Moduli Spaces of Sheaves on General Blow-ups of $\mathbb{P}^{2}$

In contrast to (semi)stable sheaves, the families of prioritary sheaves are easier to construct. Let $D$ be an effective divisor on $X$. A torsion-free coherent sheaf $\mathcal{E}$ on $X$ is called $D$-prioritary if

$$\operatorname{Ext}^{2}(\mathcal{E},\mathcal{E}(-D))=0.$$

For a character ${\bf{v}}\in K(X)_{\mathbb{Q}}$, let $\mathcal{P}_{D}({\bf{v}})\subseteq\operatorname{Coh}({\bf{v}})$ be the open substack of $D$-prioritary sheaves. If $X$ is some blow-up of $\mathbb{P}^{2}$ with an exceptional divisor $E$ and a fibre $F:=H-E$, where $H$ is the pullback of a hyperplane section on $\mathbb{P}^{2}$, then stack $\mathcal{P}_{F}({\bf{v}})$ of $F$-prioritary sheaves is irreducible by a theorem of Walter [Wal98]. Let $E_{i}$ be the exceptional divisors. It is natural to take the polarization $A$ to be $H-\sum\varepsilon_{i}E_{i}$ with $0<\varepsilon_{i}\ll 1$. Then every $\mu_{A}$-semistable sheaf is automatically $H$-prioritary. Therefore, if $M_{X,A}({\bf{v}})$ is nonempty, then it is an open dense substack of $\mathcal{P}_{H}({\bf{v}})$. We prove the following result.

The higher rank Brill-Noether theory aims to classify non-special Chern characters (Definition 1.2) on a polarized surface $(X,A)$. The applications have been found in classifying globally generated Chern characters ([CH18b]), describing effective cones of moduli spaces ([Hui16][CHW17]), and classifying Chern characters with non-empty moduli spaces $M_{X,A}({\bf{v}})$ ([CH21]).

The classification of non-special Chern characters was worked out for $\mathbb{P}^{2}$ in [GH94], and for Hirzebruch surfaces, including $\mathbb{P}^{1}\times\mathbb{P}^{1}$ and $\mathbb{F}_{1}$, in [CH18b]. For del Pezzo surfaces and arbitrary blow-ups, partial results were obtained in [CH20] under the condition that the Chern character ${\bf{v}}$ satisfies $\chi({\bf{v}})=0$. For del Pezzo surfaces of degree $\geq 4$, a classification of all non-special Chern characters is given in [LZ19]. In this paper, we classify the Chern characters for the blow-ups of $\mathbb{P}^{2}$ along $m$ general points which satisfies the weak Brill-Noether property.

Let $X$ be the blow-up of $\mathbb{P}^{2}$ along $m$ distinct points $p_{1},...,p_{m}$, and $E_{1},...,E_{m}$ be the corresponding exceptional divisors. When $p_{1},...,p_{m}$ are collinear, we have the following.

We will see in Section 9 that stable sheaves can deform to nearby surfaces. Applying the semicontinuity of the dimensions of cohomology groups, one obtains the following result on general blow-ups.

Recall that an exceptional bundle is a simple vector bundle $\mathcal{E}$ with $\operatorname{Ext}^{i}(\mathcal{E},\mathcal{E})=0$ for $i>0$. On $\mathbb{P}^{2}$, there is a beautiful description of the Chern characters of exceptional bundles [LP97]. When $X$ is a del Pezzo surface, it is known that every torsion-free exceptional sheaf is locally free, constructible (see Definition 6.9) and stable with respect to the anticanonical polarization. See [KO95] for a thorough study of exceptional objects on del Pezzo surfaces.

On the blow-up of $\mathbb{P}^{2}$ along $m$ distinct points, we don’t know whether there are non-constructible exceptional bundles. However, we still have the following result on the stability of constructible ones.

