  𝔐_{ℙ^{2}}(2, c_{1}, c_{2}) = ∐_{(a,b)∈ℑ} 𝔐(a, b)

where 𝔐(a, b) is the constructible subset of 𝔐_{ℙ^{2}}(2, c_{1}, c_{2}) of vector bundles with associated pair (a, b).

Proposition 2.3. For all (a, b) ∈ ℑ, the closed set \overline{𝔐(a, b)} is an irreducible algebraic subset of 𝔐_{ℙ^{2}}(2, c_{1}, c_{2}) of dimension

dim \overline{𝔐(a, b)} = dim Hom(F_{1}, F_{0}) + dim Hom(F_{0}, F_{1})+
− dim End(F_{1}) − dim End(F_{0}) + 1 − #{(i, j) : a_{i} = b_{j}},

(7)

where F_{0} = ⊕_{j=1}^{k+2}𝒪(−b_{j}), F_{1} = ⊕_{i=1}^{k}𝒪(−a_{i}).

Proof. Proposition 3.3 in [BS92]. □

In the following lemma we find an upper bound on the regularity of semistable vector bundles on ℙ^{2} of rank-2. A lower bound is given in corollary 3.7.

Theorem 2.4. A normalized semistable rank-2 bundle ℰ on ℙ^{2} is c_{2}-regular.

Proof. For brevity’s sake, we set ξ_{i} := a_{i} − b_{i+2} and t_{i} := b_{i+2} − b_{2}. Obviously, ξ_{i} ≥ 1 and t_{i} ≥ 0. We rewrite (3) as

(8)  ∑_{i=1}^{k} ξ_{i} = b_{1} + b_{2} + c_{1}.

By equation (4) and theorem 2.2, using inequalities (2) and (5), we get

\begin{aligned}b_{1}^{2} + b_{2}^{2} + 2c_{2} − c_{1}^{2} & = ∑_{i=1}^{k}(a_{i}^{2} − b_{i+2}^{2}) = ∑_{i=1}^{k} ξ_{i}(2b_{2} + 2t_{i} + ξ_{i}) ≥ \\ & ≥ 2b_{2} ∑_{i=1}^{k} ξ_{i} + ∑_{i=1}^{k}(2t_{i} + ξ_{i}) = \\ & = (2b_{2} + 1)(b_{1} + b_{2} + c_{1}) + 2 ∑_{i=1}^{k} t_{i}. \end{aligned} ; (9)

If we suppose that b_{2} + ∑_{i=1}^{k} t_{i} ≥ c_{2} + 1, we have

b_{1}^{2} + b_{2}^{2} + 2b_{2} − c_{1}^{2} − (2b_{2} + 1)(b_{1} + b_{2} + c_{1}) ≥ 2.

The left side of the above inequality is non-increasing with respect to b_{1}, hence it remains true after substituting −c_{1} to b_{1}; but b_{2} − b_{2}^{2} ≥ 2 has no solutions. Thus ∑_{i=1}^{k} t_{i} must be at most c_{2} − b_{2} and in particular

(10)  b_{k+2} = b_{2} + t_{k} ≤ b_{2} + ∑ t_{i} ≤ c_{2}.

Now, we must show that a_{k} ≤ c_{2} + 1. We rewrite (3) as ∑_{i=1}^{k−1} ξ_{i} = b_{1} + b_{2} + b_{k+2} − a_{k} + c_{1} and by (4)

\begin{aligned}b_{1}^{2} + b_{2}^{2} + b_{k+2}^{2} − a_{k}^{2} + 2c_{2} − c_{1}^{2} & = ∑_{i=1}^{k−1}(a_{i}^{2} − b_{i+2}^{2}) = ∑_{i=1}^{k−1} ξ_{i}(2b_{2} + 2t_{i} + ξ_{i}) ≥ \\ & ≥ 2b_{2} ∑_{i=1}^{k−1} ξ_{i} + ∑_{i=1}^{k−1} ξ_{i} ≥ \\ & ≥ (2b_{2} + 1)(b_{1} + b_{2} + b_{k+2} − a_{k} + c_{1}) \end{aligned} ; (11)

that can be put in the form

(12) b_{1}^{2} + b_{2}^{2} − (2b_{2} + 1)(b_{1} + b_{2} + c_{1}) + 2c_{2} − c_{1}^{2} ≥

≥ (a_{k} − b_{k+2})(a_{k} + b_{k+2} − 2b_{2} − 1).

