Moduli Stacks of Bundles on Local Surfaces

Abstract

We give an explicit groupoid presentation of certain stacks of vector bundles on formal neighborhoods of rational curves inside algebraic surfaces. The presentation involves a Möbius type action of an automorphism group on a space of extensions.

Appendices

Appendix A: Some Cohomology Groups

The ring of global functions on $\widehat{Z_{k}}$ is

$$\displaystyle{\mathcal{O}(\widehat{Z_{k}}) = \mathbb{C}[[x_{0},x_{1},\mathop{\ldots },x_{k}]]\bigl /\sum _{i=0}^{k-2}\sum _{ j=i+2}^{k}\bigl (x_{ i}x_{j} - x_{i+1}x_{j-1}\bigr )\text{,}}$$

and for $Z_{k}^{(n)}$ one gets $\mathcal{O}(Z_{k}^{(n)}) = \mathcal{O}(Z_{k})/m^{n+1}$ where m is the ideal $(x_{0},\ldots,x_{k})$. Note that here $x_{i} = z^{i}u$ in terms of the original coordinates on U and U ⁽ⁿ⁾. The zeroth cohomology is the torsion-free $\mathcal{O}(\widehat{Z_{k}})$ module

$$\displaystyle{H^{0}(\widehat{Z_{ k}},\mathcal{O}(s)) =\bigoplus _{\mathit{ki}+s-l\geq 0,l\geq 0,i\geq 0}\mathbb{C}z^{l}u^{i} \subset \mathcal{O}(\hat{U}).}$$

Similarly, we have the $\mathcal{O}(Z_{k}^{(n)})$ module

$$\displaystyle{H^{0}(Z_{ k}^{(n)},\mathcal{O}(s)) =\bigoplus _{\mathit{ ki}+s-l\geq 0,l\geq 0,n\geq i\geq 0}\mathbb{C}z^{l}u^{i} \subset \mathcal{O}(U^{(n)}).}$$

Remark 8.

The set $H^{0}(Z_{k}^{(n)},\mathcal{O}(s))$ is the $\mathbb{C}$ points of the spectrum of the polynomial algebra freely generated over $\mathbb{C}$ by variables indexed by pairs (l, i) such that $\mathit{ki} + s - l \geq 0,l \geq 0,n \geq i \geq 0$. It is also easy to see that $H^{1}(Z_{k}^{(n)},\mathcal{O}(s))$ vanishes for s ≥ 0.

Appendix B: The Cohomology Spectral Sequence of $\mathcal{H}\mathbf{\mathit{om}}(E,F)$

Consider a scheme Z covered by just two affine open sets U ₁ and U ₂ and two rank 2 vector bundles E and F on Z which trivialize on the U _i . Assume also that $H^{1}(Z,\mathcal{O}) = 0$. The Čech complex for computing the cohomology of $\mathcal{H}\mathit{om}(E,F)$ on Z looks like

$$\displaystyle{\mathrm{Hom}_{U_{1}}(E\vert _{U_{1}},F\vert _{U_{1}})\oplus \mathrm{Hom}_{U_{2}}(E\vert _{U_{2}},F\vert _{U_{2}}) \rightarrow \mathrm{Hom}_{U_{1}\cap U_{2}}(E\vert _{U_{1}\cap U_{2}},F\vert _{U_{1}\cap U_{2}}).}$$

If we choose local trivializations for $E\vert _{U_{1}},E\vert _{U_{2}}$ and $F\vert _{U_{1}},F\vert _{U_{2}}$ then the complex becomes

$$\displaystyle{\mathrm{Hom}_{U_{1}}(\mathcal{O}^{\oplus 2},\mathcal{O}^{\oplus 2}) \oplus \mathrm{Hom}_{ U_{2}}(\mathcal{O}^{\oplus 2},\mathcal{O}^{\oplus 2}) \rightarrow \mathrm{Hom}_{ U_{1}\cap U_{2}}(\mathcal{O}^{\oplus 2},\mathcal{O}^{\oplus 2})}$$

with differential

$$\displaystyle{(A,B)\mapsto G_{E}A - BG_{F}}$$

where $G_{E},G_{F}$ are the transition matrices of E and F. On the other hand suppose we know that E and F can be written on Z as extensions of line bundles L ₂ by L ₁. By choosing local splittings the Čech complex becomes