On $\mathbb{P}^{2}$, the existence of stable sheaves is controlled by the exceptional bundles. Drézet and Le Potier construct a function $\delta:\mathbb{R}\rightarrow\mathbb{R}$ whose graph in the $(\mu,\Delta)$-plane, which completely determines when $M_{H}({\bf{v}})$ is nonempty. If $(\mu({\bf{v}}),\delta({\bf{v}}))$ lies above the graph of $\delta$, then $M_{H}({\bf{v}})$ is nonempty. Otherwise, $M_{H}({\bf{v}})$ is empty or ${\bf{v}}$ is semi-exceptional. See [DLP85] and [LP97] for the argument and [CH21] for a figurative illustration. The classification of semistable characters on $\mathbb{P}^{1}\times\mathbb{P}^{1}$ is worked out in [Rud94]. For Hirzebruch surfaces and generic polarizations, the classification of non-empty moduli spaces is worked out by [CH21]. The existence theorems for del Pezzo surfaces of degree $\geq 3$ with the anti-canonical polarization is given by [LZ19].

As a consequence of the classification on $\mathbb{P}^{2}$, we are able to construct a family of stable bundles on general blow-ups of $\mathbb{P}^{2}$ by analyzing the special blow-up along collinear points. We define the weak DL-condition for a Chern character ${\bf{v}}$ in Section 8. Roughly speaking, it means that for a constructible exceptional bundle $\mathcal{E}$ whose slope is close to the slope of ${\bf{v}}$, the Euler characteristic $\chi({\bf{v}},\mathcal{E})$ or $\chi(\mathcal{E},{\bf{v}})$ has the expected sign.

In Section 2, we recall the preliminary facts needed in the rest of the paper. In Section 3, we study the basic properties of the blow-up of $\mathbb{P}^{2}$, especially the cohomology of line bundles.

Section 4 compiles some useful properties and constructions of special prioritary sheaves. We construct a family of prioritary sheaves and characterize Chern characters on the blow-up of $\mathbb{P}^{2}$ along distinct collinear points that satisfy the weak Brill-Noether property. In Section 5, we prove the stability of constructible exceptional bundles, which is crucial in the classification of stable Chern characters.

In sections 6, we define a sharp Bogomolov function and determine its basic properties. We primarily concentrate on the characters on the blow-up of $\mathbb{P}^{2}$ along collinear points. In this case, we study the stability of sheaves with respect to generic polarization in detail. In Section 7, using deformation theory, we generalize our results to general blow-ups of $\mathbb{P}^{2}$.

Finally, in Section 8, we compute its ample cone of the Hilbert scheme of points on the blow-up of $\mathbb{P}^{2}$ along distinct collinear points.  

Acknowledgements The author is greatly indebted to his advisor, Izzet Coskun, for suggesting the problem and many stimulating conversations. The author would also like to thank Lawrence Ein, Benjamin Gould, Shizhuo Zhang, Yeqin Liu, Sixuan Lou for helpful discussions and suggestions.

Let $\mathcal{E}$ be a torsion-free sheaf on a polarized surface $(X,A)$. Let $K(X)_{\mathbb{Q}}$ be the Grothendieck group of $X$ with $\mathbb{Q}$-coefficients. The Chern character $\operatorname{ch}(\mathcal{E})=(\operatorname{ch}_{0}(\mathcal{E}),\operatorname{ch}_{1}(\mathcal{E}),\operatorname{ch}_{2}(\mathcal{E}))$ is given by

$$\operatorname{ch}_{0}(\mathcal{E})=r(\mathcal{E}),\quad\operatorname{ch}_{1}(\mathcal{E})=c_{1}(\mathcal{E}),\quad\operatorname{ch}_{2}(\mathcal{E})=\frac{c_{1}^{2}-2c_{2}}{2}.$$

We define the total slope $\nu$, the $A$-slope $\mu_{A}$ and the discriminant $\Delta$ by

$$\nu(\mathcal{E})=\frac{c_{1}(\mathcal{E})}{r(\mathcal{E})},\quad\mu_{A}(\mathcal{E})=\frac{c_{1}(\mathcal{E}).A}{r(\mathcal{E})},\quad\Delta(\mathcal{E})=\frac{\mu(\mathcal{E})^{2}}{2}-\frac{\operatorname{ch}_{2}(\mathcal{E})}{r(\mathcal{E})}.$$

These quantities depend only on the Chern character of $\mathcal{E}$ (and the polarization) but not on the particular sheaf. Given a Chern character ${\bf{v}}$, we define the total slope $\nu$, the $A$-slope $\mu_{A}$ and the discriminant $\Delta$ of ${\bf{v}}$ by the same formulae.