Suppose that a_{k} ≥ c_{2} + 2. By (10) we have a_{k} − b_{k+2} ≥ c_{2} + 2 − c_{2} = 2 and we observe also that a_{k} + b_{k+2} − 2b_{2} − 1 ≥ c_{2} − b_{2} + 1. Substituting and simplifying, equation (12) becomes

(13)  b_{1}^{2} + b_{2}^{2} − (2b_{2} + 1)(b_{1} + b_{2} + c_{1}) − c_{1}^{2} ≥ c_{1}^{2} + 2

As before, we can restrict ourselves to the case b_{1} = −c_{1} obtaining

b_{1}^{2} + b_{2}^{2} − (2b_{2} + 1)(b_{1} + b_{2}) ≥ 2

which has no solution for b_{i} positive. Then a_{k} ≤ c_{2} + 1. □

Remark 2.5. The above theorem is sharp. Indeed ((c_{2} + 1), (0, 1, c_{2})) and ((c_{2} + 1), (1, 1, c_{2})) are admissible pairs associated to rank-2 semistable bundles with Chern classes c_{1}, c_{2} and regularity c_{2}.

Remark 2.6. It is also possible to prove that a semistable rank-2 bundle on ℙ^{2} is c_{2} regular if c_{1} = 0 and (c_{2} + 1)-regular if c_{1} = −1 using the bounds on dimension of cohomology groups proved by Elencwajg and Forster (proposition 2.18 in [EF80]) and the Grauert-Mülich theorem.

Remark 2.7. From (8) and the thesis of the previous theorem, the value k in (1) is bounded by:

(14)  k ≤ ∑_{i=1}^{k} ξ_{i} = b_{1} + b_{2} + c_{1} ≤ 2c_{2} + c_{1}

Hence, for fixed Chern classes c_{1}, c_{2}, there are only a finite number of admissible pairs of rank-2 vector bundles and we can write an algorithm to enumerate such pairs restricting the search to a finite domain.

3. Natural pairs and general vector bundles

We say that (a, b) = ((a_{1}, …, a_{k}), (b_{1}, …, b_{k+2})) is a natural pair if it is admissible and

(15)  b_{k+2} < a_{1},  a_{k} ≤ b_{1} + 2.

The above inequalities imply a_{i} ≠ b_{j} for all i and j.

We observe that natural pairs are parametrized by three integers s, k, α such that

(16)  k ≥ 1  and  − k + 1 ≤ α ≤ k + 2

as follows: the pair (a, b)_{s,k,α} corresponding to the triple (s, k, α) is the pair associated to a resolution of the form

(17)  0 → 𝒪(−s − 1)^{k} → 𝒪(−s)^{α} ⊕ 𝒪(−s + 1)^{k−α+2} → ℰ → 0

if α ≥ 0, or of the form

(18)  0 → 𝒪(−s − 1)^{k+α} ⊕ 𝒪(−s)^{−α} → 𝒪(−s + 1)^{k+2} → ℰ → 0

if α < 0. We have excluded the case α = −k so that s is the regularity of the pair, i.e., s = max(a_{k} − 1, b_{k+2}).

In this section we are going to show that resolutions of general vector bundles have natural pairs.

Theorem 3.1. One has codim 𝔐(a, b) = 0 if and only if (a, b) is a natural pair.

As a remarkable consequence we will derive a quite simple proof of the irreducibility of moduli spaces of rank-2 stable vector bundles on ℙ^{2}, and we will compute regularity and cohomology of their general elements.

We recall that, since dim Ext^{2}(ℱ, ℱ) = 0 for any stable vector bundle ℱ on ℙ^{2}, the corresponding moduli space 𝔐_{ℙ^{2}}(2, c_{1}, c_{2}) is smooth of dimension

(19)  dim Ext^{1}(ℱ, ℱ) = 4c_{2} − c_{1}^{2} − 3.

Let us consider the function A(t) := h^{2}(𝒪(t)) and its finite differences of first and second order (Δ_{u}A)(t) := A(t + u) − A(t) and (Δ_{v}Δ_{u}A)(t) := (Δ_{u}A)(t + v) − (Δ_{u}A)(t).

Lemma 3.2. If ℰ ∈ 𝔐_{ℙ^{2}}(2; c_{1}, c_{2}) has associated admissible pair (a, b), then

\begin{aligned}codim \overline{𝔐(a, b)} & = h^{1}(ℰ(b_{1})) + h^{1}(ℰ(b_{2})) + #\{(i, j) : a_{i} = b_{j}\}+ \\ & + ∑_{i,j=1}^{k}(Δ_{b_{i+2}−a_{i}}Δ_{b_{j+2}−a_{j}}A)(a_{i} − b_{i+2}). \end{aligned} ; (20)

Proof. Let

(21)  0 → F_{1} → F_{0} → ℰ → 0

be the minimal resolution of ℰ where

(22)  F_{0} = ⨁_{j=1}^{k+2} 𝒪(−b_{j}),  F_{1} = ⨁_{i=1}^{k} 𝒪(−a_{i}).