$$\displaystyle{\mathrm{End}_{U_{1}}(\mathcal{O}^{\oplus 2}) \oplus \mathrm{End}_{ U_{2}}(\mathcal{O}^{\oplus 2})\stackrel{D_{ 1}}{\rightarrow }\mathrm{End}_{U_{1}\cap U_{2}}(\mathcal{O}^{\oplus 2})}$$

$$\displaystyle{D_{1}(N_{1},N_{2}) = \left (\begin{array}{*{10}c} g_{1} & 0 \\ 0 &g_{2}\\ \end{array} \right )N_{1}-N_{2}\left (\begin{array}{*{10}c} g_{1} & 0 \\ 0 &g_{2}\\ \end{array} \right ),}$$

$$\displaystyle{D_{2}(M_{1},M_{2}) = \left (\begin{array}{*{10}c} g_{1} & p_{E} \\ 0 & g_{2}\\ \end{array} \right )M_{1}-M_{2}\left (\begin{array}{*{10}c} g_{1} & p_{F} \\ 0 & g_{2}\\ \end{array} \right ).}$$

$$\displaystyle{\ker (D_{1})\stackrel{\overline{D_{2}}}{\longrightarrow }\mathrm{coker}(D_{1})}$$

Let us compute the cohomology groups

$$\displaystyle{\ker (\overline{D_{2}}) = \mathrm{Hom}(E,F)\mathop{\cong}H^{0}(X,\mathcal{H}\mathit{om}(E,F))}$$

and

$$\displaystyle{\mathrm{coker}(\overline{D_{2}}) = \mathrm{Ext}^{1}(E,F)\mathop{\cong}H^{1}(X,\mathcal{H}\mathit{om}(E,F))}$$

in terms of the extension and cohomology groups of the L _i . The filtration on $\mathcal{H}\mathit{om}(E,F)$ reads

$$\displaystyle{0 \subset \mathcal{H}\mathit{om}(L_{2},L_{1}) \subset \mathcal{H}\mathit{om}(E,L_{1}) + \mathcal{H}\mathit{om}(L_{2},F) \subset \mathcal{H}\mathit{om}(E,F)}$$

with associated graded pieces $\mathcal{H}\mathit{om}(L_{2},L_{1})$, $\mathcal{E}\mathit{nd}(L_{1}) \oplus \mathcal{E}\mathit{nd}(L_{2})$, and $\mathcal{H}\mathit{om}(L_{1},L_{2})$. The associated spectral sequence computing the cohomology $\mathcal{H}\mathit{om}(E,F)$ has an E ₁ term which looks like

The E ₂ term looks like

The E ₃ term looks like

The first differential we consider is

$$\displaystyle{H^{0}(X,\mathcal{O})^{\oplus 2} = \mathrm{End}(L_{ 1}) \oplus \mathrm{End}(L_{2})\stackrel{d_{1}^{1,-1}}{\rightarrow }\mathrm{Ext}^{1}(L_{ 2},L_{1}).}$$

It is the connecting map for the cohomology of the short exact sequence

$$\displaystyle\begin{array}{rcl} 0& \rightarrow & \mathcal{H}\mathit{om}(L_{2},L_{1}) \rightarrow \mathcal{H}\mathit{om}(L_{2},F) + \mathcal{H}\mathit{om}(E,L_{1}) {}\\ & \rightarrow & \mathcal{E}\mathit{nd}(L_{1}) \oplus \mathcal{E}\mathit{nd}(L_{2}) \rightarrow 0 {}\\ \end{array}$$

Consider the induced filtration on Hom(E, F) given by

$$\displaystyle{0 \subset \mathrm{Hom}(L_{2},L_{1}) \subset \mathrm{Hom}(E,L_{1}) + \mathrm{Hom}(L_{2},F) \subset \mathrm{Hom}(E,F).}$$