Then we have the following Riemann-Roch formula for torsion-free sheaves $\mathcal{E}$ and $\mathcal{F}$:

$$\chi(\mathcal{E},\mathcal{F})=r(\mathcal{E})r(\mathcal{F})(P(\nu(\mathcal{F})-\nu(\mathcal{E}))-\Delta(\mathcal{E})-\Delta(\mathcal{F})),$$

where

$$P(\nu):=\chi(\mathcal{O}_{X})+\frac{\nu.(\nu-K_{X})}{2}$$

is the Hilbert polynomial of $\mathcal{O}_{X}$. In particular, taking $\mathcal{E}=\mathcal{O}_{X}$, this reduces to the usual Riemann-Roch formula

$$\chi(\mathcal{F})=r(\mathcal{F})(P(\nu(\mathcal{F}))-\Delta(\mathcal{F})).$$

We now introduce the basic notions and properties about stability conditions. For more details, see [HL10] or [LP97].

The sheaf $\mathcal{E}$ is called $\mu_{A}$-semistable if every proper subsheaf $0\neq\mathcal{F}\subseteq\mathcal{E}$ of smaller rank satisfies

$$\mu_{A}(\mathcal{F})\leq\mu_{A}(\mathcal{E}).$$

If the inequality is strict for every such $\mathcal{F}$, then $\mathcal{E}$ is called $\mu_{A}$-stable.

Now suppose that $A$ is an integral divisor. Define the Hilbert polynomial $P_{\mathcal{E}}(m)$ and the reduced Hilbert polynomial $p_{\mathcal{E}}(m)$ of $\mathcal{E}$ by

$$P_{\mathcal{E}}(m)=\chi(\mathcal{E}(m)),\quad p_{\mathcal{E}}(m)=\frac{\chi(\mathcal{E}(m))}{r(\mathcal{E})}.$$

Then $\mathcal{E}$ is $A$-semistable (or Gieseker semistable) if every proper subsheaf $0\neq\mathcal{F}\subseteq\mathcal{E}$ of smaller rank satisfies that

$$p_{\mathcal{F}}(m)\leq p_{\mathcal{E}}(m)$$

for all $m\gg 0$. If the inequality is strict for every such $\mathcal{F}$, then $\mathcal{E}$ is called $A$-stable (or Gieseker $A$-stable). It follows immediately from the Riemann-Roch formula that

$$\mu_{A}\textup{-stable}\Rightarrow A\textup{-stable}\Rightarrow A\textup{-semistable}\Rightarrow\mu_{A}\textup{-semistable}.$$

If $\mathcal{E}$ is $\mu_{A}$-semistable for some polarization $A$, then $\Delta(\mathcal{E})\geq 0$ by Bogomolov inequality.

Every torsion free sheaf $\mathcal{E}$ admits a Harder-Narasimhan filtration with respect to both $\mu_{A}$- and $A$-stability, that is there is a finite filtration

$$0=\mathcal{E}_{0}\subset\mathcal{E}_{1}\subset\mathcal{E}_{2}\subset\cdots\subset\mathcal{E}_{n}=\mathcal{E}$$

such that the quotients

$$\mathcal{G}_{i}=\mathcal{E}_{i}/\mathcal{E}_{i-1},$$

called Harder-Narasimhan factors, are $\mu_{A}$- (respectively $A$-) semistable and

$$\mu_{A}(\mathcal{G}_{i})>\mu_{A}(\mathcal{G}_{i+1})\quad\quad\textup{(respectively, }p_{\mathcal{G}_{i}}(m)>p_{\mathcal{G}_{i+1}}(m)\textup{ for }m\gg 0\textup{)}$$

for $1\leq i\leq n-1$. Moreover, the Harder-Narasimhan filtration is unique.

In this section, we recall some results by Walter [Wal98].

In this section, we recall some known results of exceptional bundles.

