Factoriality Properties of Moduli Spaces of Sheaves on Abelian and K3 Surfaces

Arvid Perego (1) , Antonio Rapagnetta (2)

1 IECL - Institut Élie Cartan de Lorraine
2 Università degli Studi di Roma Tor Vergata [Roma, Italia] = University of Rome Tor Vergata [Rome, Italy] = Université de Rome Tor Vergata [Rome, Italie]

In this paper we complete the determination of the index of factoriality of moduli spaces of semistable sheaves on an abelian or projective K3 surface S. If v " 2w is a Mukai vector, w is primitive, w 2 " 2 and H is a generic polarization, let MvpS, Hq be the moduli space of H´semistable sheaves on S with Mukai vector v. First, we describe in terms of v the pure weight-two Hodge structure and the Beauville form on the second integral cohomology of the symplectic resolutions of MvpS, Hq (when S is K3) and of the fiber KvpS, Hq of the Albanese map of MvpS, Hq (when S is abelian). Then, if S is K3 we show that MvpS, Hq is either locally factorial or 2´factorial, and we give an example of both cases. If S is abelian, we show that MvpS, Hq and KvpS, Hq are 2´factorial.

Mots clés: Moduli spaces of sheaves; K3 surfaces; abelian surfaces; factoriality

Domaines: Géométrie algébrique [math.AG]

Arvid Perego, Antonio Rapagnetta. Factoriality Properties of Moduli Spaces of Sheaves on Abelian and K3 Surfaces. International Mathematics Research Notices, 2012, 2014 (3), pp.643--680. ⟨10.1093/imrn/rns233⟩. ⟨hal-01266784⟩