One has

$$\displaystyle{ \frac{\mathrm{Hom}(E,F)} {\mathrm{Hom}(E,L_{1}) + \mathrm{Hom}(L_{2},F)}\mathop{\cong}\text{ker}(d_{2}^{0,0}) \subset \mathrm{Hom}(L_{ 1},L_{2}),}$$

and

$$\displaystyle{\frac{\mathrm{Hom}(E,L_{1}) + \mathrm{Hom}(L_{2},F)} {\mathrm{Hom}(L_{2},L_{1})} \mathop{\cong}\text{ker}(d_{1}^{1,-1}) \subset H^{0}(X,\mathcal{O})^{\oplus 2}.}$$

For any choices of splittings

$$\displaystyle{\mathrm{Hom}(E,F)\stackrel{\psi }{\leftarrow }\text{ker}(d_{2}^{0,0}) \subset \mathrm{Hom}(L_{ 1},L_{2})}$$

and

$$\displaystyle{\mathrm{Hom}(E,L_{1}) + \mathrm{Hom}(L_{2},F)\stackrel{\phi }{\leftarrow }\text{ker}(d_{1}^{1,-1}) \subset H^{0}(X,\mathcal{O})^{\oplus 2}}$$

we get a decomposition

$$\displaystyle{ \mathrm{Hom}(E,F) = \mathrm{Hom}(L_{2},L_{1}) \oplus \phi (\text{ker}(d_{1}^{1,-1})) \oplus \psi (\text{ker}(d_{ 2}^{0,0})). }$$

(43)

We record formulas for $d_{1}^{1,-1}$ and $d_{2}^{0,0}$ in the case that $X = Z_{k}^{(n)} \times T$ for some affine scheme T, $L_{1} = \mathcal{O}(-j)$, $L_{2} = \mathcal{O}(j)$, E = E _p , $F = E_{p^{{\prime}}}$.

$$\displaystyle{d_{1}^{1,-1}: H^{0}(X,(L_{ 1} \otimes L_{1}^{\vee }) \oplus (L_{ 2} \otimes L_{2}^{\vee })) \rightarrow \mathrm{Ext}^{1}(L_{ 2},L_{1})}$$

We compute

$$\displaystyle{\left (\begin{array}{*{10}c} z^{j}& p^{{\prime}} \\ 0 &z^{-j}\\ \end{array} \right )\left (\begin{array}{*{10}c} \underline{a}&0\\ 0 &\underline{d}\\ \end{array} \right )-\left (\begin{array}{*{10}c} \underline{a}&0\\ 0 &\underline{d}\\ \end{array} \right )\left (\begin{array}{*{10}c} z^{j}& p \\ 0 &z^{-j}\\ \end{array} \right ) = \left (\begin{array}{*{10}c} 0&\underline{d}p^{{\prime}}-\underline{ a}p \\ 0& 0\\ \end{array} \right ).}$$

Therefore the element of $\mathrm{Ext}^{1}(L_{2},L_{1})$ to which the pair $(\underline{a},\underline{d})$ maps is represented by $(\underline{d}p^{{\prime}}-\underline{ a}p)\vert _{(U^{(n)}\cap V ^{(n)})\times T}$. The differential

$$\displaystyle\begin{array}{rcl} & & d_{1}^{1,-1}: H^{0}(X,\mathcal{O}^{\oplus 2}) \rightarrow \mathrm{Ext}^{1}(\mathcal{O}(j),\mathcal{O}(-j)) {}\\ & & \qquad \qquad (\underline{a},\underline{d})\mapsto \underline{d}p^{{\prime}}-\underline{ a}p. {}\\ \end{array}$$

In order to write down the next differential

$$\displaystyle{d_{2}^{0,0}: \mathrm{Hom}(\mathcal{O}(-j),\mathcal{O}(j)) \rightarrow \mathrm{Ext}^{1}(\mathcal{O}(j),\mathcal{O}(-j))/\text{image}(d_{ 1}^{1,-1})\text{,}}$$

we choose regular functions $\alpha _{U},\delta _{U}$ on U and $\alpha _{V },\delta _{V }$ on V such that

$$\displaystyle\begin{array}{rcl} -z^{-j}p^{{\prime}}\underline{c}_{ U}& =& \alpha _{U} -\alpha _{V } {}\\ z^{j}p\underline{c}_{ U}& =& \delta _{U} -\delta _{V } {}\\ \end{array}$$

so

$$\displaystyle{d_{2}^{0,0}(\underline{c}) =\delta _{ U}p^{{\prime}}-\alpha _{ V }p.}$$