A coherent sheaf $\mathcal{E}$ on $X$ is called exceptional if $\operatorname{Hom}(\mathcal{E},\mathcal{E})=\mathbb{C}$ and $\operatorname{Ext}^{i}(\mathcal{E},\mathcal{E})=0$ for any $i>0$. The Mukai’s lemma ([Muk84] [KO95]) implies that every torsion free exceptional sheaves are locally free. Thus we call torsion free exceptional sheaves exceptional bundles. There do exist torsion exceptional sheaves. For example, the structure sheaves $\mathcal{O}_{E_{i}}$ of exceptional divisors on the blow-up of $\mathbb{P}^{2}$ along general points are exceptional.

Let $A=H-\sum\varepsilon_{i}E_{i}$ be a polarization on $X$, i.e. $\varepsilon_{i}>0$ and $\sum\varepsilon_{i}<1$. We say that $A$ is generic (or $\varepsilon$ is generic) if $(\varepsilon_{1},...,\varepsilon_{m})$ is a generic point in the region defined by $\varepsilon_{i}>0$ and $\sum\varepsilon_{i}<1$. Sometimes, we prefer to take $A=H-\varepsilon\sum E_{i}$, that is, $\varepsilon_{1}=\cdots=\varepsilon_{m}$, whence we mean $\varepsilon$ is a generic number in $(0,1/m)$ by saying $A$ is generic or $\varepsilon$ is generic.

The following lemma is proved in [CH21] on Hirzebruch surfaces. For the reader’s convenience, we give the proof on our surface here.

The simplest examples of exceptional bundles on blow-ups of $\mathbb{P}^{2}$ are line bundles. Now given an ordered pair of sheaves $(\mathcal{E},\mathcal{F})$, we form the evaluation and coevaluation maps

$$\operatorname{ev}:\mathcal{E}\otimes\operatorname{Hom}(\mathcal{E},\mathcal{F})\longrightarrow\mathcal{F},\quad\operatorname{coev}:\mathcal{E}\longrightarrow\mathcal{F}\otimes\operatorname{Hom}(\mathcal{E},\mathcal{F})^{*},$$

each of which is associated to the identity element of the space $\operatorname{Hom}(\mathcal{E},\mathcal{F})\otimes\operatorname{Hom}(\mathcal{E},\mathcal{F})^{*}$. If the evaluation map is surjective, then we consider the kernel

$$0\longrightarrow L_{\mathcal{E}}\mathcal{F}\longrightarrow\mathcal{E}\otimes\operatorname{Hom}(\mathcal{E},\mathcal{F})\longrightarrow\mathcal{F}\longrightarrow 0;$$

if the coevaluation map is injective, then we consider the cokernel

$$0\longrightarrow\mathcal{E}\longrightarrow\mathcal{F}\otimes\operatorname{Hom}(\mathcal{E},\mathcal{F})^{*}\longrightarrow R_{\mathcal{F}}\mathcal{E}\longrightarrow 0.$$

If $(\mathcal{E},\mathcal{F})$ is an ordered pair of exceptional bundles, then the left and right mutations are exceptional whenever they are defined. This gives us a way of producing exceptional bundles.

Start with a strong exceptional collection $\sigma_{0}=(\mathcal{E}_{0},...,\mathcal{E}_{n})$ on a surface $X$. A transformation of the exceptional collection $\sigma_{0}$ is defined as a transformation of a pair of neighboring objects in this collection. One extends $\sigma$ to an infinite periodic collection $(\mathcal{E}_{i})_{i\in\mathbb{Z}}$ by setting $\mathcal{E}_{i+(n+1)k}=\mathcal{E}_{i}\otimes(\omega_{X}^{*})^{\otimes k}$ for $i=0,1,...,n$. We can also do mutations in collections: if $(\mathcal{E}_{i},\mathcal{E}_{i+1})$ has a surjective evaluation map (resp. injective coevaluation map), then one replaces $(\mathcal{E}_{i},\mathcal{E}_{i+1})$ by $(L_{\mathcal{E}_{i}}\mathcal{E}_{i+1},\mathcal{E}_{i})$ (resp. $(\mathcal{E}_{i+1},R_{\mathcal{E}_{i}+1}\mathcal{E}_{i})$). When the operations are defined, we can iterate mutations. Write $L_{j}(\mathcal{E}_{i})_{i\in\mathbb{Z}}$ for the left mutation $L_{\mathcal{E}_{j}}$.

