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O’Grady’s singularity. (La singularité de O’Grady.) (French) Zbl 1156.14030
Let \((X,H)\) be a polarized smooth projective surface over the complex numbers which is symplectic, that is either \(K3\) or abelian. Fix a primitive Mukai vector \(v \in H^{\text even}(X, \mathbb Z)\) with \(\langle v, v \rangle =2\).
Denote by \(M_{2v}\) the moduli space of semistable sheaves on \(X\) with respect to the polarization \(H\) with Mukai vector \(2v\).
The main result of the article is Theorem 1.1 which states that the blow up \(\tilde M_{2v} \to M_{2v}\) of \(M_{2v}\) along the reduced singular locus provides a symplectic resolution.
This simplifies the description of the symplectic varieties constructed by K. G. O’Grady [J. Reine Angew. Math. 512, 47–117 (1999; Zbl 0928.14029); J. Algebr. Geom. 12, No. 3, 435–505 (2003; Zbl 1068.53058)] which do not appear in Beauville’s two series.
The authors consider the stratification by automorphism type \(M_{2v} \supset S^2M_v \supset \Delta_{M_v}\) of the moduli space \(M_{2v}\). The dense open set \(M_{2v} \setminus S^2M_v\) parameterizes stable sheaves \(E\). The stratum \(S^2M_v \setminus \Delta_{M_v}\) parameterizes polystable sheaves \(E=F_1 \oplus F_2\) with \([F_i] \in M_V\) and \(F_1 \not \cong F_2\). Finally the closed stratum \(\Delta_{M_v}\) parameterizes sheaves \(F \oplus F\) for \([F] \in M_v\).
On the other hand they consider the affine space \(Z\) which is explicitly given as subvariety \(Z \subset \mathfrak {sp}(V,\omega)\) where \((V,\omega)\) is a four dimensional symplectic vector space, and \(Z\) consists of those \(B \in \mathfrak {sp}(V,\omega)\) with \(B^2=0\). \(Z\) is singular but possesses a semi-small resolution \(\tilde Z \to Z\) which can be described as a blow up (Theorem 2.1). The key result (Theorem 4.5) is that there exists an isomorphism of germs of analytic spaces \((M_{2v},[F\oplus F] ) \cong (\mathbb C^4 \times Z ,0)\). Thus, the resolution \(\tilde Z\) of \(Z\) yields a resolution \(\tilde M_{2v} \to M_{2v}\).
Reviewer: Georg Hein (Essen)

MSC:
14J10 Families, moduli, classification: algebraic theory
14J28 \(K3\) surfaces and Enriques surfaces
14K10 Algebraic moduli of abelian varieties, classification
14D20 Algebraic moduli problems, moduli of vector bundles
Software:
SINGULAR
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References:
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