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Assume $X$ is a smooth projective scheme over some algebraically closed field $k$. Let $M_1$ and $M_2$ be two moduli spaces of vector bundles on $X$.

The first space contains just one point, this point is given by a bundle $F$ with $dim(Ext^1(F,F))=0$ and $dim(Ext^2(F,F))=1$. So we have $M_1=Spec(k)$.

The second space also contains just one point. This point is given by a bundle $E$, such that $dim(Ext^1(E,E))=dim(Ext^2(E,E))=1$.

How can i determine the structure of $M_2$?

Since $dim(M_2)=0 < dim(Ext^1(E,E)=1$ i think $M_2$ cannot be smooth. And since $Ext^2(E,E)=1$ the bundle is possibly obstructed.

I think $M_2$ is possibly a non reduced point $Spec(k[x]/(x^n))$ for some $n\in \mathbb{N}$.

Is this idea corerct? How to find the number $n$? Is there a general way to determine the local structure of a moduli space at some point by just knowing the $Ext$-groups of that point? What are other possible structures for $M_2$?

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  • $\begingroup$ Being smooth and unobstructed are the same. Why is $\dim(M_2) = 0$? With $\dim (M_2) \leq 1$, $M_2$ is either a smooth curve or equal to $Spec(k[x]/(x^n))$ for some $n>1$ around $E$. I think finding $n$ might be difficult. $\endgroup$
    – Piotr Achinger
    Commented Mar 4, 2013 at 21:12
  • $\begingroup$ If one knows that $M_2$ contains just one point, could the space really be one-dimensional? May be that's where i am thinking in the wrong direction $\endgroup$
    – TonyS
    Commented Mar 5, 2013 at 12:11
  • $\begingroup$ Also take a look at Ravi Vakil's "Murphy Law" paper, namely singularities of moduli spaces of stable torsion free sheaves. I do not know though if the same could be done for bundles. $\endgroup$
    – Misha
    Commented Mar 5, 2013 at 16:03

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The moduli space is locally cut out by the zero set of the Kuranishi map $Ext^1(E,E)\to Ext^2(E,E)$. I believe that the quadradic term of the Kuranishi map is given by the Yoneda pairing, $ Ext^1(E,E)\otimes Ext^1(E,E)\to Ext^2(E,E)$, but in general, there are higher order terms which are more complicated to compute. If those higher terms are zero (and so that the moduli space is locally determined by these Ext groups, and the Yoneda pairing), then the moduli problem is said to be formal. For example, Goldman and Millson prove that the moduli space of certain flat vector bundles on Kahler manifolds is formal:

http://www2.math.umd.edu/~wmg/PMIHES_1988__67__43_0.pdf

(BTW, To really get the moduli stack, you also need to divide the zero set of the Kuranishi map by the automorphism group of $E$).

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  • $\begingroup$ Thank you for this answer. I will a have a look at this article. $\endgroup$
    – TonyS
    Commented Mar 5, 2013 at 12:11
  • $\begingroup$ The article should help a little with understanding the generalities of moduli spaces and Kuranishi maps, but probably won't help answer your specific question of what $n$ is. If you can compute the Yoneda pairing and you get a non-zero number, then n=2. If the pairing is 0 then $n>2$, but it will be hard to determine in general. $\endgroup$
    – Jim Bryan
    Commented Mar 5, 2013 at 15:51

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