Stability of ℰ ensures the vanishing dim(Ext^{2}(ℰ, ℰ)) = h^{2}(ℰ^{∗} ⊗ ℰ) = 0 so that h^{2}(F_{0}^{∗} ⊗ ℰ) = h^{2}(F_{1}^{∗} ⊗ ℰ). Then from (21) we easily find the following data:

\begin{aligned}h^{0}(F_{0}^{∗} ⊗ ℰ) & = h^{0}(F_{0}^{∗} ⊗ F_{0}) − h^{0}(F_{0}^{∗} ⊗ F_{1}), \\ h^{0}(F_{1}^{∗} ⊗ ℰ) & = h^{0}(F_{1}^{∗} ⊗ F_{0}) − h^{0}(F_{1}^{∗} ⊗ F_{1}), \\ dim(Ext^{1}(ℰ, ℰ)) & = h^{1}(ℰ^{∗} ⊗ ℰ) = \\ & = h^{1}(F_{0}^{∗} ⊗ ℰ) − h^{1}(F_{1}^{∗} ⊗ ℰ)+ \\ & + h^{0}(F_{1}^{∗} ⊗ ℰ) − h^{0}(F_{0}^{∗} ⊗ ℰ) + 1 \end{aligned} ; (23)

and from (7) we have

codim \overline{𝔐(a, b)} = dim(Ext^{1}(ℰ, ℰ)) − dim \overline{𝔐(a, b)} =
= h^{1}(F_{0}^{∗} ⊗ ℰ) − h^{1}(F_{1}^{∗} ⊗ ℰ) + #{(i, j) : a_{i} = b_{j}}

(24)

Now, by splitting F_{0} as 𝒪(−b_{1}) ⊕ 𝒪(−b_{2}) ⊕ F̃_{0} with F̃_{0} := ⊕_{i=3}^{k+2}𝒪(−b_{i}), the above formula becomes

\begin{aligned}codim \overline{𝔐(a, b)} & = h^{1}(ℰ(b_{1})) + h^{1}(ℰ(b_{2})) + #\{(i, j) : a_{i} = b_{j}\}+ \\ & + h^{1}(F̃_{0}^{∗} ⊗ ℰ) − h^{1}(F_{1}^{∗} ⊗ ℰ). \end{aligned} ; (25)

Since h^{2}(F̃_{0}^{∗} ⊗ F_{0}) = h^{2}(F̃_{0}^{∗} ⊗ F̃_{0}) and h^{2}(F_{1}^{∗} ⊗ F_{0}) = h^{2}(F_{1}^{∗} ⊗ F̃_{0}) the following identity holds:

(26) h^{1}(F̃_{0}^{∗} ⊗ ℰ) − h^{1}(F_{1}^{∗} ⊗ ℰ) =

= h^{2}(F̃_{0}^{∗} ⊗ F_{1}) − h^{2}(F̃_{0}^{∗} ⊗ F_{0}) − h^{2}(F_{1}^{∗} ⊗ F_{1}) + h^{2}(F_{1}^{∗} ⊗ F_{0}) =

= h^{2}(F̃_{0}^{∗} ⊗ F_{1}) − h^{2}(F̃_{0}^{∗} ⊗ F̃_{0}) − h^{2}(F_{1}^{∗} ⊗ F_{1}) + h^{2}(F_{1}^{∗} ⊗ F̃_{0}) =

= ∑_{i,j=1}^{2} [h^{2}(𝒪(b_{i+2} − a_{j})) − h^{2}(𝒪(b_{i+2} − b_{j+2}))+

− h^{2}(𝒪(a_{i} − a_{j})) + h^{2}(𝒪(a_{i} − b_{j+2}))].

Then equation (20) follows by substitution of (26) in (25). □

Proof of theorem 3.1. It can be verified by direct computation from proposition 2.3 that, if ℰ has natural pair, then the codimension of \overline{𝔐(a, b)} is zero. Conversely, let u, v be two non-negative integers. Since all finite differences (Δ_{u}A)(t) := A(t + u) − A(t) are non decreasing functions of t, then

(27)  (Δ_{v}Δ_{u}A)(t) ≥ 0

and by the previous lemma

(28)  codim \overline{𝔐(a, b)} ≥ h^{1}(ℰ(b_{1})) + h^{1}(ℰ(b_{2})) + #\{(i, j) : a_{i} = b_{j}\}.