On the surface $X$ obtained by blow up $\mathbb{P}^{2}$ along $m$ distinct points, one has a standard exceptional collection

$$\sigma_{0}=\left(\mathcal{O}(-2H),\mathcal{O}(-H),\mathcal{O}(-E_{1}),\cdots,\mathcal{O}(-E_{m}),\mathcal{O}\right).$$

If we know the dimension of the global sections of a line bundle $\mathcal{O}_{X}(D)$, then by Serre duality, one has

$$h^{2}(X,\mathcal{O}_{X}(D))=h^{0}(X,\mathcal{O}_{X}(K_{X}-D)).$$

One can also compute $h^{1}(X,\mathcal{O}_{X}(D))$ via Riemann-Roch formula:

$$h^{1}(X,\mathcal{O}_{X}(D))=h^{0}(X,\mathcal{O}_{X}(D))+h^{0}(X,\mathcal{O}_{X}(K_{X}-D))-\frac{D.(D-K_{X})}{2}-1.$$

In this way, we know all the cohomology of a line bundle.

Now we compute the global sections of line bundles on $X$. Let $D=aH-\sum b_{i}E_{i}$ be a divisor with $a\geq\max\{b_{1},0\}$. If any $b_{i}<0$, then chasing the exact sequence

$$0\longrightarrow\mathcal{O}_{X}(D-E_{i})\longrightarrow\mathcal{O}_{X}(D)\longrightarrow\mathcal{O}_{E_{i}}(b_{i})\longrightarrow 0,$$

one has $h^{0}(X,\mathcal{O}_{X}(D))=h^{0}(X,\mathcal{O}_{X}(D-E_{i}))$, so one can replace $D$ by $D-E_{i}$. Repeating this, we may assume that $b_{i}\geq 0$ for any $i$.

Assume first that $D$ is nef. Chasing the exact sequence

$$0\longrightarrow\mathcal{O}_{X}((n-1)H)\longrightarrow\mathcal{O}_{X}(nH)\longrightarrow\mathcal{O}_{H}(n)\longrightarrow 0,$$

and using the vanishing of $h^{1}(X,\mathcal{O}_{X}(nH))$ for $n\geq 0$, one obtains that

$$h^{0}(X,\mathcal{O}_{X}(aH))=\frac{(a+1)(a+2)}{2}.$$

Now consider the exact sequence

$$0\longrightarrow\mathcal{O}_{X}(aH-bE_{1})\longrightarrow\mathcal{O}_{X}(aH-(b-1)E_{1})\longrightarrow\mathcal{O}_{E_{1}}(b-1)\longrightarrow 0$$

for $1\leq b\leq b_{1}$. Since $a\geq b_{1}$, one always has the surjection

$$H^{0}(X,aH-(b-1)E_{1})\twoheadrightarrow H^{0}(E_{1},\mathcal{O}_{E_{1}}(b-1)).$$

As a consequence, we get

$$h^{0}(X,\mathcal{O}_{X}(aH-b_{1}E_{1}))=\frac{(a+1)(a+2)}{2}-\frac{b_{1}(b_{1}+1)}{2}.$$

Repeating this for $E_{2},...,E_{m}$ and using the assumption that $a\geq\sum b_{1}$, we deduce that

$$h^{0}(X,\mathcal{O}_{X}(D))=\frac{(a+1)(a+2)}{2}-\sum_{i}^{m}\frac{b_{i}(b_{i}+1)}{2}.$$

(1)

If $D$ is not nef, then the short exact sequence

$$0\longrightarrow\mathcal{O}_{X}(D-L)\longrightarrow\mathcal{O}_{X}(D)\longrightarrow\mathcal{O}_{L}\left(a-\sum b_{i}\right)\longrightarrow 0$$

together with $a<\sum b_{i}$ implies that $h^{0}(X,\mathcal{O}_{X}(D))=h^{0}(X,\mathcal{O}_{X}(D-L))$. Thus we can replace $D$ by $D-L=(a-1)-\sum(b_{i}-1)E_{i}$. If some coefficient $b_{i}$ becomes negative, one uses the previous reduction to place $b_{i}$ by $0$. Repeating this, we reduce the computation to the case when $D$ is nef.

Now we can prove the following useful properties.