Projectivity and birational geometry of Bridgeland moduli spaces on an Enriques surface (Q2831999)

The moduli space of Gieseker semistable sheaves on a projective surface with fixed numerical invariants was classically constructed via geometric invariant theory. A recent revolutionary tool to study the birational geometry of these moduli spaces is Bridgeland stability conditions on the derived category of the underlying surface, defined in [\textit{T. Bridgeland}, Ann. Math. (2) 166, No. 2, 317--345 (2007; Zbl 1137.18008)]. When non-empty, the space of (full numerical) stability conditions can be divided into chambers for any fixed numerical invariants, one of which corresponds to Gieseker semistable sheaves, as proved in [\textit{T. Bridgeland}, Duke Math. J. 141, No. 2, 241--291 (2008; Zbl 1138.14022)]. The guiding principle in such a picture is that crossing a wall should induce a birational transformation on the moduli space of semistable objects. This principle has been extensively studied in a few examples, in particular with a most recent breakthrough in the case of \(K3\) surfaces by \textit{A. Bayer} and \textit{E. Macrì} [J. Am. Math. Soc. 27, No. 3, 707--752 (2014; Zbl 1314.14020); Invent. Math. 198, No. 3, 505--590 (2014; Zbl 1308.14011)].NEWLINENEWLINEThe paper under review aims to generalize the techniques used by Bayer and Macrì to the case of Enriques surfaces and contains a few very interesting results. First of all, the author proved the existence of a coarse moduli space for a generic Bridgeland stability condition and constructed a natural ample divisor on it, by relating the stability conditions of an Enriques surface to those of its \(K3\) cover. Moreover, the author also verified the birationality of the moduli spaces for two adjacent chambers, under the assumption that the Mukai vector is primitive and the locus of stable objects on the wall has a complement of codimension at least 2. Finally, the author gave three applications of the machinery, respectively on effective bounds of the Gieseker chamber for an arbitrary primitive Mukai vector, the nef cone of the Hilbert scheme of points on an unnodal Enriques surface, and the base-point freeness and very ampleness of multiples of an ample class on an unnodal Enriques surface. Some open questions were also mentioned for further investigation.