Abstract
We prove the non-emptiness and irreducibility of \(M_H(v,L)\), the moduli space of Gieseker semistable sheaves on an unnodal Enriques surface Y with primitive Mukai vector v of positive rank and determinant L with respect to a generic polarization H. This completes the chain of progress initiated by Kim (Nagoya Math J 150:85–94, 1998). We also show that the stable locus \(M^s_H(v)\ne \varnothing \) for non-primitive v with \(v^2>0\).
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Notes
We work over \({\mathbb {C}}\) in this paper, so this is the correct definition.
We can drop the prefix quasi by applying [9, Theorem 4.6.5] with B in their notation running through the sheaves \({\mathcal {O}}_Y,{\mathbb {C}}([\mathrm {pt}])\) and \({\mathcal {O}}_Y(D)\) for D such that \(D\cdot c_1(v)\) is minimal. Indeed, using the fact that \(\gcd (r,c_1,2s)=2\) and \(r+2s\equiv 2 (\mathop {\mathrm {mod}}\nolimits 4)\), one sees that \(\gcd ((v,v(B)))=1\) as B runs through these sheaves.
Since \(\mu \)-stability and \(\mu \)-semistability are equivalent in our case, the fibers of the morphism \(M\rightarrow M^{\mu s s}\), in the notation of [9, Chapter 8], are precisely the fibers of the Hilbert–Chow morphism \(Y^{[l]}\rightarrow Y^{(l)}\). Thus the usual stratification of the Donaldson–Uhlenbeck compactification applies to M with the symmetric product replaced by the Hilbert scheme of points. See [9] for a more detailed discussion of what sheaves are identified in \(M^{\mu s s}\).
When \(n_Q\) is odd, we instead use the estimate \(j\le n_Q+1+a\) as shown above in (11).
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Acknowledgments
I would like to thank my advisor, Lev Borisov, for his constant support and guidance, especially in talking out ideas. I would also like to thank Dan Abramovich for a suggestion that helped the clarity of the exposition. Finally, I am very grateful to an anonymous referee for finding a mistake in the original argument. This led me not only to fix the mistake but also to prove irreducibility using a more geometric technique. The author was partially supported by NSF Grant DMS 1201466.
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Appendix: Dimension estimates for Brill–Noether loci on Hilbert schemes of points
Appendix: Dimension estimates for Brill–Noether loci on Hilbert schemes of points
We prove in this appendix the dimension estimate we used in the body of the paper. The result concerns bounding the dimension of the locus of 0-cycles with given cohomology with respect to a linear system.
Lemma 8.1
Let L be an effective divisor and \(S^i:=\{Z|h^0(I_Z(L))=i\}\subset Y^{(l)}\). Then for \(i>0\),
In particular, if L is ample, then \(\mathop {\mathrm {dim}}\nolimits |L|=\frac{1}{2}L^2\), so
Proof
Denote by \(S:=\{(Z,{\mathbb {C}}v)|{\mathbb {C}}v\in |I_Z(L)|\}\subset Y^{(l)}\times |L|\). Then the second projection
is surjective with fibers of dimension l(Z). Indeed, for any \(C\in |L|\) the fiber over C is \(C^{(l)}\). Thus \(\mathop {\mathrm {dim}}\nolimits S=\mathop {\mathrm {dim}}\nolimits |L|+l(Z)\). The image of S under the first projection is
where the fiber over \(Z\in Y^{(l)}\) is \(|I_Z(L)|\). Thus \(p_1^{-1}(S^i)\) is a \({\mathbb {P}}^{i-1}\) bundle over \(S^i\), from which it follows that
\(\square \)
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Nuer, H. A note on the existence of stable vector bundles on Enriques surfaces. Sel. Math. New Ser. 22, 1117–1156 (2016). https://doi.org/10.1007/s00029-015-0218-6
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DOI: https://doi.org/10.1007/s00029-015-0218-6