TY - JOUR
T1 - MMP via wall-crossing for moduli spaces of stable sheaves on an Enriques surface
AU - Nuer, Howard
AU - Yoshioka, Kōta
N1 - Publisher Copyright:
© 2020 Elsevier Inc.
PY - 2020/10/7
Y1 - 2020/10/7
N2 - We use wall-crossing in the Bridgeland stability manifold to systematically study the birational geometry of the moduli space Mσ(v) of σ-semistable objects of class v for a generic stability condition σ on an arbitrary Enriques surface X. In particular, we show that for any other generic stability condition τ, the two moduli spaces Mτ(v) and Mσ(v) are birational. As a consequence, we show that for primitive v of odd rank Mσ(v) is birational to a Hilbert scheme of points. Similarly, in even rank we show that Mσ(v) is birational to a moduli space of torsion sheaves supported on a hyperelliptic curve when ℓ(v)=1. As an added bonus of our work, we prove that the Donaldson-Mukai map θv,σ:v⊥→Pic(Mσ(v)) is an isomorphism for these classes. Finally, we use our classification to fully describe the geometry of the only two examples of moduli of stable sheaves on X that are uniruled (and thus not K-trivial).
AB - We use wall-crossing in the Bridgeland stability manifold to systematically study the birational geometry of the moduli space Mσ(v) of σ-semistable objects of class v for a generic stability condition σ on an arbitrary Enriques surface X. In particular, we show that for any other generic stability condition τ, the two moduli spaces Mτ(v) and Mσ(v) are birational. As a consequence, we show that for primitive v of odd rank Mσ(v) is birational to a Hilbert scheme of points. Similarly, in even rank we show that Mσ(v) is birational to a moduli space of torsion sheaves supported on a hyperelliptic curve when ℓ(v)=1. As an added bonus of our work, we prove that the Donaldson-Mukai map θv,σ:v⊥→Pic(Mσ(v)) is an isomorphism for these classes. Finally, we use our classification to fully describe the geometry of the only two examples of moduli of stable sheaves on X that are uniruled (and thus not K-trivial).
KW - Bridgeland stability
KW - Enriques surfaces
KW - Moduli of sheaves
UR - http://www.scopus.com/inward/record.url?scp=85088112474&partnerID=8YFLogxK
U2 - 10.1016/j.aim.2020.107283
DO - 10.1016/j.aim.2020.107283
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AN - SCOPUS:85088112474
SN - 0001-8708
VL - 372
JO - Advances in Mathematics
JF - Advances in Mathematics
M1 - 107283
ER -