If codim \overline{𝔐(a, b)} = 0, we have a_{k} ≤ b_{1} + 2 and #\{(i, j) : a_{i} = b_{j}\} = 0, since h^{1}(ℰ(b_{1})) = 0 implies h^{2}(F_{1}(b_{1})) = 0. This forces (a, b) to be a natural pair. □

Proposition 3.3. Suppose that 𝔐_{ℙ^{2}}(2, c_{1}, c_{2}) be nonempty and let

(29)  s := max{ρ ∈ ℤ : 2ρ^{2} + 2c_{1}ρ − 2ρ ≤ 2c_{2} − c_{1}^{2} + c_{1} − 1},

If α and k are defined by

(30)  \begin{aligned}α & := 2c_{2} − c_{1}^{2} + 2 − 2s^{2} − 2c_{1}s, \\ k & := (2s + c_{1} − 2 + |α|)/2, \end{aligned}

then (a, b)_{s,k,α} is the only natural pair of 𝔐_{ℙ^{2}}(2, c_{1}, c_{2}).

Proof. Note that, since 𝔐_{ℙ^{2}}(2, c_{1}, c_{2}) is nonempty, from theorem 3.1 there exists a vector bundle associated to a natural pair. It is easy to verify that equations (3) and (4) are equivalent to (30) and conditions (16) are equivalent to

2s^{2} + 2c_{1}s − c_{1} − 2s + 1 ≤ 2c_{2} − c_{1}^{2} ≤ 2s^{2} + 2c_{1}s + c_{1} + 2s.

The intervals

[2s^{2} + 2c_{1}s − c_{1} − 2s + 1, 2s^{2} + 2c_{1}s + c_{1} + 2s]

are disjoint for s varying in ℤ. Hence s is uniquely determined from c_{1}, c_{2} and satisfies (29). □

Remark 3.4. Equation (29) in proposition 3.3 is also equivalent to

(29bis)  s := min{ρ ∈ ℤ : 2ρ^{2} + 2c_{1}ρ + 2ρ ≥ 2c_{2} − c_{1}^{2} − c_{1}}.

Theorem 3.5. Moduli spaces of stable rank-2 vector bundles on ℙ^{2} are irreducible.

Proof. Moduli spaces of stable rank-2 vector bundles on ℙ^{2} are smooth. By theorem 3.1 and the above proposition they can have only one connected component. □

Corollary 3.6. The general element of 𝔐_{ℙ^{2}}(2, c_{1}, c_{2}) has natural cohomology.

The above corollary justify the terminology “natural pair”. A different proof for it, working also for rank greater than 2, can be found in [HL93], by using sophisticated techniques of stacks theory.

Using proposition 3.3 we are going to give some bounds on regularity and cohomology of stable vector bundles. In particular, for rank-2 vector bundles, the next two corollaries give respectively a refined version of corollary 5.4 in [Bru80] and proposition 7.1 in [Har78].

Corollary 3.7. The general vector bundle ℰ in 𝔐_{ℙ^{2}}(2, c_{1}, c_{2}) has regularity s given by (29).

Corollary 3.8. Let [ℰ] be a vector bundle in 𝔐 = 𝔐_{ℙ^{2}}(2, c_{1}, c_{2}) and let s be defined by (29). Then H^{0}(ℰ(t)) ≠ 0 if

t ≥ s  when 2s^{2} + 2c_{1}s + 2s = 2c_{2} − c_{1}^{2} − c_{1},
t ≥ s − 1  otherwise.

The above inequality is sharp, in the sense that it gives a necessary and sufficient condition for ℰ general.

Proof. Let ((a_{1}, …, a_{k}), (b_{1}, …, b_{k+2})) be the admissible pair associated to a vector bundle ℰ in 𝔐. Then one has H^{0}(ℰ(t)) ≠ 0 if and only if t − b_{1} ≥ 0. By the semicontinuity of cohomology groups and theorem 3.5, it is enough to restrict ourselves to the case where ℰ is general. So, by (17) and (18) one has H^{0}(ℰ(t)) ≠ 0 if and only if

t ≥ s  when α = k + 2
t ≥ s − 1  otherwise

and the condition α = k + 2 is equivalent to 2s^{2} + 2c_{1}s + 2s = 2c_{2} − c_{1}^{2} − c_{1} by (30). □

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Carla Dionisi – Dipartimento di Matematica Applicata “G. Sansone” Via S. Marta, 3, 50139, Firenze, Italy

                     email: dionisi@math.unifi.it

Marco Maggesi – Dipartimento di Matematica “U. Dini” Viale Morgagni, 67/a, 50134, Firenze, Italy

                     email: maggesi@math.unifi.it